THE THEORY OF
DIFFERENCE SCHEMES
Alexander A. Samarskii

THE THEORY OF
DIFFERENCE SCHEMES

THE THEORY OF
DIFFERENCE SCHEMES
Alexander A. Samarskii
l\Aoscow M. v. Lomonosov State University
l\Aoscow, Russia
VI A R C E L
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1. K. Yano, Integral Formulas in Riemannian Geometry A970)
2. S. Kobayashi, Hyperbolic Manifolds and Holomorphic Mappings A970)
3. V. S. Vladimirov, Equations of Mathematical Physics (A. Jeffrey, ed.; A. Littlewood,
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4. 6. N. Pshenichnyi, Necessary Conditions for an Extremum (L. Neustadt, translation
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5. L A/anc) et al., Functional Analysis and Valuation Theory A971)
6. S. S. Passman, Infinite Group Rings A971)
7. L. Domhoff, Group Representation Theory. Part A: Ordinary Representation Theory.
Part B: Modular Representation Theory A971, 1972)
8. W. Boothby and G. L. Weiss, eds.. Symmetric Spaces A972)
9. y. Matsushima, Differentiable Manifolds (E. T. Kobayashi, trans.) A972)
10. L. E. Ward Jr., Topology A972)
11. A. Babakhanian, Cohomological Methods in Group Theory A972)
12. R. Gilmer, Multiplicative Ideal Theory A972)
13. J. Yeh, Stochastic Processes and the Wiener Integral A973)
14. J. Barros-Neto, Introduction to the Theory of Distributions A973)
15. R. Larsen, Functional Analysis A973)
16. K. Yano and S. Ishihara, Tangent and Cotangent Bundles A973)
17. C. Procesi, Rings with Polynomial Identities A973)
18. R. Hermann, Geometry, Physics, and Systems A973)
19. N. R. Wallach, Harmonic Analysis on Homogeneous Spaces A973)
20. J. Dieudonne, Introduction to the Theory of Formal Groups A973)
21. /. Vaisman, Cohomology and Differential Forms A973)
22. B.-Y. Chen, Geometry of Submanifolds A973)
23. M. Marcus, Finite Dimensional Multilinear Algebra (in two parts) A973, 1975)
24. R. Larsen, Banach Algebras A973)
25. R. O. Kujala and A. L. Vitter, eds.. Value Distribution Theory: Part A; Part B: Deficit
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26. K. 6. Stolarsky, Algebraic Numbers and Diophantine Approximation A974)
27. A. R. Magid, The Separable Galois Theory of Commutative Rings A974)
28. e. R. McDonald, Finite Rings with Identity A974)
29. J. Satake, Linear Algebra (S. Koh et al., trans.) A975)
30. J. S. Golan, Localization of Noncommutative Rings A975)
31. G. Klambauer, Mathematical Analysis A975)
32. M. K. Agoston, Algebraic Topology A976)
33. K. R. Goodearl, Ring Theory A976)
34. L. E. Mansfield, Linear Algebra with Geometric Applications A976)
35. N. J. Pullman, Matrix Theory and Its Applications A976)
36. B. R. McDonald, Geometric Algebra Over Local Rings A976)
37. C. W. Groetsch, Generalized Inverses of Linear Operators A977)
38. J. E. Kuczkowski and J. L. Gersting, Abstract Algebra A977)
39. C. O. Christenson and W. L. Voxman, Aspects of Topology A977)
40. M. Nagata, Field Theory A977)
41. R. L. Long, Algebraic Number Theory A977)
42. W. F. Pfeffer, Integrals and Measures A977)
43. R. L. Wheeden and A. Zygmund, Measure and Integral A977)
44. J. H. Curtiss, Introduction to Functions of a Complex Variable A978)
45. K. Hrbacek and T. Jech, Introduction to Set Theory A978)
46. W. S. Massey, Homology and Cohomology Theory A978)
47. M. Marcus, Introduction to Modern Algebra A978)
48. E. C. Young, Vector and Tensor Analysis A978)
49. S. B. Nadler, Jr., Hyperspaces of Sets A978)
50. S. K. Segal, Topics in Group Kings A978)
51. A. C. M. van RooiJ, Non-Archimedean Functional Analysis A978)
52. L. Corwin and R. Szczarba, Calculus in Vector Spaces A979)
53. C. Sadosky, Interpolation of Operators and Singular Integrals A979)
54. J. Cronin, Differential Equations A980)
55. C. W. Groetsch, Elements of Applicable Functional Analysis A980)

56. /. Vaisman, Foundations of Three-Dimensional Euclidean Geometry A980)
57. H. I. Freedan, Deterministic Mathematical Models in Population Ecology A980)
58. S. e. Chae, Lebesgue Integration A980)
59. 0. S. Rees et al., Theory and Applications of Fourier Analysis A981)
60. L Nachbin, Introduction to Functional Analysis (R. M. Aron, trans.) A981)
61. G. Orzech and M. Orzech, Plane Algebraic Curves A981)
62. R. Johnsonbaugh and W. E. Pfaffenberger, Foundations of Mathematical Analysis
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63. W. L. Voxman and R. H. Goetschel, Advanced Calculus A981)
64. L. J. Corwin and R. H. Szczarba, Multivariable Calculus A982)
65. V. I. Isiraiescu, Introduction to Linear Operator Theory A981)
66. R. D. Jarvinen, Finite and Infinite Dimensional Linear Spaces A981)
67. J. K. Beam and P. E. Ehrlich, Global Lorentzian Geometry A981)
68. D. L Anvacost, The Structure of Locally Compact Abelian Groups A981)
69. J. W. Brewer and M. K. Smith, eds., Emmy Noether: A Tribute A981)
70. K. H. Kim, Boolean Matrix Theory and Applications A982)
71. T. W. Wieting, The Mathematical Theory of Chromatic Plane Ornaments A982)
72. D. B.Gauld, Differential Topology A982)
73. R. L Faber, Foundations of Euclidean and Non-Euclidean Geometry A983)
74. M. Carmeli, Statistical Theory and Random Matrices A983)
75. J. H. Carruth et al.. The Theory of Topological Semigroups A983)
76. R. L Faber, Differential Geometry and Relativity Theory A983)
77. S. Barnett, Polynomials and Linear Control Systems A983)
78. G. Karpilovsky, Commutative Group Algebras A983)
79. F. Van Oystaeyen and A. Verschoren, Relative Invariants of Rings A983)
80. /. Vaisman, A First Course in Differential Geometry A984)
81. G. W. Swan, Applications of Optimal Control Theory in Biomedicine A984)
82. T. Petrie and J. D. Randall, Transformation Groups on Manifolds A984)
83. K. Goebel and S. Reich, IJniform Convexity, Hyperbolic Geometry, and Nonexpansive
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84. T. Albu and C. Nastasescu, Relative Finiteness in Module Theory A984)
85. K. Hrbacek and T. Jech, Introduction to Set Theory: Second Edition A984)
86. F. Van Oystaeyen and A. Verschoren, Relative Invariants of Rings A984)
87. 6. R. McDonald, Linear Algebra Over Commutative Rings A984)
88. M. Namba, Geometry of Projective Algebraic Curves A984)
89. G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics A985)
90. M. R. Bremner et al.. Tables of Dominant Weight Multiplicities for Representations of
Simple Lie Algebras A985)
91. A. E. Fekete, Real Linear Algebra A985)
92. S. 6. Chae, Holomorphy and Calculus in Normed Spaces A985)
93. A. J. Jerri, Introduction to Integral Equations with Applications A985)
94. G. Karpilovsky, Projective Representations of Finite Groups A985)
95. L. Narici and E. Beckenstein, Topological Vector Spaces A985)
96. J. Weeks, The Shape of Space A985)
97. P. R. Gribik and K. O. Kortanek, Extremal Methods of Operations Research A985)
98. J.-A. Chao and W. A. Woyczynski, eds.. Probability Theory and Hamnonic Analysis
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99. G. D. Crown et al.. Abstract Algebra A986)
100. J. H. Carruth et al., The Theory of Topological Semigroups, Volume 2 A986)
101. R.S. Doran and V. A. Belfi, Characterizations of C*-Algebras A986)
102. M. W. Jeter, Mathematical Programming A986)
103. M. Altman, A Unified Theory of Nonlinear Operator and Evolution Equations with
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104. A. Verschoren, Relative Invariants of Sheaves A987)
105. R. A. Usmani, Applied Linear Algebra A987)
106. P. Blass and J. Lang, Zarisl<i Surfaces and Differential Equations in Characteristic p >
0A987)
107. J. A. Reneke et al.. Structured Hereditary Systems A987)
108. H. Busemann and B. B. Phadke, Spaces with Distinguished Geodesies A987)
109. R. Harte, Invertibility and Singularity for Bounded Linear Operators A988)
110. G. S. Ladde et al.. Oscillation Theory of Differential Equations with Deviating
Arguments A987)
111. L. Dudkin et al.. Iterative Aggregation Theory A987)
112. T. Okubo, Differential Geometry A987)

113. D. L Stand and M. L. Stand, Real Analysis with Point-Set Topology A987)
114. T. C. Gard, Introduction to Stochastic Differential Equations A988)
115. S. S. Abhyankar, Enumeratlve Combinatorics of Young Tableaux A988)
116. H. Strade and R. Farnsteiner, Modular Lie Algebras and Their Representations A988)
117. J. A. Huckaba, Commutative Rings with Zero Divisors A988)
118. W. D. Wallis, Combinatorial Designs A988)
119. W. Wi^sfaw, Topological Fields A988)
120. G. Karpilovsky, Field Theory A988)
121. S. Caenepeel and F. Van Oystaeyen, Brauer Groups and the Cohomology of Graded
Rings A989)
122. W. Kozlowski, Modular Function Spaces A988)
123. E. Lowen-Colebunders, Function Classes of Cauchy Continuous Maps A989)
124. M. Pavel, Fundamentals of Pattern Recognition A989)
125. V. Lakshmikantham et ai. Stability Analysis of Nonlinear Systems A989)
126. R. Sivaramakrishnan, The Classical Theory of Arithmetic Functions A989)
127. N. A. Watson, Parabolic Equations on an Infinite Strip A989)
128. K. J. Hastings, Introduction to the Mathematics of Operations Research A989)
129. e. Fine, Algebraic Theory of the Bianchi Groups A989)
130. D. N. Dikranjan et a/., Topological Groups A989)
131. J.C. Morgan II, Point Set Theory A990)
132. P. BilerandA. Witkowski, Problems in Mathematical Analysis A990)
133. H. J. Sussmann, Nonlinear Controllability and Optimal Control A990)
134. J.-P. Florens et al.. Elements of Bayesian Statistics A990)
135. N. Shell, Topological Fields and Near Valuations A990)
136. 6. F. Doolin and C. F. Martin, Introduction to Differential Geometry for Engineers
A990)
137. S. S. Holland, Jr., Applied Analysis by the Hilbert Space Method A990)
138. J. Okninski, Semigroup Algebras A990)
139. K. Zhu, Operator Theory in Function Spaces A990)
140. G. 6. Price, An Introduction to Multicomplex Spaces and Functions A991)
141. R. B. Darst, Introduction to Linear Programming A991)
142. P. L. Sachdev, Nonlinear Ordinary Differential Equations and Their Applications A991)
143. T. Husain, Orthogonal Schauder Bases A991)
144. J. Foran, Fundamentals of Real Analysis A991)
145. W. C. Brown, Matrices and Vector Spaces A991)
146. M. M. Rao and Z. D. Ren, Theory of Oriicz Spaces A991)
147. J. S. Golan and T. Head, Modules and the Structures of Rings A991)
148. C. Small, Arithmetic of Finite Fields A991)
149. K. Yang, Complex Algebraic Geometry A991)
150. D. G. Hoffman et al.. Coding Theory A991)
151. M. O. Gonzalez, Classical Complex Analysis A992)
152. M. O. Gonzalez, Complex Analysis A992)
153. L W. Baggett, Functional Analysis A992)
154. M. Sniedovich, Dynamic Programming A992)
155. R. P. Agarwal, Difference Equations and Inequalities A992)
156. C. Brezinski, Biorthogonality and Its Applications to Numerical Analysis A992)
157. C Swartz, An Introduction to Functional Analysis A992)
158. S. B. Nadler, Jr., Continuum Theory A992)
159. M. A. Al-Gwaiz, Theory of Distributions A992)
160. E Perry, Geometry: Axiomatic Developments with Problem Solving A992)
161. E. Castillo and M. R. Ruiz-Cobo, Functional Equations and Modelling in Science and
Engineering A992)
162. A. J. Jerri, Integral and Discrete Transforms with Applications and Error Analysis
A992)
163. A. Chariieret al., Tensors and the Clifford Algebra A992)
164. P. Biler and T. Nadzieja, Problems and Examples in Differential Equations A992)
165. E Hansen, Global Optimization Using Interval Analysis A992)
166. S. Guerre-Delabriere, Classical Sequences in Banach Spaces A992)
167. y. C. Wong, Introductory Theory of Topological Vector Spaces A992)
168. S. H. Kulkarni and B. V. Limaye, Real Function Algebras A992)
169. W. C. Brown, Matrices Over Commutative Rings A993)
170. J. Loustau and M. Dillon, Linear Geometry with Computer Graphics A993)
171. W. V. Petryshyn, Approximation-Solvability of Nonlinear Functional and Differential
Equations A993)

172. E. C. Young, Vector and Tensor Analysis: Second Edition A993)
173. T. A. Bick, Elementary Boundary Value Problems A993)
174. M. Pavel, Fundamentals of Pattern Recognition: Second Edition A993)
175. S. A. Albeverio et al., Noncommutative Distributions A993)
176. W. Fulks, Complex Variables A993)
177. M. M. Rao, Conditional Measures and Applications A993)
178. A. Janicki and A. Weron, Simulation and Chaotic Behavior of a-Stable Stochastic
Processes A994)
179. P. Neittaanmaki and D. Tiba, Optimal Control of Nonlinear Parabolic Systems A994)
180. J. Cronin, Differential Equations: Introduction and Qualitative Theory, Second Edition
A994)
181. S. Heikkila and V. Lakshmikantham, Monotone Iterative Techniques for Discontinuous
Nonlinear Differential Equations A994)
182. X. Mao, Exponential Stability of Stochastic Differential Equations A994)
183. 6. S. Thomson, Symmetric Properties of Real Functions A994)
184. J. E. Rubio, Optimization and Nonstandard Analysis A994)
185. J. L. Bueso et al.. Compatibility, Stability, and Sheaves A995)
186. A. N. !\/lichel and K. Wang, Qualitative Theory of Dynamical Systems A995)
187. M. R. Darnel, Theory of Lattice-Ordered Groups A995)
188. Z. Naniewicz and P. D. Panagiotopoulos, Mathematical Theory of Hemivariational
Inequalities and Applications A995)
189. L. J. Corwin and /?. H. Szczarba, Calculus in Vector Spaces: Second Edition A995)
190. L. H. Erbe etal.. Oscillation Theory for Functional Differential Equations A995)
191. S. Agaian et al.. Binary Polynomial Transforms and Nonlinear Digital Filters A995)
192. M. /. Gil', Norm Estimations for Operation-Valued Functions and Applications A995)
193. P. A. Grillet, Semigroups: An Introduction to the Structure Theory A995)
194. S. Kichenassamy, Nonlinear Wave Equations A996)
195. V. F. Krotov, Global Methods in Optimal Control Theory A996)
196. K. I. Beidaret al.. Rings with Generalized Identities A996)
197. V. I. Amautov et al.. Introduction to the Theory of Topological Rings and Modules
A996)
198. G. Sierksma, Linear and Integer Programming A996)
199. R. Lasser, Introduction to Fourier Series A996)
200. V. Sima, Algorithms for Linear-Quadratic Optimization A996)
201. D. Redmond, Number Theory A996)
202. J. K. Beem et al., Global Lorentzian Geometry: Second Edition A996)
203. M. Fontana et al., Prufer Domains A997)
204. H. Tanabe, Functional Analytic Methods for Partial Differential Equations A997)
205. C. Q. Zhang, Integer Flows and Cycle Covers of Graphs A997)
206. E. Spiegel and C. J. O'Donnell, Incidence Algebras A997)
207. 6. Jakubczyk and W. Respondek, Geometry of Feedbacl< and Optimal Control A998)
208. T. W. Haynes et al.. Fundamentals of Domination in Graphs A998)
209. T. W. Haynes et al.. Domination in Graphs; Advanced Topics A998)
210. L A. DAIotto et al., A Unified Signal Algebra Approach to Two-Dimensional Parallel
Digital Signal Processing A998)
211. F. Halter-Koch, Ideal Systems A998)
212. N. K. Govil et al.. Approximation Theory A998)
213. R. Cross, Multivalued Linear Operators A998)
214. A A Martynyuk, Stability by Liapunov's Matrix Function Method with Applications
A998)
215. A. Favini and A. Yagi, Degenerate Differential Equations in Banach Spaces A999)
216. A lllanes and S. Nadler, Jr, Hyperspaces: Fundamentals and Recent Advances
A999)
217. G. Kato and D. Struppa, Fundamentals of Algebraic Microlocal Analysis A999)
218. G. X.-Z. Yuan, KKM Theory and Applications in Nonlinear Analysis A999)
219. D. Motreanu and N. H. Pavel, Tangency, Flow Invariance for Differential Equations,
and Optimization Problems A999)
220. K. Hrbacek and T. Jech, Introduction to Set Theory, Third Edition A999)
221. G. E. Kolosov, Optimal Design of Control Systems A999)
222. N. L. Johnson, Subplane Covered Nets B000)
223. 6. Fine and G. Rosenberger, Algebraic Generalizations of Discrete Groups A999)
224. M. Vath, Volterra and Integral Equations of Vector Functions B000)
225. S. S. Miller and P. T. Mocanu, Differential Subordinations B000)

226. R. Li et al.. Generalized Difference Methods for Differential Equations: Numerical
Analysis of Finite Volume Methods B000)
227. H. Li and F. Van Oystaeyen, A Primer of Algebraic Geometry B000)
228. R. P. Agarwal, Difference Equations and Inequalities: Theory, Methods, and
Applications, Second Edition B000)
229. A. B. Kharazishvili, Strange Functions in Real Analysis B000)
230. J. M. Appell et al.. Partial Integral Operators and Integro-Differential Equations B000)
231. A. I. Pnlepko et al.. Methods for Solving Inverse Problems in Mathematical Physics
B000)
232. F. Van Oystaeyen, Algebraic Geometry for Associative Algebras B000)
233. D. L. Jagerman, Difference Equations with Applications to Queues B000)
234. D. R. Hankerson et al., Coding Theory and Cryptography: The Essentials, Second
Edition, Revised and Expanded B000)
235. S. Dascalescu et al., Hopf Algebras: An Introduction B001)
236. R. Hagen et al., C*-Algebras and Numerical Analysis B001)
237. y. Talpaert, Differential Geometry: With Applications to Mechanics and Physics B001)
238. R. H. Villan^al, Monomial Algebras B001)
239. A. N. Michel et al., Qualitative Theory of Dynamical Systems, Second Edition B001)
240. A. A. Samarskii, The Theory of Difference Schemes B001)
Additional Volumes in Preparation

Preface
In the early 1960s, when computers first became freely available to academic
scientists, they spawned a whole new area of mathematics. Suddenly, for
example, the solution of a system of 30 linear equations in 30 unknowns
reduced to a problem that could be solved in a straightforward fashion on a
computer using Gaussian elimination. A whole host of other applications,
whose computational efforts were previously prohibitive, became feasible
with the new technology. In addition, the availability of the technology
itself spurred mathematicians to invent new methods of solution for existing
and emerging problems. As we begin the first decade of this century, the
exploitation of computers in mathematics has become very diverse. Some
researchers use "blackbox" library routine.? to solve problems arising from
physical situations. Others are engaged in developing software for such
routines. Yet others are carrying out fundamental research in the capacities
of existing methods and the design of new ones. But perhaps the best
evidence of the effect the computer has had on mathematics in recent years
can be seen in the undergraduate curriculum. Most university degrees
in mathematics would now be regarded as incomplete without courses in
discrete mathematics and numerical analysis - both subjects that have
come to prominence because of modern computer technology.
A rough description of the term modeling is the process of applying
methods well-developed in computational mathematics to real-life situa-

IV
Preface
tions. Quite often, the interest in modeling centers as much on getting
an appropriate mathematical description of the problem as it does on the
computational methods employed. Nevertheless there is a fruitful interplay
between modeling and computational mathematics, as will be evident from
a perusal of the first few chapters of this book.
One can now say that a new method arose for theoretical
investigations of rather general and complex processes which allow for mathematical
descriptions or mathematical modeling. This is the method of numerical
experiment, which makes it possible better to understand real-world
phenomena using exploratory devices of computational mathematics. When
investigating some physical process by means of this method, an early step
is the mathematical statement of a problem or the well-founded choice of
a mathematical model capable of describing this process. It is preceded by
a proper choice of physical approximations, that is, to decide which factors
should be taken into account and which may be neglected or omitted in a
general setting. This choice is the privilege of physicists.
What is a mathematical model? The group of unknown physical
quantities which interest us and the group of available data are closely
interconnected. This link may be embodied in algebraic or differential equations. A
proper choice of the mathematical model facilitates solving these equations
and providing the subsidiary information on the coefficients of equations as
well as on the initial and boundary data.
Mathematical physics deals with a variety of mathematical models
arising in physics. Equations of mathematical physics are mainly partial
differential equations, integral, and integro-differential equations. Usually
these equations reflect the conservation laws of the basic physical quantities
(energy, angular momentum, mass, etc.) and, as a rule, turn out to be
nonlinear.
After writing a system of equations capable of describing the physical
process of interest, it is necessary for the investigator to consider the
resulting mathematical model by methods of the general theory of
differential and integral equations and to make sure that a problem has been
completely posed and the available data are sufficient and consistent, to
derive the conditions under which the problem has a unique solution, to
find out whether this solution may be written in explicit form, and whether
particular solutions are possible. Particular solutions are important in
giving the preliminaries regarding the nature of the physical process. They
also can serve as "goodness-of-fit" tests for the desired quality of numerical
methods. At the second stage, one is to construct an approximate
(numerical) method and a computer algorithm for solving the problem. The third
stage is computer programming of the algorithm under such a choice. At

Preface v
the fourth stage the computational procedures need to be investigated and
implemented. The fifth stage focuses on the review of their delivery and
necessary corrections to the approved mathematical model governing what
can happen.
It may turn out that the mathematical model is too rough, meaning
numerical results of computations are not consistent with physical
experiments, or the model is extremely cumbersome for everyday use and its
solution can be obtained with a prescribed accuracy on the basis of
simpler models. Then the same work should be started all over again and the
remaining stages should he repeated once again.
The stages of numerical experiments for theoretical investigations of
physical problems were explained above. Greatest attention is being paid
to a new technology for research with rather complicated mathematical
models.
Classical methods of mathematical physics are employed at the first
stage. Numerous physical problems lead to mathematical models having
no advanced methods for solving them. Quite often in practice, the user is
forced to solve such nonlinear problems of mathematical physics for which
even the theorems of existence and uniqueness have not yet been proven
and some relevant issues are still open.
For the moment, we are interested only in the second stage of
numerical experiments. A computational algorithm usually means a sequence
of logic and arithmetic operations that enables us to solve a problem. A
computational algorithm is developed to solve the problem with any degree
of accuracy 5 > 0 in a finite number of operations Q{£). This is the
general requirement, thus raising many mathematical questions. What does
the expression "with any degree of accuracy" mean? A possibility, at least
in principle, of obtaining a solution with a prescribed accuracy should be
substantiated. However, sometimes it may happen that Q(£) is finite for
some £, but it is so great that in practice it is unrealistic to produce a
solution with such a degree of accuracy. For any problem an infinite number
of various computational algorithms may be constructed possessing, for
example, similar asymptotic properties, so that Q{s) will be of the same order
in £ as £ —> 0. Not all such methods should be involved, but only those
suitable for computational restrictions. Naturally, the methods should be
used that require the minimal execution time for solving the problem with
a prescribed accuracy. The execution time must be reasonable, that is,
measurable in minutes or hours if such calculations are encountered very
rarely. Of course, the computing process must be as inexpensive as possible.
The time for solving the problem depends on the algorithms, qualitj' of the
program and computer performance. It is difficult to evaluate the latter

vi Preface
when only a priori estimates of the algorithm quality are available. The
same applies equally well to the total number of logic operations regardless
of how computers will be chosen. Therefore, in theoretical investigations it
is fairly common to compare computer algorithms by means of the number
of arithmetic operations Qo(£)- An economical algorithm having the
minimal number of operations is sought in the class of possible algorithms (for
instance, with the same asymptotics for Qois) as e --+0). This is one of
the main problems in the theory of numerical methods.
At the initial stage of such an analysis of algorithms it is usually
supposed that a computing process is ideal, that is, computations are carried
out with an infinite number of significant digits. But any computer makes
calculations with a finite speed and a finite number of digits. Not all
numbers are accessible to computers; there are computer null and computer
infinity. For instance, abnormal termination occurs when computer infinity
arises during the course of execution. A computing process may become
unstable, thus causing difficulties. In such cases rounding eiTors may
accumulate to a considerable extent so that the algorithm will be useless in
practical applications (several examples of unstable algorithms are
available in Chapter 2). A real, that is, a proper computer algorithm should
be stable and should not allow excessively large intermediate values
leading to abnormal termination. From such reasoning it seems clear that an
ideal computer algorithm may be optimal with respect to the number Qo(£)
and quite unfit for computers. These requirements necessitate seeking an
optimal real computer algorithm.
Any numerical experiment is not a one-time calculation by standard
formulas. First and foremost, it is the computation of a number of
possibilities for various mathematical models. For instance, it is required to find
the optimal conditions for a chemical process, that is, the conditions under
which the reaction is completed most rapidly. A solution of this problem
depends on a number of parameters (for instance, temperature, pressure,
composition of the reacting mixture, etc.). In order to find the optimal
workable conditions, it is necessary to carry out computations for different
values of those parameters, thereby exhausting all possibilities. Of course,
some situations exist in wdiich an algorithm is to be used only several times
or even once.
The statement of the problem of finding an optimal algorithm depends
on how it is to be apphed (for individual variants or a great number of
variants).
Here various issues surrounding programming, organization and
realization of computing are of little interest and will be excluded from
further consideration. It is worth noting, however, that programming

Preface vii
should be closely linked with further development of numerical algorihms.
Mathematical-physics problems are fairly complicated and algorithms for
solving them are extremely cumbersome. In mastering the difficulties
involved, one might reasonably try to split them into blocks or "modules".
Fortunately, quite often processes having different physical backgrounds
will be described by the same equations (for instance, the processes of
diffusion, heat conduction, and magnetization). One and the same
mathematical model could be suitable for various physical problems even if each
one has its own physical particulars. On the other hand, a mathematical
model may change substantially a number of times during the course of
numerical experiments and this obstacle necessitates improving the
algorithm constructed and the corresponding program. Therefore, it becomes
extremely important to create programs (a program package) in conformity
with the modulus principle, whose use permits us efficiently to conduct
numerical experiments and solve various types of problems of miscellaneous
physical genesis. This is one, of the modern trends in programming and
solving major mathematical-physics problems leading to the development
of new numerical methods.
Usually the finite difference method or the grid method is aimed at
numerical solution of various problems in mathematical physics. Under
such an approach the solution of partial differential equations amounts to
solving systems of algebraic equations.
This book is devoted to the theory of difference methods (schemes)
applied to typical problems in mathematical physics. There seem to be at
least two widespread approaches within the theory;
A) Composition of discrete (difference) approximations to equations of
mathematical physics and verifying a priori quality characteristics of these
approximations, mainly the error of approximation, stability, convergence,
and accuracy of the difference schemes obtained;
B) Solution of difference equations by direct or iterative methods
selected on the basis of the economy criteria for the corresponding
computational algorithms.
Because of the enormous range of difference approximations to an
equation having similar asymptotic properties with respect to a grid step
(the same order of accuracy or the number of necessary operations), their
numerical realizations resulted in the appearance of different schemes for
solving the basic problems in mathematical physics.
Of course, one strives to develop the best possible method, whose
use permits us to obtain the desired solution in minimal computing time.
Indeed, the search for such numerical procedures among admissible
methods is the main goal of such theory. In designing an optimal method (its

viii Preface
choice depends on the type of problems being solved) the set of
admissible methods is gradually reduced by successive introduction of necessary
constraints and requirements regarding approximation, stability, economy,
convergence, etc. The following general principle plays a crucial role in such
matters; a difference scheme (a discrete scheme) must reproduce properties
of the initial differential equation as well as possible.
In practical implementations, in order to construct difference schemes
of a desired quality one should formulate the general approaches, heuristic
tricks, turns, and rules for later use. The design of any method for
solving a problem is stipulated by the homogeneity and conservatism of the
appropriate difference scheme. The meaning of conservatism is that the
difference scheme reproduces some conservation law (a balance equation)
on a grid. Conservatism of homogeneous schemes is a necessary condition
for convergence in the class of discontinuous coefficients for stationary and
nonstationary problems in mathematical physics.
The property of the difference scheme conservatism for linear
equations is generally equivalent to the self-adjointness of the relevant difference
operator (see Chapters 3~4).
The basic notions of the theory of difference schemes are the error of
approximation, stability, convergence, and accuracy of difference scheme.
A more detailed exposition of these concepts will appear in Chapter 2. They
are illustrated by considering a number of difference schemes for ordinary
differential equations. In the same chapter we also outline the approach
to the general formulations without regard to the particular form of the
difference operator.
The question of the accuracy of the scheme, being of principal
importance in the theory, amounts to studying the error of approximation and
stability of the scheme. Stability analysis necessitates imposing a priori
estimates for the difference problem solution in light of available input data.
This is a problem in itself and needs investigation.
On the other hand, neither is the estimation of the approximation
error a trivial issue. Even the simplest example of the scheme on a nonequidis-
tant grid for a second-order equation shows that it is desirable to evaluate
the approximation error not in the G'-norm, but in a weaker norm of
special type (in one of the negative norms), thus imposing the need for a priori
estimates for a solution of a difference problem from the right-hand side
in a weaker norm. Various estimates of this sort arising in Chapter 3
apply equally well to establishing the convergence of homogeneous difference
schemes in the case of discontinuous coefficients.
This book includes many remarkable examples illustrating different
approaches to the stability analysis of difference scheme. Thus, in Chapter

Preface ix
5 which is devoted to difference schemes for nonstationary equations with
constant coefficients we discover asymptotic stability of difference schemes
for the heat conduction equation, a property intrinsic to the differential
equation.
Chapters 2-5 are concerned with concrete difference schemes for equa.-
tions of elliptic, parabolic, and hyperbolic types. Chapter 3 focuses on
homogeneous difference schemes for ordinary differential equations, by means
of which we try to solve the canonical problem of the theory of difference
schemes in which a primary family of difference schemes is specified (in such
a case the availabihty of the family is provided by pattern functionals) and
schemes of a desired quality should be selected within the primary family.
This problem is solved in Chapter 3 using a particular form of the scheme
and its solution leads us to conservative homogeneous schemes.
Chapter 4 provides the general theory of difference schemes in which
it seems reasonable to eliminate constraints on the structure and imphcit
forms of difference operators. Such a theory treats difference schemes as
operator equations (an analog of grid approximations for elliptic and
integral equations) and operator-difference equations (difference equations
in t with operator coefficients), which are analogs of difference schemes
for time-dependent equations of mathematical physics (for instance,
equations of parabolic and hyperbolic types). The operators of schemes are
viewed in such a setting as linear operators in an abstract normed vector
space Hh depending on the vector parameter h (an analog of a grid step
in X = (x-j, x, . . . , X )) equipped with the norm | /i | > 0. Thus, in the
sequel two types of schemes will be given special investigation.
The operator scheme is associated with an operator equation of the
first kind
Ay = f,
which can always be parametrized by a real variable h, where A = Ah,
Ak: Hk (—> Hh (the operator Ah depending on h really acts from Hh into
Hh), i = ih ^ f^h is a known vector and y = y^. G Hh is the vector of
unknowns.
The two-layer operator-difference scheme we are interested in acquires
the canonical form
j+i „ „j
+ Ay^=<p\ i = 0,l,..., y°eHh,
where r is the step in t: tj = JT, j = 0,1,...; A, B: Hh >—* Hh are
operators depending on h and r and, generally, on t^; y^ = yhr(tj) £ Hu
and (/?■' = (fh ri^j) £ -^ft '^^'S, respectively, the sought and given functions

^ Preface
of the discrete argument L — JT with the values in the space Hk (below
we omit the subscripts h and r).
Multilayer schemes, that is, the schemes containing the values y{t) at
some moments t = tj, ij+i, ^j+2) ■ ■ ■ rnay be reduced to two-layer schemes,
where A and B are operator matrices.
Stability theory is the central part of the theory of difference schemes.
Recent years have seen a great number of papers dedicated to
investigating stability of such schemes. Many works are based on applications of
spectral methods and include ineffective results given certain restrictions
on the structure of difference operators. For schemes with non-self-adjoint
operators the spectral theory guides only the choice of necessary stability
conditions, but sufficient conditions and a priori estimates are of no less
importance. An energy approach connected with the above definitions of
the scheme permits one to carry out an exhaustive stability analysis for
operators in a prescribed Hilbert space Hi,.
Clearly, stability is an intrinsic property of schemes regardless of
approximations and interrelations between the resulting schemes and relevant
differential equations. Because of this, any stability condition should be
imposed as the relationship between the operators A and 5, More
specifically, let a family of schemes specified by the restrictions on the operators
A and B be given: A = A* > 0 or {Ay, v) = {y, Av) and {Ay, y) > 0 for
any y, v & H, where ( , ) is an inner product in H, B > 0 and B ^ B*
{B is non-self-adjoint). The problem statement consists of extracting from
that family a set of schemes that are stable with respect to the initial data,
having the form
J-1-1 „ „i
+ Ay^=0, i = 0,l,,,,, y°&HH
and satisfying the inequality
The meaning of stability here is the validity of the preceding estimate.
A necessary and sufficient condition for a two-layer scheme to be stable
can be written as the operator inequality
B > 0.5tA
{By, y) > 0.5 T {Ay, y) for any y <^ H .

Preface xi
This condition can easily be verified in the case of discrete schemes for
equations of mathematical physics. It allows one to extract from the
primary family of schemes the set of stable schemes, within which one should
look for schemes with a prescribed accuracy, volume of computations, and
other desirable properties and parameters.
One of the important corollaries to stability theory is the general
method of regularization in the class of stable schemes (by changing the
operators A and B) for the design of schemes of a desired quality.
Chapter 6 includes a priori estimates expressing stability of two-layer
and three-layer schemes in terms of the initial data and the right-hand side
of the corresponding equations. It is worth noting here that relevant
elements of functional analysis and linear algebra, such as the operator norm,
self-adjoint operator, operator inequality, and others are much involved in
the theory of difference schemes. For the reader's convenience the necessary
prerequisities for reading the book are available in Chapters 1-2.
The book includes many good examples illustrating the practical use of
general stability theory with regard to particular schemes to assist the users
in subsequent implementations. Stability is probably the most pressing
problem in any algorithm, since it is a necessary rather than a sufficient
condition for accuracy.
Despite the great generality of the research presented in this hook, it
is of a constructive nature and gives the reader an under.staiiding of relevant
special cases as well as providing one with insight into the general theory.
I hope it proves to be useful and stimulating.
One of the popular branches of modern mathematics is the theory of
difference schemes for the numerical solution of the differential equations
of mathematical physics. Difference schemes are also widely used in the
general theory of differential equations as an apparatus available for proving
existence theorems and investigating the differential properties of solutions.
The theory of difference schemes has a number of special problems.
In the final analysis, of greatest importance from the viewpoint of
numerical analysis is the design of algorithms permitting one to obtain a
solution of a differential equation on a computer with a prescribed accuracy
in a finite number of operations. The user can encounter in this connection
the question of the quality of an algorithm, that is, the manner in which
the accuracy of the algorithm depends on
1) the available information on the original problem,
2) the amount of calculation (viz. the machine time spent in .solving
the problem with a prescribed accuracy).
Experience with computers has stimulated the formulation of a number of

xii Preface
special (for the theory of difference methods) problems:
1) the determination of the attainable order of accuracy of difference
schemes for various classes of problems,
2) the design of schemes for the solution of a wide class of problems
with a certain guaranteed accm'acy,
3) the construction of schemes with increased accuracy in narrower
classes of problems,
4) the development of methods for investigating stability and
convergence of difference schemes,
5) the formulation of the general principles for constructing stable
difference schemes and economizing the amount of calculation
(economical schemes)
and others.
The main purpose of the final chapters is to show how the results
of the general theory of difference schemes are aimed at stating principles
for constructing difference schemes of a prescribed quality. This approach
requires forsaking a more detailed description of the structure of difference
operators for concrete classes of differential equations and presenting the
theory in the language of functional analysis. The difference schemes
relating to analogs of nonstationary equations of mathematical physics are
treated in this connection as difference (with respect to the variable t)
equations with operator coefficients in an abstract space of any dimension. The
difference schemes for elliptic equations are viewed as operator equations of
the first Icind. It should be emphasized, however, that the indicated notions
of schemes have a much more general meaning.
Stability theory is quite applicable to formulate a general principle
for regularizing difference schemes in order to obtain stable schemes of a
prescribed quality.
The theory of iterative methods for solving the equation Au = f,
where A £ G7 i-^ 77) is a linear operator in a Hilbert space 77, is treated as
a contemporary part of the general stability theory of operator-difference
schemes. Our main concern is with obtaining effective estimates for the
rate of convergence of the iterations and with choosing an optimal set of
iteration parameters. Special attention is being paid to a class of
implicit schemes with a factorized operator B on the upper level of the form
B =: {E + ojRi) {E + U1R2), where E is the identity operator, w > 0 is
a numerical parameter, and Ri and R2 = /?* are adjoint or "triangular"
(with a triangular matrix) linear operators. A formula for the parameter ui
is obtained through such an analysis from the condition that the number
of iterations be minimized.

Preface xiii
An estimate of the convergence rate for the method of minimal
corrections is derived in the case when A is a non-self-adjoint operator and in
others situations.
Also, we consider the total approximation method as a constructive
method for creating economical difference schemes for the multidimensional
equations of mathematical physics. The notion of additive scheme is
introduced as a system of operator difference equations that approximates the
original differential equation in the total sense. Two quite general heuristic
methods (proposed earlier by the author) for obtaining additive economical
.schemes are discussed in full details. The additive schemes require a new
technique for investigating convergence and a new type of a priori estimates
that take into account the definition of the property of approximation.
We have not had the chance to discuss some works on difference
methods of an applied character, although such works best illustrate the real
possibilities of difference methods and provide constant sources of stimulation
for the formulation of new problems.
When economical schemes for multidimensional problems in
mathematical physics are developed in Chapter 9, we shall need a revised concept
of approximation error, thereby changing the definition of scheme. The
notion of summed (in t) approximation in Section 3 of Chapter 9 is of a
constructive nature, making it possible to produce economical schemes for
various problems.
Last, but not least, I wish to thank those who have assisted me in
this enterprise.
Alexander Samarskii

Contents
Preface iii
1 Prehniinaries 1
1.1 Difference equations 1
1.2 .Some variants of the elimination method 34
2 Basic concepts of the theory of difference schemes 41
2.1 Relevant elements of functional analysis 41
2.2 Difference approximation of elementary differential operators 50
2.3 Stability of a difference scheme 87
2.4 Mathematical apparatus in the theory of difference schemes 98
2.5 Difference schemes as operator equations.
General formulations 116
3 Homogeneous difference schemes 145
3.1 Homogeneous schemes for second-order equations with
variable coefficients 145

xvi Contents
3.2 Conservative schemes 147
3.3 Convergence and accuracy of homogeneous conservative
schemes 159
3.4 Homogeneous difference schemes on non-equidistant grids 168
3.5 Other problems 178
3.6 Difference Green's function 199
3.7 Higher-accuracy schemes 207
3.8 Methods for designing difference schemes 214
3.9 Stability with respect to coefficients 229
4 Difference schemes for eUiptic equations 237
4.1 The Dirichlet difference problem for Poisson's equation 237
4.2 The maximum principle 258
4.3 Stability and convergence of the Dirichlet difference problem 265
4.4 Some properties of difference elliptic operators 272
4.5 Higher-accuracy scheme.? for Poisson's equation 290
5 Difference schemes for time-dependent equations
with constant coefficients 299
5.1 One-dimensional heat conduction equation with constant
coefficients 299
5.2 Asymptotic stability 327
5.3 Schemes for the heat conduction equation with .several
spatial variables 340
5.4 Schrodinger time-dependent equation 349
5.5 The transfer equation 354
5.6 Difference schemes for the equation of vibrations of a string 364
5.7 Selected problems 378
6 Stabihty theory of difference schemes 383
6.1 Operator-difference schemes 383
6.2 Cla.sses of stable two-layer schemes 397

Contents xvii
6.3 Classes of stable three-layer schemes 428
7 Homogeneous difference schemes for time-dependent
equations of mathematical physics 459
7.1 Homogeneous difference schemes for the heat conduction
equation with variable coefficients 459
7.2 Homogeneous difference schemes for hyperbolic equations 499
8 Difference methods for solving nonlinear equations
of mathematical physics 507
8.1 Difference methods for solving the quasilinear heat
conduction equation 507
8.2 Conservative difference schemes of nonstationary
gas dynamics 525
9 Economical difference schemes for multidimensional
problems in mathematical physics 543
9.1 The alternating direction method (the longitudinal-transverse
scheme) for the heat conduction equation 543
9.2 Economical factorized schemes 564
9.3 The summarized approximation method 591
10 Methods for solving grid equations 643
10.1 Direct methods 643
10.2 Two-layer iteration schemes 653
10.3 The alternative-triangular method 676
10.4 Iterative alternating direction methods 710
10.5 Other iterative methods 729
Symbols 745
Concluding remarks 749
References 753
Index 759

Preliminaries
Contemporary methods for solving problems of mathematical physics are
gaining increasing popularity. They are being used more and more in
solving applied problems not only by professional mathematicians, but also by
investigators and users working in other branches of science, engineering
and technology. In order to make this book accessible not only to
specialists, but also to graduate and post-graduate students, we give a complete
account of notions and definitions which will be used in the sequel. The
concepts and theorems presented below are of an auxiliary nature and are
included for references rather than for primary study. For this reason the
majority of statements are quoted without proofs. We will also cite
bibliographical sources for further, more detailed, information.
1.1 DIFFERENCE EQUATIONS
1. Preliminary comments. By applying approximate methods the problem
of solving differential equations leads to the systems of linear algebraic
equations:
Au = f
where A = ( «■ ) is a square (iV x vV)-matrix, u is the vector of unknowns
and f is the right-hand side vector.

2 Preliminaries
Most of the well-developed methods available for solving such a system
falls within the categories of direct or "exact-fitted" methods and iterative
Or successive-approximate methods which are gaining increasing popularity.
The starting point in more a detailed exploration is the simplest
systems of linear algebraic equations, namely, difference equations with special
matrices in simplified form, for example, with tridiagonal matrices.
The history of difference equations contains plenty of good examples
when such equations emerged during the course of direct descriptions of real
processes in science and technology. Equations of mathematical physics,
that is, partial differential equations are sources for a broad class of
difference equations approximating integral and differential equations and give a
substantial contribution to the continuing development of the theory of
difference schemes. The main feature of difference equations is stipulated by
the fact that the matrices of the high-order corresponding systems (about
lO'^-lO^) are sparse.
It is natural from the viewpoint of applications to treat various
difference equations regardless of the initial differential equations which have
induced them. All of the resulting schemes and properties are invariant to
a concrete differential equation.
In the present section a direct method for solving the boundary-value
problems associated with second-order difference equations will be the
subject of special investigations,
2. Examples of difference equations. Undoubtedly, the reader has already
encountered the simplest examples of first-order difference equations in
connection with the formulae for the terms of an arithmetic or a geometric
progressions: ak+i = Uk + d or Uk+i -2ak + at-i - 0 and ak+i = qak,
respectively, where the argument of the members aj, = a(k) takes only
positive integer values.
We briefly touch upon the basic concepts of the grid methods. The
basic notions such as grids and grid functions will be studied in more
detail in Chapter 2. A discrete set of points (nodes) is called a grid or
lattice. Let a grid function yi = y(i) of the integer-valued argument
i = 0, ±1, ±2, ., ., be given. The right and left differences are defined
at a point i to be:
A vi :- y{i + 1) - y{i) = yi+i - yt,
Vt/i ~ yii) - y{i - 1) = yi - Vi-i ■
By definition, At/j.i = yi — yi_i = Vyi- It should be noted that the
expressions above may be viewed as formal analogs of the first derivative

Difference equations 3
Lu = u' known from differential calculus. The same procedure works for
the second, the third and other differences
A^t/i = A(Ayi) = A(yi+i - yi)
~ {yi+2 - Vi+i) - {Vi+i -yi) = yi+2 - 2 yi+i + yi,
A^y. =A(A'"-^k)-
We see that one more right point is captured every time when the operator
A is applied. Consequently, applying A for 7n times we justify that A™ yi
contains the values yi, yi^i, ■ ■ ■ , yi+m at the points i,i + I, ... , i + m.
A very simple rule could be useful: the left difference operation at any
point i coincides with the right difference operation at the point i — ] so
that
AV yi = Af yi.^ = yi+^ - 2 y,- + y^.i .
Let a linear equation with the entering differences and coefficients
Q;o(i), Q;j(i), . . . ,Q;m(J) be composed:
a„{i) A" yi + a,{i) A""! y,; + ■ • • + a,„_i(i) A y, + a„(i) yi = fi .
Substituting the expressions for differences A'^yi, k = 1,2, ... ,171, one can
modify it to an mth order linear difference equation related to an
unknown yi:
ao{i)yi+m +a^{i)yi+rn-i + • • • + am-i{i)yi+i + am{i)yi = fi ,
ag{i) 7^ 0 , a„(i) ^ 0.
This definition is a formal analog of an mth order ordinary differential
equation
d'^u d"^~^u du
ao{x) 7^ 0.
As one possible example we consider the simplest ordinary differential
equation
du
Tx = ^(")

Preliminaries
and offer below some of its approximations with new notations Ui = u(xi),
Wj_l_i = u{xi^i) and Xi^i = Xi + h, where /i > 0 is the distance from Xi to
Xi + h, namely
du \ u(xi + h) — u(xi) Wi+i — Ui A Ui
dx j h
du \ Ui — Ui-i V Ui
dx I h h
du \ Wi+i - Mi_i A Wj -I- V Ui
dx J 2 h 2 h
All of the preceding difference expressions approach 4^ as /i ^ 0. The
symbol ~ means, as usual, approximation or correspondence. In what
follows we say that the expression
At(; _ Mj + l - Ui
~ir ~ h
approximates the first derivative ^ = u'.
The very definition implies that the equation
7 — Ji J Ji — J\^i) )
is a first-order difference equation in one or another form
A Hi - h fi or yi+i z:z yi + h fi .
There is no difficulty to solve this equation for a given initial value y^.
It should be noted here that a second-order difference equation also may
appear in approximating a first-order differential equation, for example, in
connection with these expressions
uix,+i) = u{xi) + h u'(xi} + i /i2 u"{x,) +-h^ u"'{x,) + 0{h^).
«(a;i_i) = u{xi) - h u'{x,.) + \h'^ u"{xi) - ^ h^ u"'{xi) + 0{h'^) ,

Difference equations 5
the sum of which gives
Neglecting 0{h'^) we derive rough expressions for u" within the notations:
(Pu\ Wj 11 — 2u,-+ Wj_i A^Wj-i AVwi
da;2 ; '~ /i2 /j2 /j
^ —a;,;
Substituting the first expression into the expansion Ui^i = Ui -\- hu[ -\-
I /i^ w" + 0{h^) instead of w", we deduce that
J _ Mj + l - Mi _ /l Mj + l - 2M; + M;_l ^ ^^^2^
1^  /l2
A) «:• = ^^i±i-:ii^ - '; "'+^~;-'^"-^ + oi/^^).
Replacing in A) w^ by /j- we cancel 0{h'^) and multiply the resulting
equation by 2/i. As a final result we get the second-order difference equation
AVt;i-2At;i = -2/i/, .
Its modification gives the approximation of the first-order differential
equation
^ = f
dx
Difference equations possess remarkable properties which will be given
special investigation in the near future.
3. The first-order difference equations and inequalities. Of our concern is
the first-order difference equation
B) feAt/j + ayi - fi,
which is a formal analog of the first-order differential equation
du
b — -|- au = f.
dt ■'
Alternative forms of equation B) are
b {Vi+i -yi) + ayi = fi or b t/i+i = cyi+ fi, c-b-a.

6 Preliminaries
In the general case b = bi, a = Ui and c = Ci, that is, we deal with known
functions of the argument i. If bi :^ 0, then
Vi+i = qi Vi + Vi ■
It is evident that
hi hi
Thus, we see that this problem has a unique solution if the value y is given
for some i. For the sake of simplicity let y^ be known in advance for «' = 0.
With this, one can determine all the values y^,y^, ... by the recurrence
formula just established. In the case qi = q = const and (pi — 0 this
provides support for the view that the whole collection of tji constitutes a
geometric progression. If qi = q and 99^ ^ 0, then
Vi+i = qvi + fi = q{qVi-i + Vi-i) + Vi = q"^Vi-i + Vi + 9 fi-i ■
The outcome of this is
C) Vi+i =9'+^ t/o + fi + qfi-i + ••• +9'"^ Vi +9V0
k = 0
Adopting those ideas, it performs no difficulty to find a solution to the
equation yij^i = gj j/i + (/9i, ?' = 0, 1, 2,..., in the case qi ^ const.
In tackling the first-order difference inequalities
D) yi+i < qy^ + fi, i = 0, 1, 2, ... ,
with known members q > 0 and fi and a given value y^ we prefer an
alternative form of writing
E) Vi+i = q Vi + fi , Va=yo-
Its solution can be most readily found with further reference to the relation
yi <Vi. Indeed, subtracting equality E) from inequality D) we get
t/i+i - Vi+i < q{yi ~Vi) < q^{yi-i - Vi_i) < •• • < q'+^ (t/o - v^) = 0 ,
yielding t/j+i < Vi^i for any q, where Vi has been expressed in q, Vg, f} by
analogy with C).

Difference equations 7
4. The second-order difference equations. The Caiichy problem.
Boundary-value problems. The second-order difference equation transforms into
a more transparent form
F) Ai yi-i - Ci Vi + Bi yi+i = -Fi, i=l,2, ... ,
Ai^O, Bi^O,
which in the notation Atji = tji^i — tji becomes is modified as follows
G) Bi A yi -A, A yi_i - {d - Bi - A,) yi = -Fi .
By virtue of the relations
A J/, - V J/, = A t/i - A yi„i = A^ j/i_i = 2/, + i -'2yi + yi_i ,
At;,_i = -A'^yi_i +Ayi,
equation G), in turn, can be rewritten as
Ai A^ y,_i + {B, - A,) Ay,- {Q - Ai ~ Bi) y, = -Fi, Ai ^ 0 ,
or, what amounts to the same,
Bi A^ yi_i + {Bi - A,) A yi_i - {Q - Ai - Bi) yi = -Fi.
The latter difference equation clarifies that F) is an analog of a second-
order differential equation.
It is necessary to specify two conditions for the complete posing of this
or that problem. The assigned values of y and A y suit us perfectly and
lie in the background a widespread classification which will be used in the
sequel. When equation F) is put together with the values t/j and A j/j given
at one point, they are referred to as the Cauchy problem. Combination
of two conditions at different nonneighboring points with equation F) leads
to a boundary-value problem.
For the sake of definiteness, we concentrate primarily on Cauchy
problem with given y^ and Ay^. Knowing y^ and t/j = y^ + Ay^, yi for
i = 2,3,... serves as a basis for the formulae
C'iyi - Aiy,.i- Fi
Vi+i = 5 , -Bi / 0 .
Therefore, it is concluded that problem F) has a unique solution for given
value j/o and y^.

8 Preliminaries
For the second-order difference equations capable of describing the
basic mathematical-physics problems, boundary-value problems with
additional conditions given at different points are more typical. For example, if
we know the value y^ for i = 0 and the value y^ for i = N, the
corresponding boundary-value problem can be formulated as follows: it is necessary
to find the solution yi, 0 < i < N, of problem F) satisfying the boundary
conditions
(8)
A^i
J/a
A'2
with known numbers ^j and fi^.
Common boundary conditions may be specified by
Vo = Xiyi+ f^i
y^
^a^A
^2
that is, at the boundary nodes i = 0 and i — N not only the function values,
but also the first difference values or linear combinations of the function
and difference values are yet to be known. Substituting y^ ~ y^ + Ay^ into
the first condition (8') yields
Xj A;
A
Vo
-1^1
and needs certain clarification. The case Xj = 0 corresponds to the first
kind boundary condition: y^ is given at the boundary node i = 0.
When Xj = 1 we deal with the second kind boundary condition: Ay^
is given at the same node. All the cases with Xj ^ 0; 1 reflect the third
kind boundary condition as a linear combination of the function and
the first difference at the node i = 0.
Due to serious achievements of the Russian and foreign
mathematicians in applied mathematics the majority of mathematical-physics
problems may be reduced to computational algorithms, at every step of which
3-point equations like F) with conditions (8') must be solved.
Moreover, a lot of rather complicated problems in numerical analysis
gives rise to the canonical problem, where a square (A^-|-l) x (A^-|-l)-matrix
of the corresponding system acquires a tridiagonal form
/
1
0
JT--1 0-1 -O^
0 \
0
0
jT-i — Oj'
0
0
\ 0
A.
-C,
N- I
V

Difference equations 9
In the case of boundary conditions of the second or third kinds its
order is A'^ + 1, while for the system F) with the supplementary conditions
(8) the order is A'^ — 1. All the matrices of interest possess the main feature:
they have nonzero elements only on the three diagonals (the main and two
adjacent ones).
With the aid of effective Gauss method for solving linear equations
with such matrices a direct method known as the elimination method has
been designed and unveils its potential in solving difference equations.
5. The elimination method. The problem we must solve take now the form
A, yi_i - Ci t/i + Bi y,,+i =-Fi, i = 1, 2, .. . , N - 1 ;
(9)
where A, 7^ 0 and 5, 7^ 0 for all i = 1, 2, .. . , A^ - 1.
Other ideas are connected with reduction of the original second-order
difference equation (9) to three first-order ones, which may be, generally
speaking, nonlinear. First of all, the recurrence relation with indeterminate
coefficients aj and Pi is supposed to be valid:
A0) iji = tti+i ?A+i +Pi+i .
Substituting tji-i = aiyi + Pi into (9) yields
[Ai cYi - C'i) yi+ Ai Pi -I- Bi j/i+i - -Fi,
which leads, because of A0), to
[ {A, cxi-C) a,+i -F B, ] jA+i -F Ai P, -F (A, a,; - C,) A+i = -Fi ■
If the conditions
{Ai ai - Ci) tti+i +Bi = Q, AiPi + {Ai rn - d) Pi+i + Fi = 0
are fulfilled simultaneously, then the equation in view holds true for any j/j-.
Thus, assuming d — ai Ai ^ 0, we establish the recurrence formulae for
determination of both ai+i and /3j'+i:
A1)
A2)
Oj- - ai Ai
A, P, + Fi
P'+^ -^ r A. '
i=l,2, ..
?:= 1,2,..
.,N^l,
.,N^l,

10 Preliminaries
under agreement A0).
By going through the matter chronologically the forward elimination
path for finding cvi, Pi and the backward path for finding tji do arise during
the course of the elimination method. Let us clarify the situation. With
knowledge of aj and Pi one can determine, according to A0), all the values
xji moving from i + 1 to i. To find Oi and Pi, we succeed the reverse order
(from i to i + 1) according to formulae A1)-A2).
Two relevant aspects are worth noting in this context:
1) equations A1)-A2) for finding rii and Pi, being nonlinear, reproduce the
relationships between the values at two adjacent points;
2) for each unknown a, P or y, it is necessary to solve the corresponding
Cauchy problem.
These requirements necessitate imposing auxiliary values which can
serve as the initial conditions. The assignment of boundary conditions is
aimed at specifying these or tho,se values. Having substituted « = 0 into
A0), we get t/g = ttj t/j +/3j. On the other hand, y^ = x^ y^ +A'ii gi^ii^g
A3) ttj = Xj ,
A4) /?i=A^i.
Thus, we set up for a and P the Cauchy problems described by A1), A3)
and A2), A4), respectively, thereby completing the forward path of the
elimination.
At the second stage, knowing cxi and Pi, the boundary value y,^ is
recovered from the system of the equations
under the constraint 1 — a^ x^ ^ 0.
Hence, the initial condition in question becomes for A0)
A5) ..-^^= + "=^-
1 - "n ^2
The computational formulae A0) and A5) constitute what is called the
backward elimination path.
The algorithm presented below as the sequence of applied formulae is
called the right elimination method and is showing the gateway for the

H
a ,
H
'.+1
■+i
-
=
^2
a
Ai
5,
- ai Ai
Pi + Fi
- cti Ai
+ ^2 /^N
Difference equations 11
general case:
R.
i = 1,2,... ,Af- 1 , ttj = Xj ,
i = l,2,... ,iV-l, /3i=Aii,
i - «„ Xj
«/» = ai+iyi+i+A+i, i = iV-l,iV-2,... ,1,0.
Here the symbols (—>) and (^) indicate the directions of index count: either
from i to i + 1 or from i + 1 to i.
In connection with the preceding algorithm, it is natural to raise the
question of correctness and stability providing a possibility of applying the
method and obtaining a solution with a prescribed accuracy. Special
investigations give definite answers to to these questions.
6. Stability of the eliixunation method. Let us stress that the conditions
Ci — cti Ai ^ Q and 1 — «„ Xj 7^ 0 cannot be excluded or relaxed during the
course of the right elimination method. Just for this reason the restrictions
on coefficients for well-posed and stability conditions needs investigation.
Common practice involves sufficient conditions
A6)
|x„|<l, a=l,2, IxjI-fIxjI <2,
yielding \ai\ < 1 for all i = 1, 2,. . . , A^.
The proof is carried out by induction. Assuming \ai\ < 1 we will show
that |Q!i+i| < 1. Since jaj = |xj| < 1, the same propertry will cover all
i = 2, 3,. .. , N on account the chain of the inequalities:
\C, ^ a, Ai\ - \B,\ > \a\ - \a,\ ■ |A,| - |5,| > |A,| ■ (l-|«i|) >0,
from which it follows that \C'i — ai Ai\ > 0, since Bi ^ 0.
Granted |ajj < 1, observe that
Y, + ,\ = ^:r^^ r<l
\Ci-Aia,\

12 Preliminaries
as well as |ai+i| < 1 are ensured by \ai\ < 1. The assumption \a^\ — |xj| <
1 implies |ai| < 1 for alH = 1, 2,. ,, , n. The lower estimate holds true for
the denominator in formula A5):
|l-a^v><2l > 1 - \aj ■ IxjI > 1 " 1^2! > 0,
because either \>c^\ < 1 or |a„| < 1.
Thus, we have proved that under conditions A6) the right elimination
algorithm is correct, meaning nonzero denominators m formulae A1), A2)
and A5). So, under conditions A6) problem (9) has a unique solution given
by formulae A0)-A5).
It is necessary to point out that calculations by these formulae may
induce accumulation of rounding errors arising in arithmetic operations. As
a result we actually solve the same problem but with perturbed coefficients
Ai, Bi, d, Xj, ><2 and right parts Fi, 'jl^^Jl^- UN is sufficiently large, the
growth of rounding errors may cause large deviations of the computational
solution iii from the proper solution yi.
The trivial example shows how instability may arise in the process of
calculations of t/j by the formula yij^i = qyi, q > 1. One expects that, for
any y^, there exists a number n^ such that overflow occur for y„ = q" Vq , n =
Uq, thus causing a abnormal termination. An important obstacle in dealing
with this problem is that iji satisfies the equation j/i+i ^ qiji + r] with a
rounding error 77. Indeed, for the error 6 yi = yi — yi the equation is valid:
6yi+i = qSyi+i], 6yg =1-],
from which it follows that the value
q' - 1
S Vi - q'v+ r ^ ' 9 > 1 .
q- 1
increases exponentially along with increasing i.
Returning to the right elimination method, we show that the
conditions \ai\ < 1 guarantee that the error Syi^i = j/i+i — j/i+i arising when
computing yi does not increase. Indeed, the equations
yi = tti+i iji+i + Pi+i , yi = tti+i 2/i+i + Pi+i
imply that
E iji = tti+i 8 iji+i , |<5 t/i I < |ai+i I ■ |E j/i+i I,
meaning \6 yi\ < |Et;i_|_i| because |ai+i| < 1.
Taking into account perturbations of the coefficients tti+i and /3j+i,
another conclusion can be drawn that the accuracy e in determination of
the solution yi of problem (9) is
max \8yi\ « e^N'^ ,
l<i<N
where £g is a rounding error and A' is the total number of grid nodes.

Difference equations 13
7. The left elimination method. The counter elimination method. Still
using the framework of the right elimination method (formulae A0)-A5))
in reverse order, we obtain through such an analysis the computational
formulae of the left elimination method:
A7) ?.= ^ f „ , i = 7V-l,7V^2,...,2,l, e.
Bj r]i+i + Fj
Ci — ii+i Bi
A8) %\^pl±l±Il^ i = N^l,N-2,...,2,l
A9) y,+i =e«+i K+ ??«+!, i = 0,l,... ,N-l,
and
B0) Vo
l-h + Xl 111
1 - ^1 ^1
Indeed, with the relation j/j_|_i = 6_|_i j/f + rji^i granted we eliminate
successively jji+i, jji = iiVi-x + Tji from equation (9). The outcome of this
is
-Fi = Ai j/,:_i + {Bi 6+i - d) yi + Bi rn+i
= [Ai - {Ci - fl, (,i+i) (,i] K_i + Bi, r]i+i - (C, - B,(,i+i) r]i .
Equation (9) is valid if we agree to consider
A. - {Ci - Bi 6-+i) e. = 0 , -Fi = Bi ?y.+i - {d - Bi ^i+i) in ,
from which formulae A7) and A8) immediately follow. The value j/g can
be found from the condition j/g = Xj j/j + ji^ and the formula j/j = ■?! J/j +??o-
The presented inequalities
ia-i?,6-+i|>|c,|-|fl,l • fe+ii, 11-^1 ^il>i-ieil • I^J
confirm that the correctness and stability of the left elimination method
are ensured by conditions A6), since |6| < 1 for all i = 1, 2,. .. , TV.
Joint use of the left and right elimination methods refers to the counter
elimination method. The essence of this method is to consider a fixed inner

14 Preliminaries
node f = fg, 0 < ig < TV, and to compute the coefficients Ui and Pi for
0 £ *' £ *o + 1 according to formulae A0)-A5):
tti+i = 7; '-—7- , «■ = 1, 2, . . . , «o . "i = ^1 ,
Ci - ai Ai
Pi+i = -p^ T' « = 1, 2, ... , lo, A=/ii,
and for fg < i < N the coefficients ,Jj- and rji can be found by formulae
A7H20):
e, = ^, f „ , i=7V^l,7V-2, ...,io, C = ^2,
« = TV - 1, TV -2, ... , Zo , Vn = fh
Ci — 6+1 -Sj-
The next step is to put the first solution of the form A0) together with the
second one of the form (9) the node i = ig. The outcome of this is
yielding
t - "io + l '^^'o + l
As far as 1 — O-i^+i "QiQ + i > 0, this formula is meaningful because under
conditions A6) at least one of the modules: either |ajg_|_i| or |?yig+i| is less
than 1. With knowledge of j/i^ the remaining values j/i arise in the process
of parallel calculations: for i < i^ by formula A0) and for i > ig by formula
A9).
Of course, the counter elimination method could be especially effective
in an attempt to determine yi merely at only one node i = ig.
8. Maximum principle. To make our exposition more transparent, the case
of interest is related to the first kind boundary-value problem with Xj = 0
and Xg = 0:
-C [vi] = Ai yi_i - Ciy, + Bi j/i+i =:; -Fi ,
B1)
z = 1, 2, ... , tV- 1, J/o = /^i . Vn = l-h-

Difference equations 15
Theorem 1 (The maximum principle). Let the conditions
B2) Ai>0, Bi>0, D, = Q-A^-Bi>0
be fulfilled for all i = 1,2,... , TV — 1 and the grid function yi, yi ^ const,
satisfy at all the inner nodes i = 1, 2,. .. , TV — 1 the condition C [yi] > 0
(or jC[yi] < 0). Then yi cannot take the maximal positive value (or the
minimal negative value) at the inner points, that is, for i = 1,2,... , TV — 1.
Proof Given a grid function yi, let £ [yi] > 0 for all i = 1, 2,... , A^ — 1
and yi attain its maximum at one of the inner nodes i = i^, 0 < f^, < TV,
so that
yi^ - max y^ - M^ > Q .
0<i<N
As yi ^ const, there exists an inner point fg (may be, coincident with i^)
such that yi = yi^ = M^ > 0 and at one of the adjacent points, say for
?' = ig — 1, we have j/ig-i < M^. We may attempt C [yi] in simplified form
-C [yi] = Bi (yt+i -iji) - Ai (iji -yi-i) - (Ci - Ai~ B,)yi,
leaving us under conditions B2) s at the point i = i^ with
^yio] = %(yio+i -vio)~^io(y^o -Ko-i)-(Cio -^io -Bio)y^o
< -Bi^ (yi^ - j/,g+i) - A,g (j/ig - Kg_i) < 0
by virtue of the relations
yio > yio+i' yio > y^o-^ > "^^o > "^> Bi^ > o.
The result obtained is not consistent with the condition: jC [yi] > 0, valid
for all i = 1, 2,... , A^ — 1 including i — i^. Thus, we proved the first
statement of the theorem. The second one can be established in a same
way with further replacement of yi by —yi.
Corollary 1 Under conditions B2) and
£M<0, i^l,2,...,N-l; %>0, y„>0,
the function yi is nonnegative: yi > 0 for all i = 0,1,. .. , N. The case
^ [Vi] > Oi % ^ 0 ^^'^ Vn ^ 0 leads to iji < 0 for other subscripts i =
l,2,...,iV-l.
Indeed, let £ [yi] < 0 and yi < 0 at least at one point i = i^. Then yi
should attain its minimal negative value at an inner point i = i^, 0 < ig <
A'^. But this fact contradicts Theorem 1.

16 Preliminaries
Corollary 2 Under conditions B2) the problem
B3) £[k] = 0, f=l,2, ... ,7V-1; j/„ = 0, j/„ =:0,
has the unique solution yi = 0, thereby justifyingg that problem B1) js
uniquely solvable for aJiy ingredients Fi, /ij and ji^.
In fact, assuming that the solution yi of problem B3) becomes nonzero
at least at one point i = i^ we come to a contradiction with the maximum
principle: if j/j, > 0, then yi attains its maximal positive value at some
point fg, 0 < fg < TV, violating with Theorem 1; the case yi^ < 0 may be
viewed on the same footing.
Theorem 2 (Comparison theorem). Let conditions B2) hold, yi be the
solution of problem B1) and y^ be the solution to the following problem:
^yi] = -Fi, f = 1,2,... ,7V - 1 ; J/o =/^i , yN^P-2,
with
\Fi\<F,, i=l,2,...,N -1; |/ij<//i, Uh\<fi2-
Then the relations occur:
\y^\<y^, f = 0,1,... ,7V.
Proof Due to Corollary 1 we have f/i > 0 for 0 < i < 7V, since -C [j/j] =
— Fi < 0 and % > 0, y„ > 0. Observe that the functions Ui = yi ~ yi
and Vi = yi + yi satisfy equation B1) with the right parts Fi ~ Fi > 0 and
Fi + Fi > 0, and boundary values u^ = fij^ ~ fij^ > 0, u^ = jl^ — H2 ^ ^ ^^'^
v^ = fi^ + H^ > 0, v^ =1^2+1^2^ 0, respectively. By applying Corollary 1
to such a setting we get Ui >0, Vi yQ ot —iji < yi < i/i, meaning |j/i| < f/j-.
The function jji is called a majorant for the solution of problem
B1). A first step towards the solution of problem B1) is connected with a
majorant j/,; \\y^ < \\y^.
Corollary 3 The solution to the problem
-C [ J/i ] = 0 , 0 < z < 7V ; % = /^i ' Vn ~ l-h'
can be majorized as

Difference equations
17
The problem of auxiliary character such as
£[k] = 0, 0<i<7V; j/o = j/„ =/i = max (l^til, l/igl) .
will complement our studiea, for which the comparison theorem gives ||j/^ <
WVq- On the same grounds. Theorem 1 implies \\y^ < /I, since the function
f/j > 0 takes its maximal positive value only at a boundary point: either
for i = 0 or for i = TV.
Theorem 3 Let the conditions
B4) \Ai\>0, |fl,|>0, Di = \Q\-\A\-\Bi\>0
hold for all i — 1,2,... , A^ — 1. A solution of the problem
B5) £[k] = -F,, z = 1,2,... ,7V-1; % = 0, j/„ = 0 ,
admits the estimate
B6)
IVc <
To prove the desired estimate, the intention is to use an alternative
form of equation B1)
B7)
Ci Vi = Ai yi^i + Bi j/i+i + Fi
Let \yi\ take its maximal value \yi^\ > 0 at a point i = ig, 0 < ig < A^,
so that |j/jg I > \yi\ for all i = 0,1,. .. , A^. We are led by equation B7) for
I = ig to
|C',g y,g I = \Qj ■ |j/ig J = J^.^ ^ ^ _ i + B^g ^ ^ + 1 + ^^ ^ |
<(KI+l^^ol) ■\y^o\ + \F^o\■
because of this
{\QJ-\A,J--\B,J) •|y>g| = A-o -IKoI^IF.J

18 Preliminaries
and, consequently,
II I I <. I^'-'
0<i-<A' Di
— — 'o
Before going further, problem B5) with coefficients subject to
conditions B2) for 0 < i < A^ may be of help in achieving the final aims:
A, > 0 , 5i > 0, L>, = C, - Ai - 5, > 0.
When the condition Di > 0 fails to be true j/; arranges itself as a sum
0 0
i/i =: yi -\- Ui, where J/,; solves
B8) Bi{yi,+i~y,)-A{yi-h-i) = -Fi, 0<i<N,
0 0
y, = y^=0.
The statement of the problem for m; is
B9) L [u,] = Bi Ui+i - Ci Ui + Ai Mi_i = -A Vi, 0<i< N ,
Wo = "w = 0 .
whose solution satisfies the inequality
C0) ||u|L = max \u,\ < \\°y\\^.
0
This result is a corollary to the next lemma with if>i " y i involved.
Lemma. For the sohition of problem B5) with coefficients subject
to conditions B2) and the right part Ft " Di (pi, i = 1,2, .. . , N — 1, the
estimate is valid:
C1) ll%<ll^c-
Proof In the case Di = 0 the estimate is obvious, since yi = 0 due to
Corollary 2 to Tlieorem 1. By relating Di > 0 at least at one point we have
occasion to use the function Yi > 0, being a solution to the problem
C2) C[Yi]=~D,\<pi\, i= 1,2,... ,7V^1; Y„ = Y^ = 0 .

Difference equations 19
The solution of problem C2) can be most readily evaluated with the aid of
the relation \\y^ < \\Yq, valid an account of Theorem 2. Let the maximum
of the function Yi be arrived at a point i = i^. Then
Bi^{Y,^ + ,~Y^) <0, A,(y,„~l^„_i)>0
and C2) implies the double inequality
Aoy.o<Aj^.ol<Aoll^c-
In the case Di^ > 0, it follows from the foregoing that
C3) \\Y,, < 11^^ .
For Di^ = 0 we deduce from C2) that
With the relations Yi^ > Yi^-i and Yi^ > Yio + i in view, we find that
Ji'o + i = Yi^ = yio-i ,
it being understood that the same maximal value is attained at the adjacent
to I'o points.
By merely setting i = i^ =1^ + 1 (or I'j = i^ — 1) we follow established
practice and obtain, as a final result, the inequality
A, Y^ < A,• Wfc .
giving either inequality C3) or the equality YJ'.+i = ^^,-1 = Yi^. As
Di ^ 0, we get Di^ > 0 and inequality C3) for some i = i , thereby
completing the proof.
Theorem 4 The solution m of problem B5) with coefficients B2) admits
the estimate
C4,) \\y^<2\\y^
0
where y is the solution of problem B8).

20 Preliminaries
0
Proof Recall that the difference Ui = Hi — V i is the solution of problem
0
B9) with the right part Fi = DiV i. The inequality
ll% = ll^+«c <ll^c+ll«c
in combination with estimate C0) leads to estimate C4).
This provides reason enough to reduce proper evaluation of the
solution of the general problem B5) to that of the simpler equation B8), whose
0
solution y i can be found in explicit form.
9. Maximum principle for the third kind boundary-value problem. The
maximum principle and its corollaries remain valid for the general
boundary-value problem F), (8'), whose statement is
C[y,] = ~F,, i = 0, 1,2, ...,iV;
B1*) 'C [ j/o ] = ~j/o-F Xi j/i , i^o = Ml ,
'C [ l/,v ] = -Vn + ^2 Vn-i . Fn = A''2 .
with the members Cg = 1, A^ = 0, B^ = x^, C„ = 1, A^ = Xr^ and 5„ = 0.
Theorem 1 (The maximum principle). Let under the set of the restrictions
.4, >0, 5,>0, C\>A,+B,, i= 1, 2, ... , iV~l;
B2*)
0 < Xl <l , 0<>f2<l. 0<Xj-|->f2<2,
tile function y^, yi^ const, satisfy the conditions
C[y^]>0 (£[k]<0), i=0, 1, 2, ...,iV.
Then yi cannot take the maximal positive {the minimal negative) value at
any node i = 0,1, 2,.. , ,N, that is, ih < 0 (;/,: > 0).
The proof is analogous to that of Theorem 1 from Section 8. It is
necessary only to consider, in addition, the following two cases:
a) if I'o = 0, that is, mcix^ j/; = J/q ~ ^o > 0 ^'^'^ Vi < ^oi then
^[yo] = -Vo + ^il/i = ~A ~^i) Vo -^i(l/o -Vi) <0
for 0 < J<i < 1;

Difference equations
21
b) if ig = N, that is, maxi j/,: = y^ = Mo > 0, but j/„_i < Mg, then
'C[l/„] = -l/„ +^2J/«-i = -A -^2) J/« -^2(i/n /n-i) < 0
for 0 < J<2 < 1,
In both cases we came to a contradiction with the condition £ [jji] > 0
for all i = 0, 1, 2, ,. . , A'^. It is therefore concluded that j/j < 0, since if
j/i > 0 at least at one point i = i», its maximal positive value should be
attained at some point i = i^ (for example, at ig = i^) that in principle is
impossible.
Corollary 2a Under conditions B2*) the problem
£[yi] = 0, i = 0,l,...,7V
has only the trivial solution.
There is no difficulty to reformulate the remaining assertions from the
previous sections, but we do not dwell on precise statements,
10. Solution estimation for difference boundary-value problems by the
elimination method. In tackling the first boundary-value problem difference
equation B1) has the tridiagonal matrix of order A'^ — 1
0
B,
0
0
A.
0
0
Bn-2
\
being symmetric for the case
C5)
5,: = A.
i+\
Difference equations with a symmetric matrix are typical in numerical
solution of boundary-value problems associated with self-adjoint differential
equations of second order. In what follows we will show that the condition
Bi = Ai + i is necessary and sufficient for the operator £ [j/,] be self-acljoint.
As can readily be observed, any difference equation of the form
C6) Si j/i+i - Q j/i-F A,; j/i_i =-i^i, 1/0 = A'l >
AiT^O, Bii^O, 1=1,2,... ,N -I,
Vn = IJ-2,

22 Preliminaries
can be made self-adjoint:
Aj/i = Oi+i (j/i+i - j/i) - Ui {yi - yi^^) - diyi = -<pi ,
i=l,2,...,N-l.
Indeed, we multiply equation C6) by the function rji ^ 0 and involve
the conditions Ai rji = a^ and Bi rji = Oi^i, whose use permits us to establish
Ai^i T]i+i = BiT^i- Oi+i and
Bi T-T Bk
where rj^ is an arbitrary constant. As a result we get the equation A j/i —
~(pi with ipi = rii Fi and a^ = Ai ??i, d^ = (C,- - A,- - 5i) ??,' = Q - flj - fli+i,
Ci = C'i rji.
By app:ilying the elimination formulae to the problem
C7) Aj/i = a, j/i_i -Ci j/i + ai+i j/i+i = ~(pi ,
i:=l,2,...,7V-l, j,„==0, j/„=0,
we deduce the estimate
C8) IMIc< E ^^El^.l
under the natural premises
C9) |ai|>0, |c,|> |a,| + ja,+i|.
The computational formulae of the right elimination method help derive
estimate C8):
j/i = fti+i j/i+i-|-/?i+i , I = 0, 1,... , A^-1 , 1/n = 0.
D0) Qi+i = "'+'—, i= 1,2,... ,Af-1, ai=0,
a - ai ai
A+i="'^'^'^' . i=l,2,...,yV-l, A=0.

Difference equations 23
Since jai+il < 1 under conditions C9), formulae D0) imply that
\yi\ < \o^i+i\\yi+i\ + \Pi+i\ < li/i+il + |A+il.
yielding, in turn,
D1) IkI < E \Pk\, 0<i<7V~l, y^=0.
k=i+l
Substitution ji = a^ Pi leads to the useful relations
Ji + l - Oi+i {ji +(pi) ,
i i
\7i+l\ < \7i\+ \<P^\ < l7ll+ E I'Pkl = E \^k\
k=l k=l
and, consequently,
m< ^ E Ifkl, i = 2,3,...,N, /?, =0.
r»| k=i
Combination of D1) with the foregoing allows us to establish C8).
Before going further, observe that a solution to the problem
D2) Ayi = -<pi, i-[,2,...,N-l; l/o = Mi . Vn = l-h '
can be majorized by
N-l i
WVc < max(|;ui|, IM2I) + X] I i YI ^'Pkl ■
i=i l"^+il k=L
As stated in Corollary 3, a solution to the problem
Ay^-O, i = 1,2,... ,Af- 1 ; l/o = Mi , 2/n = A'2 .
with a, > 0 and c^ > a^ + a^^i can be most readily evaluated by the
quantities
D3) \\y^ <max{\n,\,\fi^\) .
The solution of problem D2) arranges itself as a sum yi = iji + Vi, where
the second summand is a .solution of problem C7). Applying estimates C8)
and D3), and the inequality \yi\ < |j/i|+ \vi\ we come to the desired result.

24
Preliminaries
11. The second-order difference equations with constant coefficients. If
the coefficients of the difi'erence equation
D4)
-C [yi] = Ai y,^i - C'i yi + Bi ja+i = 0
do not depend on i, meaning Ai = a, C'i = c and Bi — b for all i = 1, 2,
a solution of the following equation
D5)
b yi+i - c iji + a j/,:„ 1 = 0 , b^ 0, a j^ 0 ,
may be found in explicit form. In so doing, let y^ and yl"' be two solutions
of difference equation D4) that are linearly independent if the equalities
C:2/f^ + Q2/f^ = 0.
i = 0,l,2,..
hold only for Cj = Cj = 0. This condition can be replaced by the usual
requirement for the determinant of the system of algebraic equations
c ,/'^ + r ,P^ - 0
saying that, for all i and m
^i, i-\-7n
Vi
„B)
A) . B)
m= 1,2,... ,
^ 0.
By pursuing the parallel we see, in particular, that the condition
Ai,
:+i
A) B)
Vi Vi
A) B)
Vi + l Vi + 1
„A)
„B)
Aj/P^ Ayf)
7^ 0
is analogous to that of the linear independence for solutions u{x) and u{x)
of an ordinary second-order differential equation:
u(x) u(x)
{u{x)y mx))'
+ 0
Eliminating j/,., j and yily with the aid of equation D4) we get
^t,i+i — n yj^tyi^iy, -Oj (/j+i ^j
A ^/l) J2) _ A) B)

Difference equations 25
Ai
A,: ,:_i_i — —-— A,:
ij,!+l
Bi
Due to this fact the condition Aij_ij_|_i ^ 0 for some i = i^ yields Aij^i ^ 0
for all possibilities of i.
Taking into account the equation C [y^] = 0 we see that, for any m > 1,
^i,i+m can be expressed through Aij^i and, hence, Aj^i+m 7^ 0 is ensured
by the condition Aj^j-i-i ^ 0.
If y\ ' and yi ' are linearly independent solutions of the homogeneous
equation D4), then its general solution can be designed as a linear
combination of y\ and yS ' with arbitrary constants Cj and Cj!
yi = C,y\'^ + C,yf\
The constants are free to be chosen from the initial or boundary conditions,
since Aj^i+m 7^ 0 for all admissible subscripts i and m.
The general solution of any nonhomogeneous difference equation
C [yi\ = —Fi is representable by
Vi = C'lJ/j- +C^yf' +yi ,
where iji is a particular solution to the equation C [iji] = —Fi.
In the case of the Cauchy problem with assigned values j/q and j/j, we
have at our disposal the system of algebraic equations for constants Cj and
C,y^o^ + C,yf^ = y,,
C,y'l^ + Cj^^ = y,.
As far as y\ ' and j/j- ' are linearly independent, the condition A^ j 7^ 0
provides a possibility of finding Cj and C^- In the case of the boundary-
value problem with y^ = fi^ and y^ = fi^ incorporated, the constants Cj
and Cj can uniquely be determined under the agreement A^, ^^ ^ 0.
If the coefficients involved in equation D4) are constant, that is, A^ =
a, Ci = c and Bi = b, then particular solutions can be found in explicit
form. This can be done by attempting particular solutions to equation D5)
in the form yk =9*, while the number q ^ 0 remains as yet unknown.
Substituting y^ into D5) we obtain the quadratic equation for q:
D6) bq"^ ~-cq + a = 0

26
Preliminaries
whose roots are
c ± a/c^ ~~ 4ab
26
Three possibilities of interest in accordance with the discriminant sign are:
1) If -D = c^ — 4ab > 0, then the preceding equation has two distinct real
roots
3i = o;. ■ 32 =
26
26
Two different particular solutions y^. = ?f and y^. = ?* corresponding to
(]j and §2 are linearly independent, since
^Jt.Jc + l
?;
<
„J; + 1 „J: + 1
^1 ^2
('i2-'iiKi 'i2 y^O
and constitute what is called the general solution of D5):
where Cj and C^ are arbitrary constants,
„(i)
2) If -D = c^ - 4a6 = 0, then q^ = q^ - c/{2h) = q^ and y'l' - q^ and
y^j^ ' = k q^ can be declared to be linearly independent particular solutions.
Indeed, substituting y^, ' = k q^ into D5) we obtain by minor changes
byi% - Clip + ayf_\ =[h{k + l)ql - ck q, + a{k - I)] q'^'
= 3o'"' Hbril -cq, + a) + q'^'' {b ql - a) = q'^' {h ql - a) = 0 ,
since b q'^ ~ a = 6[c/B6)]^ — a — D/{4b) = 0. The discriminant
^jt.i+i
3o ^ 3o
-,2J; + 1
7^0
assures us that g* and k q^ are linearly independent and can be chosen as
a basis for constructing the general solution of D5):
Vk = (Ci+fcCa) q^
c
26

Difference equations 27
3) If _D = c^ — Aab < 0, equation D6) possesses two complex-conjugate
roots
c+J\D\i
9i = ^-7 = p(cosip + 2 sm (f) = pe'"^,
Q _ w'TdI i
I2 = ;r7 = P (cos (^ — f sin (^) = p e """'',
wher
e
a , X \D\
P= \ T , f = arctg
0 C
The appropriate functions
5^ = p^' e "' *-■ "^ = p* (cos A;^ - f sin kf)
y[ = p^' cos kp , Vk = P^ s''^ ^f
are just particular solutions to equation D5). In this case both pairs of
solutions are linearly independent in connection with the linear independence
of the functions sin kip and cos kip: Ak^k+i / 0. Then the general solution
acquires the form
Vk = P (C*! cos kp -\- Cj sin k(f) .
In order to illustrate our approach a little better, it seems worthwhile giving
several simple examples.
Example 1 We are interested in giving the general solution to the equation
yk+i-'^pyk + yk~-i-=^, p>o,
still using the above framework with D = 4 (p^ — 1), the following three
cases need investigation.
a) Let p < 1. Setting p = cos a, a ^ 0, and y^. = 5* we obtain the quadratic
equation for q:
5^—2 cos a q -\- \ ■= Q

28 Preliminaries
with discriminant D = 4(cos^ a — 1) = —4 sin^ a < 0 and roots q^.^ =
len g* = e^'*^", particular solutions are given by
sm ka
A) ; B) • ;
y). ' = cos fca, Vk ^"^ '^
and the general solution can be written as
Vk = C\ cos ka + 6*2 sin ka
with arbitrary constants Cj and Cj.
b) Let p > 1. Setting p = ch a and yk = g* we get
g^-2chag+l = 0
with discriminant _D = 4 (ch^ a— 1) > 0 and roots gj 2 ~ ^h a±sh a = e .
Then g* 2 ~ "^ > Particular solutions become
2/^ ■' = ch A;a , y^ -^ = sh A;a
and the general solution can be written as
yk = C\ ch ka + C^shka
with arbitrary constants Cj and Cj.
c) Let p = 1. In this case, for yk = g*, we have g^ —2g + l = 0 and gj 2 =^ -'-■
The particular solutions y^ ■' = 1 and y\' = k form the general solution
2/j, = Cj + C'2 A; as a linear function.
Example 2 It is required to calculate the integral
h{^)= /^°^'f'f#. ^ = 0,1,2,.,..
J cos l/) — COS <f>
0
First of all we note that
0
We claim that Ik is just the solution of the Cauchy problem for the second-
order difference equation:
Ik+i~'2 cos (f Ik + h-i = 0 , ^" = 1,2,..., ^ = 0, A = TT,

Difference equations 29
assuming 7? to be arbitrary fixed. Indeed, the chain of the identities occur:
[cos (k + l)xp — cos {k + l)ip] + [cos {k — 1) xp — cos (k — 1) (p]
= 2 cos kip cos xp — 2 cos k(p cos 7?
= 2 (cos kip — cos ^7?) cos (fi + 2 (cos y/" — cos ip) cos feY/",
from which the relations immediately follow
f (cos ^' - COS y?) cos kiP
Ik+i + h-i = 2 cos f Ik + 2 / ■ dip
J cos 1p — cos (fi
0
7r
— 2 cos ip Ik + 2 / cos kip dip — 2 cos "/J /fc , ^ > 1 ,
0
yielding 4+1 ~2 cos •filk + h-i = 0.
From Example 1, case a) we know that
Ik('p) = Cj cos kip + Cj sin ki^ .
The initial conditions for ^ = 0 and k =z I give C\ = 0 and C2 = tt/sin i/j
in connection with the available information that C\ cos (p + C\ sin 7? = tt.
The outcome of this is
sin k(fi
Il,{ip) - TT ~~. , fc = 0, 1,2,.. . .
sirup
12. Formulae of "difference differentiation" by parts of the product and
sum. The formula for differentiating the product of two real functions
u{x) and v{x)
d / , . , .-, , \ dv du
is well-known from differential calculus. In an attempt to establish a grid
analog of this correlation, we consider any two grid functions yi and f, of the
discrete variable i = 0, ±1, ±2, .... The following formulae of "difference
differentiation" are valid:
A {vi Vi) - yiAvi+ Vi+i A yi = yi+i A Vi + ViAyi,
D7)
V B/i Vi) = 2/i-i V Vi + Vi Vj/i = 2/i V Vi + Vi^i V yi

30 Preliminaries
with the well-established notations of the right and left differences: A j/^ =
m+i - Vi and \7yi = Vi - yt^u so that Vj/j+i = Aj/i.
With this relation established, we find in a step-by-step fashion that
yi A Vi + Vi+i A yi = tji {vi+i - Vi) + Vi+i (y,+i - yi)
= Vi+i Vi+i - Vi Vi = A {yi Vi) .
Likewise, one can check the second formula based on another relation
V(j/ifi) = A{yi^iVi^i) .
An important role in the theory of difference schemes is played by the
identities serving on this basis as grid analogs of integration by parts:
u' V dx = u V \ — tiv' dx .
For any functions yi = y{i) and fj = v{i) defined on the grid lo = {i =
0,1,2,... ,N'\, it will be sensible to introduce analogs of the integral
(u,f)o := X, uvdx:
W-l N N-l
{y, v) ■■= E ViVi, (y, v] := T,yiVi, [y, v) := E Vi ^i ■
i=l i=l i=0
With these, the summation by parts formula hholds:
D8) (y, Av)^-(v,Vy] + y^v^-y^v,.
With this aim, we replace j/j Avi by its expression D7):
2/i A ti, = A [iji Vi) - vi+i V j/i+i
in the sum
W-l N-l N-l
(y, Av)- Y^ yiAvi = E AB/, t-,)- E v^+iViji+i
= 2/n ■"« - 2/i ■"i - E Vi' V 2/i' ,

Difference equations 31
where i' = 1 + 1. For j/^ = j/^ + (j/j — j/q) = j/^ + V y^ we arrive at
N
{y, Av) = y^ v^ - y^v^ - Y^ ViV yi = -(v, Ay] + y^v^- v^ y„ .
2 = 1
If j/i vanishes at the boundary grid nodes i = 0 and i = N, that is, either
j/p = 0, y„ = 0 or Uq = 0, f„ = 0, then D8) can be rewritten in the form
D9) {y,Av) = -{v,yy].
The identities obtained above are frequently encountered in difference
transformations and calculating various finite sums and series. We give below
some examples of such applications.
Example 1 It is necessary to calculate the sum 5^ = X^i-l*'2^ By
setting yi = i and Avi — T so that
^i+i = ^, + 2^' = X: 2*^ + ^0 = 2^+' -1 + ^0
and choosing v^ in such a way that v^ = 0 we are led by formula D8) to
N N N + l
1 = 1 i = l i=\
= -'e'B'-2'^+i) =-2 B^^+! -l)+2'^+iGV + l)
= GV-1) 2^^+1 + 2.
Example 2 Of interest is the sum 5^ = Yi^i ^'^'- ^^ ^^^^ '^^^'^ Vi — *>
Avi = a\ Vj, = (a' — a^) / (a — 1), f„ =0 and
5„ = ^^^^ [a''{Nia-l)-a)+a].
13. Green's difference formulae. For the simplest operator Lu = u" the
following identities are valid:
u v" dx — — I u' v' dx + u v'
i^uv" — u" v) dx = (^uv' — u' v)\

32 Preliminaries
which are called the first and the second Green formulae. Usually the
first formula can be modified into a more general form, namely
a a a
with Lv:={k v')' — qv.
Making here the mutual replacements o{u{x) and v(x) and subtracting
the resulting equation from E0), we establish the second Green formula in
a more general form
E1) / (u Lv — V Lu) dx = k (^uv' — u' v)
If, in addition, u and v vanish at the end points x = a and x = h, then all
preceding substitutions are equal to 0, thereby reducing formulae E0)-E1)
to
E2) (u, Lv)o = -{ku',v')o - {qu,v)o , {u, Lv)^ = (v, Lu)„
where as usually (w,f)o = /„ uvdx.
In particular, we might have
{u, Lu)a = -(k, {u'f)a -((/, u^)
Observe that the equality {u, Lv)^ = [v, Lu)^ means the self-adjointness
of the operator L.
In the further development of the difference analogs of formulae E0),
E1) and E2) Vi = Aii,-_i = V Ui is put together with D8):
E3) (y, AVti)=~(Vti,V2/] + 2/„Vti„-2/oVwi.
Likewise, setting Vj = aiVui one obtains instead of E3) one more useful
result
E4) {y, A{aV u)) = -(a Vu,Vy] + y^ a„ Vu„ - y^ a^Vu^ .
Applying E4) to the difference operator
E5) Auj = A(ai Vui) - diUi - hj+i (ui+i - Ui) - ai (ui - u,-_i) - diUi

Difference equations 33
yields the first Green formula
E6) {y, Au) = ~ (a Vu, Vy] - (du, y) + [ayV u)^ - % (a Vu)i ,
which can be viewed as an analog of E0). After interchanging in E6) the
positions of Ui and yi we get the equation
(u, Ky) = -{aVy, \/u]~{dy, u) + {ayVy)^ ~u^{aVy), ,
which will be subtracted at the next stage from E6). The final expression
leads to the second Green formula
E7) {y, Ku)-{u, Ay) = a{yVu-uVy)^ ~ a^ (j/o V u^ - Ug V y^).
In the particular case when Hj- = 1 and di = 0, that is, for
Ayi := AVyi = A^^i-i
Green's difference formula admits an alternative form
(y, AV u) = (w, AVy) + (y V w - w Vy)^ -{yAu-uAy)^
= (u, A\7y) + u^y^_^ -y^u^_^ + ^o 2/i -Vo^i,
which allows a more simpler writing of the ensuing formulae, When y^ — {)
and y^ — 0, the first Green formula becomes
E8) {y,Ku)^~{aVu,Vy)~{du,y),
giving for u — y
E9) {y,Ky) = ~{aVyyy)-{d,y-').
For any y and u subject to the homogeneous boundary conditions y^ —
y^ — 0 and Ug = u^ =0, the second Green formula takes a very elegant
form
F0) {Ky,u) = {y,Ku).

34 Preliminaries
1.2 SOME VARIANTS OF THE ELIMINATION METHOD
1. The flow variant of the elimination method for difference problems with
widely varying coefficients. In numerical solution of thermal-flow
hydrodynamics and magnito-hydrodynamics problems in which the coefficients of
heat conductivity and electric conductivity depend on the thermodynamic
parameters of the medium the elimination method for the corresponding
difference equations leads to unsatisfactory results. In thermal problems
adiabatic cells with infinitely large thermal-flow reveal themseves. The
nonconductive cells or ideal conductivity may appear in magnetic
problems, thus causing obstacles in connection with widely varying coefficients
and considerable accuracy losses. In mastering the difficulties involved,
the flow variant of the elimination method is aimed at solving a revised
supplementary problem relating to the heat flow.
The statement of the boundary-value problem is
F1) Hi yi_i ~ Ci yi + fli+i 2/i+i = -/;■, i = I, 2, . . . , iV - I ,
F2) y^ = H^y^+v^, ?/„ = h^ y^^, + ^^2.
with the members
Ci = cii^i -\- cii -\- di, dj > 0, 0 < flj- < oo ,
F3)
I > Xj, X2 2^ 0 , x^ -\- x^ < 2 .
Having stipulated condition F3), the computational formulae of the right
elimination with regard to problem F1)^F2) can be written as
Vi = "i+i Vi+i + A+i , i = 0, 1, 2, . . . , Af - I,
F4) fli+i -I- flj (I - Qfi) -I- di
A+i = ^^ (fli A- + /;) , i = 1, 2, ... , yV ~ I.
O-i + l
A new unknown difference function is called a flow and is defined by
F5) Wi := Hi {yi~i~ yi) ,

Some variants of the elimination method 35
making it possible to rearrange the problem statement and boundary
conditions F1)-F2) as
F6) Wi - Wi+i ~ diiji = ~fi , i = 1, 2,... , TV - i ,
F7) Cj (l - Xj) j/i + Wj = Cj z/j , a„ (i - X2) j/„ - X2^ty„ = a^v^.
Substituting y^ = tji^i + Wi^i/ui^i from F5) into the first formula F4)
yields
F8) tti+i A - ai+i) tji+i + tui+i = ui+i Pi+i .
To make our exposition more transparent, we introduce ai = Cj A — Oj)
and 7i = aiPi. With these, equation (8) admits the simplified form
F9) cti tji + Wi - ji ,
•showing the new nota,tions to be sensible ones. Having completed the
elimination of Ui from F6) and F9), we arrive at
G0) Wi = ——-- Wi+i + —-— .
a, + di at + di
In so doing Oj and 7j are recovered from the recurrence relations
/, . ^ fli+i [ai{\ - d,:) +di]
ai+i = a,+i ( 1 - Q'i+i ) = ■
ai^i + Ui {\ - ai) + di
or
a,; -I- di
ai+i
1 -I- (oi -I- di) I CH+i
a-i+i in + fi)
G1) 7i-H = o.i+1 Pi+i = "i-H ( O'i Pi+ fi)
ai+i + tti A - Qi) + di
or
Ji + fi
G2) 7i+i =
l + {ai + di) I Ci+i
By comparing the first boundary condition F7) with F9) for i = 1 we find
that
G3) Oi = Cj A - xj, 7j = Ci v^ .

36 Preliminaries
We note in passing that formulae G1)-G2) just established are very flexible
and more convenient for computational procedures in the case a,- >> 1.
The requirement ai « 1 necessitates making some modifications for
stability of this or that difference scheme. As a final result of minor changes,
the recurrence formulae have the representations
G1 ) Q-i+i
G2*) T.+i
fli+i + (o-i + di)
«i+i {n + !i)
Ci+i + (ai + di)
Under conditions F3) formulae G1) and G1*) imply that a,- > 0. This
provides support for the view that the coefficient cii j [cui + di) in G0) is
always less than 1, providing stability in the process of calculations of the
flow Wi.
For determination of j/j we rely on the appropriate formulae: for ai >>
1
G4) K = a,+i j/i+i +/?i+i = i\-'^l±l\ y^+i + 2i±i
\ «i+l / flj + l
and for a,- << 1
75) Vi = ■ —r J/i+i + ■ —r •
Ci+i + ai + c/,: a,_,.i + a^ + ck
Formulae G4)-G5) show that the eliminatron method is stable. The
values w^ and y^^ should be known before proceeding to the applications
of G0), G4) and G5). For this reason we involve here the second boundary
condition F7) and relatron F9) for i — N:
O-N  + In ^2 In «n A - ^2) - "n «n ^2
Vn = "" ~
«N A - ^2) +^2"^ a„ A - ^2) +Xjtt^
Observe that under conditions F3) the denominators in relevant expressions
are always positive.
To distinguish two essentially different approaches, the final algorithm
is of rather complicated structure and consists of the following steps:
• specification of Oj = Cj A — Xj) and 7j = a^ u^\

Some variants of the elimination method 37
• successive refinement of the coefficients for i = 1, 2,. .. , A^ — f with the
aid of the relations
[ [Qi + di) A + {ai + di) a-/j-\ a^+i > 1 ,
7i+i =
Gi + h) {I + {^i + di) a-^\ ) , ai+i > 1 ,
fli+i (Ti + fi) (flj+i + (a, + di)) , tti+i < 1 ;
• revision of proper boundary conditions
-1
Vn =
^«,^, =
 + 7n ^2 «,
1 - ^2 + ^2 "n a;;^
0-N  + Tf. ^2
"^A- ^2) + ^2Q'n
7n A - ^2) - Q-w ^2
1-^2 + ^2Q'n«;:^
a„ > 1
a„ < 1
«« > 1 ,
, a„ < 1
«nA- ^2) + ^2aN
• successive determination of solutions for i = A^ — 1, A^ — 2, . . . , 0:
ai diji- ai fi
Wi = —7- "^i+i + —, ,
ai + di ai + di
1 Vi+i + , flj+i > 1 ,
^ . ai+i J ai+i
fli+i ^ , 7i + /i ^1
^ T^ ^'+1 + \ rr • "»+i < 1 •
Cj+i + ttj + c/j Ci+i + a, + c/i
Let us stress that the computational algorithm given above is stable.
2. tl!yclic elimination method. We now focus the reader's attention on
periodic solutions to difference schemes or systems of difference schemes
being used in approximating partial and ordinary differential equations in
spherical or cylindrical coordinates. A system of equations such as
«i Vn - Ci ?/i + &i 5/2 = -/i .
G6) ai j/i_i - Ci iji + h Vi+i = -fi , i=2/i, ... , N -1,

38 Preliminaries
was quite applicable in determination of the periodic solution yi+^ = yi of
problem F1); a; yi-i — Ci yi + bi yij^y = —fi, provided that the conditions
of periodicity
and the additional conditions
G7) Hi > 0 , &i > 0 , Ci > Ci + hi
hold.
We give below without proving the algorithm of the cyclic elimination
method which will be used in the sequel;
G8) Oi+i = — , Pi+i = —— , 7j+i = ——
Ci Gj Qj Ci tt{ C^i Ci (li
J = 2, 3,,,,, TV;
Ci Ci Ci
G9) Pi = Qi+i pi+i + A'+i , Qi = "i+i Qi+i + li+i ,
i = TV-2,7V-3,,,, ,1;
Pn-1 Pn ) Qn-1 '-^n ' In 1
/?N + 1 + «„ + ! Pi
J/a
1 - "n+i Si - 7a
I N + l
(80) K=Pi + J/«9i, i = l,2,,., ,7V-1,
This algorithm is stable because the solutions of G9) are found by
the right elimination method being stable under conditions G7) with the
denominator \ — a^j^^ q^ —J^_^_^ > 0, Indeed, it follows from G7)-G8) that
cti < I, Ji > 0 and Q'2 + 7-2 < 1- Assuming Oi + 7j- < 1 we get
(81) a,+ i+ji+i - < < 1,
Ci di CXi Ci di CXl
Combination of G9) with (81) gives 5„_j < 1 and qi < 1, thereby justifying
the relation 1 - a-^^j q^ — 7„^j > 0.

Some variants of the ehmination method 39
3. Factorization method. At the final stage other ideas are connected with
the boundary-value problem
Lijk = Akijk-i-Ckyk + Bkijk + i = -Fk, k=l,2,... ,N-1,
(82)
By introducing the transition operator T: Tyk = Vk+i and the
identity operator E: E yi =■ y^ and obvious rearranging of the left-hand side of
(82) as a product
F, T - Ak) {ak T-E) j/,_i = (&, T - A,) (a, j/, - y,_i)
- bk ak + i yk + i - {Ak ak + hk) yk + A^ tjk-i ,
where
Ok + i bk = Bk , Ak Ok+bk = Ck ,
we find during the course of the elimination of bk = Ck — Ak Ok that
(83) ak+i = -:^. , k=l,2,... ,N-1.
Ck - Ak Ok
The factorized equation
{bkT - Ak ) (ak yk - tjk-i) = -Fk
can be solved using the following algorithm:
• recovery of the function /?^ from the equation
{hkT-Ak)pk=bk Pk + i - Ak pk = Fk
(84) /?, + != ^i^^±^ , ^ = 1,2, .,,,7V-1
Ck - Akak
• determination of yk by the right elimination method. The outcome of
this procedure is
(85) yk = afc+i J/j.+ i + A+i , fc = 0, 1, , . , , iV - 1 ,

40 Preliminaries
with the coefficients at and Pk emerging from (83)-(84). The complete
posing of the problem consists of (83)-(85), which are supplemented with
the initial conditions
(86) a, =x,, A-^,, y^ ~- ^^ ^" +'^^
1 - ^2 a„
With the right and left differences Avj; = Vk + i — Vk and V Vk =
Vk — ^k-i in the usual sense, the operator
Lyk = Ak Vk-i - C'k Vk + Bk Vk+i
can be factorized by letting
L=:L,L,_, L,=:bkA + jk, i^ = V + (ai. - 1).
The equality L^L2 tjk — Ltjk holds true if
■jk =hk - Ak , ak + i bk - Bk , Ak Ok + Bk a^^j = Ck ■
The same procedure works for eliminating 7jt and hk on the basis of formulae
(83)-(86).

Basic Concepts of the Theory
of Difference Schemes
In this chapter elementary examples illustrate the basic concepts of the
theory of difference schemes: approximation, stability and convergence. The
reader is already familiar with several methods available for investigating
stability and convergence such as the method of separation of variables and
the method of energy inequalities. In Section 4 difference equations are
treated as operator equations in an abstract space. These provide a means
of studying a wide range of interesting methods in a unified manner. We
begin our exposition with a discussion of examples that make it possible to
draw fairly accurate outlines of the possible theory regarding these
questions and with a listing of the basic results of the book together with a
development desired for them.
2.1 RELEVANT ELEMENTS OF FUNCTIONAL ANALYSIS
Our account of the theory of difference schemes is mostly based on
elementary notions from functional analysis. In what follows we list briefly
widespread tools adopted in the theory of linear operators which will be
used in the body of this book.
1. Linear operators. Let X and Y be normed vector spaces and 2? be a
subspace of the space X. If to each vector x E V there corresponds by an
41

42 Basic Concepts of the Theory of Difference Schemes
approved rule governing what can happen a vector y = Ax G Y, we say
that an operator A is given on V (or in X) with values in Y. The set V is
called the domain of the operator A and is denoted by V(A). The set of all
vectors of the form y ~ Ax, x G T>(A), is called the range of the operator
A and is denoted by 71(A). It is customary to use also the well-established
notation A(x) instead of Ax,
Two operators A and B are said to be equal if their domains coincide
and for all x G 7){A) = V(B) the condition Ax = Bx holds true.
An operator A is called linear if it is:
1) additive, meaning that for all x^, x^ E 7){A)
jT-(Xi -\- X2) ~~ jT-tCi -\- /i-X2 5
2) homogeneous, meaning that for all x G T^(A) and any number A
A{\x) = \ Ax .
A linear operator A i.s .said to be bounded if there is a constant M > 0
such that for any x G T^{A)
A) \\Ax\\^<M\\xl
(here || ■ || and || ■ II2 are admissible forms of the functional norms on the
spaces X and Y, respectively).
The minimal constant M satisfying condition A) is called the norm
of the operator A and is denoted by ||j4|| „^„ or simply || ^4 ||.
It follows from the definition of norm that
B) IIA11= sup 11^:^11, or ||A|| = sup ^^ .
||x-|ll=l ^Jto II 2; 111
It is worth noting here that in a finite-dimensional space any linear operator
is bounded. All of the linear bounded operators from X into Y constitute
what is called a normed vector space, since the norm || A|| of an operator
A satisfies all of the axioms of the norm:
1) II A II > 0; if II A II = 0, then || Ax \\.^ = 0 for all x and A = 0;
2)||AA|| = |A| .||A||;
3)||A + B||<||A|| + ||B||.
We will denote by X h^ X the set of linear bounded operators with the
domain coinciding with X and the range belonging to X. On the set X ^ X

Relevant elements of functional analysis 43
it is possible to introduce the operation of multiplication AB of operators
A and B: {AB)x = A{Bx). Clearly, AB is a linear bounded operator in
light of the obvious relation || AB || < || A || • || B ||. If {AB)x = {BA)x for
all a; G X, then operators A and B are said to be commuting. In that
case we will write AB = BA.
While solving equations of the form Ax = y we shall need yet the
notion of the inverse A^^. Let A be an operator from the space X into
the space Y. By definition, this means that V{A) = X and TZ{A) ~ Y.
If to each y E Y there corresponds only one element x G X, for which
Ax =: y, then this correspondence specifies an operator ^4"^, known as the
inverse for A, with the domain Y and range X. By the definition of inverse
operator, we have for any x E X and any y EY
A-\Ax) = x, A{A-'y)=y.
It is easy to show that if an operator A is linear, then so is the inverse A~ ^
(if it exists).
Lemma 1 In order that an additive operator A with V(A) = X and
TZ{A) = Y possess an inverse, it is necessary and sufEcient that Ax ~ 0
only if X = 0.
Theorem 1 Let A be a linear operator from X into Y. In order that the
inverse operator A~^ exist and be bounded, as an operator from Y into X,
it is necessary and sufficient that there is a constant 6 > 0 such that for all
xex
C) \\Ax\\^>6\\x\\^
(II ■ ||, is the norm on the space X and \\ ■ \\^ is the norm on the space Y).
Moreover, the estimate \\ A^^ \\ < 1/6 is valid.
2. Linear bounded operators in a real Hilbert space. Let H he a real
Hilbert space equipped with an inner product (x,y) and associated norm
II a; II = \/{x, x). We consider bounded linear operators defined on the space
H (T>(A) = H). Before giving further motivations, it will be convenient to
introduce several definitions. We call an operator A
(a) nonnegative if
D) {Ax, x)>Q for all x E H ;
(b) positive if
E) {Ax, x) > Q for all x E H except for x = 0;

44 Basic Concepts of the Theory of Difference Schemes
(c) lower semibounded if
F) (Ax,x) > -c^ II X IP for any x E H ,
where c^ is a positive number;
(d) positive definite if
G) (Ax,x)>S\\x\\^ for any x E H ,
where 5 is a positive number.
For an arbitrary nonnegative operator A and x E H, the number
(Ax.x) is called the energy of the operator A. Further comparison of
operators A and B will be carried out by means of the energy. If ((A —
B)x,x) > 0 for all x, we write A> B. In particular, inequalities D)~G)
can be replaced by the following operator inequalities:
(8)
A > 0 , meaning {Ax, a-') > 0 ,
A > 0 , meaning {Ax, x) > Q ,
A>—c^,E, meaning (^2;, a;) > —c^ || a; |p.
A > 6E , meaning {Ax, x) > S\\x |p ,
where E stands for the identity operator leaving a vector x unchanged:
Ex — X.
It is straightforward to verify that the relation established on the set
of linear operators {H h+ H) po.sse.sses the following properties:
(a) A > B and C > L* imply A + C >B+ D;
(b) A > 0 and A > 0 imply \A > 0;
(c) A > B and B > C imply A>C;
(d) If A > 0 and A~^ exists, then A~^ > 0.
If j4 is a linear operator defined on H, then the operator A* on H
subject to the condition {Ax,y) = {x,A*y), x,y E H, is called the adjoint
operator to A. If A is a linear bounded operator, then its own adjoint is
uniquely defined and falls within the category of linear bounded operators
with the norm || A* || = || A ||. A linear bounded operator A is called self-
adjoint if A* — A, that is, {Ax,y) = {x,Ay) for all x,y E H. If A is an

Relevant elements of functional analysis 45
arbitrary linear operator, then A*A and AA* are self-adjoint nonnegative
operators;
[A*Ax,y) = [Ax, Ay) = [x,A*Ay), [A*Az,x)= IM^^lP > 0,
{AA*x,x)^ \\A*x\\^>^.
It is worth noting the obvious relations [A*)* = A and (A*)~i = [A~^)*.
Any nonnegative operator A in a complex Hilbert space H is self-
adjoint:
if (Aa;,a;)>0 for all a; G if, then A = A*.
For real Hilbert spaces this statement fails to be true. As far as only real
Hilbert spaces are considered, we will use the operator inequalities for aon-
self-adjoint operators as well.
Theorem 2 The product AB of two commuting nonnegaiive self-adjoint
operators A and B is also a nonnegative self-adjoint operator.
An operator B is called a square root of an operator A if B"^ = A,
Theorem 3 There exists a unique nonnegative self-adjoint square root B
of any nonnegative self-adjoint operator A commuting with any operator
which commutes with A.
We denote by A^'"^ the square root of an operator A.
Let A be a positive self-adjoint linear operator. By introducing on
the space H the inner product (a;,t/)^ = {Ax,y) and the associated norm
II ^ lU — ^(x, x)^ we obtain a Hilbert space Ha, which is usually called
the energetic space Ha. It is easy to show that the inner product
ix,y)A =■ {Ax,y)
satisfies all of the axioms of the inner product:
A) {x,y)A = iy,x)A;
B) {x + y,z)A = (a;,2)yi +{y,z)A;
C) (Xx,y)A = A(x-,y),4;
D) (x, x)a > 0 for X ^ Q and (x, x) ~ 0 only for a; =: 0.
Axioms B) and C) are met by virtue of the linearity property. The
validity of D) is stipulated by the fact that the operator A is positive. The
meaning of the self-adjointness of the operator A is that we should have

46 Basic Concepts of the Theory of Difference Schemes
{^!y)A ~ (j/''^)^ '-"' i^-^iy) — i^j^y) — i-^y^^)- The axioms of the inner
product imply the Cauchy-Bunyakovskii inequality
\{^'^,y)A\<\\x\\^ .||j/IU
and the triangle inequality
ll^ + ylU<IU'IU + ll?/IL.
This profound result is covered by the following assertion.
Lemimia 2 For any positive self-adjoint operator A in a real Hilbert space
the generalized Cauchy-Bunyakovskii inequality holds:
(9) {Ax,y)^ <{Ax,x){Ay,ij).
Remiark The preceding inequality remains valid in the case when yl is a
nonnegative operator.
If yl is a self-adjoint operator for which A~^ exists, its "negative" norm
can be defined by
A0) ||^||^_. = v'(^-v,^).
In this line, we claim that
nn/^ II II l(y.^') I
10' W'pWa-^ = sup II ,| .
Indeed, we deduce from inequality (9) that
\{^,x)\ = \{A-'^,Ax)\<\\^\\^.. .||x-||^,
yielding
l(y,^)l ^ IklU-' ■ II^L I, II
On the other hand, for x = A~^(p it is plain to show that
\{^,x)\ _ (^,A-V)
II^IU v/(AA-v,^-V)
I^IL-

Relevant elements of functional analysis 47
which confirms the equivalence of A0) and A0').
In the sequel sufficient conditions for the existence of a bounded
inverse operator A"^ defined in the entire space H, V{A~^) — H, will be of
great importance for us.
We note in passing that Lemma 1 and Theorem 1 guarantee the
existence of an inverse operator defined only on TZ{A), the range of A, which
is not obliged to coincide with H. If the range of an operator A happens
to be the entire space H, TZ{A) — H, then the condition,? of Lemma 1 or
Theorem 1 ensure the existence of an operator A~^ with V[A~^) = H. In
particular, a po.sitive operator A with the range 71{A) = H possesses an
inverse A"-' with V[A~^) — H, since the condition [Ax, x) > 0 for all x ^ 0
implies that Ax 7^ 0 for a; 7^ 0 and Lemma 1 applies equally well to such a
setting.
Theorem 4 Let A be a linear bounded operator in Hilbert space H,
T>{A) = H. In order that the operator A possess an inverse operator A~^
with the domain 'D{A~'^) = H, it is necessary and sufficient the existence
of a constant 6 > 0 such that for all x (^ H the following inequalities hold:
II Ak II > 6 II a; II, ||yl*a;|| > 6 ||a;||.
Moreover, the estimate \\ A"^ \\ < 1/6 is true.
Corollary Let A be a positive definite linear bounded operator with the
domain T>{A) = H. Then there exists a bounded inverse operator A""' with
the domain 'D{A-^) =: H.
Indeed, with the relation A > 6E, 6 > 0 in view, we arrive at the
chains of the relations
\\Ax\\ ■ \\x\\>{Ax,x)>6\\x\\\
II A*x II ■ II a; II > I {A*x, x)\^\ {x,Ax) \ = {Ax, x) >6\\x\\\
thereby providing the relations || Ax || > ^ || a; || and || A*x || > ^ || x || as well
as the validity of the conditions of Theorem 4. The norm of the inverse
admits the estimate A~^ < 1/8.
Remark For the existence of an inverse A~^ in a finite-dimensional Hilbert
space it suffices to require the positiveness of the operator A, since the
condition yl > 0 implies the existence of a constant 6 > Q such that
(yla;,a;) > 8\\x\\^

48 Basic Concepts of the Theoi'y of Difference Schemes
for all X. As a matter of fact,
{Ax, x) = (AqX, x) ,
where Ag = {A + A*)/2 is a self-adjoint operator. Hence, {Ax, x) > 6\\ x |p,
where 6 is the smallest eigenvalue of Ag. The number 6 cannot vanish due
to the positiveness of the operator A.
For the sake of simplicity we will assume in the body of this book that
H is a finite-dimensional space.
Recall that the norm of an operator A is defined by
\\A\\= sup II At II .
II ,^ 11=1
With this, for any self-adjoint operator A the following relations occur:
A1) \\A\\= sup \{Ax,x)\ = \\A\\= sup ^-^^.
IklNi IUIIt^o l|a;||
Lemma 3 If S ~ S* is a linear bounded operator and n is a positive
integer, then
\\s"\\^\\s\\r
Proof We proceed to prove thi.s assertion by induction on n. Let n = 2,
Then
115^11= sup \{S'x,x)\= sup \\Sx\\' = \\S\\\
11-^11=1 11^1=1
meaning || .^^ || = || 5' |p. A.ssuming that relation A2) is valid with n = k — l
and n = k, we are going to show that it continues to hold for n = k + I,
k > I, Indeed,
115^*^11= sup \{S'^''x,x)\= sup \{S''+^x,S''-^x)\
Ik 11=1 II .^11=1
< sup II S^+^x II . II S''-^x II < II 5*+^ II ■ II S''-' II
||.^||=i
or, what amounts to the same,
\\s'+'\\ ■\\s'-'\\>\\s"^\\ = \\s'\\' = \\s\\''.
This provides support for the view that || 5*''"'""' || > || 5 ||*^''"S since
115*^^^11 = ||5||*^-i.
On the other hand,
\\s'+'\\<\\s\\'+\
so that
||5^+i|| = ||5||^'+i.
As far as A2) holds for n = I and n = 2, it will be true for any n.

Relevant elements of functional analysis 49
Lemma 4 If A is a self-adjoint positive bounded operator, then the
estimate is valid:
A3) \\Ay\\'<\\A\\{Ay,y).
Since A* = A > 0, there always exists the operator A^^^. By merely
setting V = A^^^^ we obtain
{Ay, Ay) = {Av, v)<\\A\\.\\v |p = \\A\\ {Ay, y) .
3. Linear operators in finite-dimensional spaces. It is supposed that an
n-dimensional vector space R^ is equipped with an inner product ( , ) and
associated norm ||a;|| = \J{x,x). By the definition of finite-dimensional
space, any vector x 6 ii;„ can uniquely be represented as a linear
combination X = Cj £j -|- • • • c„ £„ of linearly independent vectors £j , . .. , £„, which
constitute a basis for the space Rn ■ The numbers Cj, are called the
coordinates of the vector x. One can always choose as a basis an orthogonal and
normed system of vectors ^j, . . . , £„:
r 0, ii^k,
by means of which it is possible to write Cj, = (x, £j,).
Let yl be a linear operator in the space R^. Any operator A in the basis
£j, . . . , £„ can be put in correspondence with an n x n matrix 21 = (a^j,),
whose element a^j. is the ith component of the vector A^-j,. Conversely, any
matrix 21 = (ajj.), i,k = \, . . . ,n, specifies a linear operator.
The matrix of a self-adjoint operator in any orthonormal basis is a
symmetric matrix.
Let us dwell on the properties of eigenvalues and eigenvectors of a
linear self-adjoint operator A. A number A such that there exists a vector
£ ^ 0 with Ai^ = A£ is called an eigenvalue of the operator A. This vector
£ is called an eigenvector corresponding to the given eigenvalue A,
L A self-adjoint operator A in the space R^ possesses n mutually
orthogonal eigenvectors £i, . . . ,£„. We assume that all the £j,'s are
normalized, that is, ||£j.|| = 1 for k = \, . ., ,n. Then (£^, £j,) = b^-^.. The
corresponding eigenvalues are ordered with respect to absolute values:
I A J < I A, I < ... < I A J .
2. If a linear operator A given on R^ possesses n mutually orthogonal
eigenvalues, then yl is a self-adjoint operator: A = A*.

50 Basic Concepts of the Theory of Difference Schemes
3. If yl* = yl > 0, then all the eigenvalues of A are nonnegative.
4. Any vector x 6 /?„ is representable by the eigenvectors of any
operator A — A*:
n n
^='Eckik> cj; = (x,£fc) and \\^\\~='Ecl-
k=:l fc = l
5. Let A* — A > 0 {A is self-adjoint and nonnegative). Then
AJ|x|p<(Ax,x)<AJ|x||2
a,nd
AJ|x|| < l|ylx|| < A„||x|| for all xeH
where Aj > 0 and A„ > 0 are, respectively, the smallest and greatest
eigenvalues of A. The norm of a self-adjoint nonnegative operator in the
space R„ is equal to its greatest eigenvalue: || j4 || = \„.
6. If self-adjoint operators A and B are commuting {AB = BA), then
they possess a common system of eigenvectors.
7. Let self-adjoint operators A and B be commuting {AB = BA).
Then the operator AB possesses the same system of eigenvectors as the
operators A and B and \\^g = -^a '^b ' ^ ~ Ij 2,... ,n, where A), , A^ ,
and Xy^g are the ktli eigenvalues of A, B and AB = BA, respectively. By
the same token,
^A+B - ^A + ^B ■
2.2 DIFFERENCE APPROXIMATION OF ELEMENTARY
DIFFERENTIAL OPERATORS
1. Grids and grid functions. The composition of a difference scheme
approximating a differential equation of interest amouts to performing the
following operations:
• to substitute the domain of discrete variation of an argument for
the domain of continuous variation;
• to replace a differential operator by some difference operator and
impose difference analogs of boundary conditions and initial data,
Following these procedures, we are led to a system of algebraic equations,
thereby reducing numerical solution of an initial (linear) differential
equation to solving an algebraic system.

Difference approximation of elementary differential operators 51
We would like to discuss the questions raised above in more detail.
Obviously, in numerical solution of mathematical problems it is unrealistic
to reproduce a difference solution for all the values of the argument varying
in a certain domain of a prescribed Euclidean space. The traditional way
of covering this is to select some finite set of points in this domain and look
for an approximate solution only at those points. Any such set of points is
called a grid and the isolated points are termed the grid nodes.
Any such function defined at nodal points is called a grid function.
Thus, the first step in this direction is to replace the domain of continuous
variation of an argument by a grid, that is, by the domain of discrete
variation of the same argument. In other words, we have approximated the
space of solutions to a differential equation by the space of grid functions.
For this reason the properties of a difference solution and, in particular, its
proximity to an exact solution depend essentially on a proper choice of the
grid. It seems worthwhile giving several simple grids to help motivate what
is done.
Example 1. An equidistant grid on a segment. The segment [0,1]
of unit length is splitted into A^ equal intervals. The spacing between the
adjacent nodes x^ — x^_^ = /i = i is termed a grid step or simply step.
The splitting points x^ = ih constitute what is called the set of grid nodes
'^h — {^i = ih, i = I,... , N — 1} and generate one possible grid on this
segment (see Fig. 1):
x„ = 0 X, x^ ... X, ... a;„ = 1 x
Figure 1.
When the boundary points x^ = 0 and Xpf = 1 are put together with
the grid cj^j, it will be denoted by w^j = {k; = I'/i, i = 0,1,... , A^ — 1, A^}.
On the segment [0,1] we are working with a new function yj^ixj) of
the discrete argument instead of a given function y{x) of the continuous
argument. The values of such a function are calculated at the grid nodes
x'j and the function itself depends on the step h as on the parameter.
Example 2. An equidistant grid in a plane. We now consider a
set of functions u{x,t) of two arguments in the rectangle 2? = {0 < a; <
1, 0 < i < T} and split up the segment [0,1] on the a;-axis and the segment
[O,^] on the i-axis into N[ and N2 parts with steps h = 1/A^i and r =

52
Basic Concepts of the Theory of Difference Schemes
T/N2, respectively. After that, we draw the straight lines parallel to the
appropriate axes through the splitting points. The points of the intersection
of those lines are adopted, as usual, as the nodes (x^^tj), which constitute
a widespread grid
with steps h and r along the directions Ox and Ot, respectively (see Fig. 2).
t
T
JT
T
0
h
X, =
iXi,tj)
zlh
X
Figure 2.
The nodes lying on one and the same straight line (horizontal or
vertical), the distance between which is equal to the grid step (/j or r), are
called adjacent grid nodes.
Example 3. A non-equidistant grid on a segment. The next example
IS devoted to the segment 0 < ,t < 1 with A^ subintervals without concern
for how the points 0 < x^ < x^ < . . . < Xj^__i < 1 will be chosen. The
nodes {x^, i = 0, . . . , N, x^ = 0, Xj^j = 1} constitute what is called a non-
equidistant giid w^[0, 1]. The distance between the adjacent nodes, being
a grid step, equals h^ = x^ — x^^^ and depends on the subscript i. Any
distance of this type falls within the category of grid functions. The steps
should satisfy the normalization condition X^^-j /i,- = 1 .

Difference approximation of elementary differential operators 53
Example 4. A grid in a two-dimensional domain. Our final example
is connected with a complex domain G in the plane x = (x^,x^) of rather
complicated configuration with the boundary F. One way of proceeding is to
subdivide into sets of rectangles by equally spaced grid lines parallel to Ox^
defined by x^ ' = ij /jj, ij = 0, ±1, ±2,.. . , /ij > 0, and equally spaced grid
lines parallel to Ox^ defined by x^ ^ = *2^2i ^i — 0,±1,±2,... ,/ij > 0,
as shown in Fig. 3. As a final result we obtain the grid with the nodes
(ij/ij, ij/I2), ?'i, ij = 0,±1,±2,... in the pla.ne (x^,x^. It seems clear
that the lattice so constructed is equidistant in each of the directions Ox^
and Ox^. We are only interested in the representative grid nodes from the
domain G ~ G + F, including the boundary F. The nodes (I'j /ij, ij /12)
inside G are called inner nodes and the notation w^, is used for the whole
set of such nodal points. The points of the intersection of the straight lines
x^ = ij /ij and x)^ '^' = ^2 ^2 > H > ^'2 ~ Oj ilj i2,... 0, with the boundary
F are known as boundary nodes. The set of all boundary nodes is denoted
by 7;j. In Fig, 3 the boundary and inner nodes are quoted with the marks
* and o, respectively,
As can readily be observed, there are bounda,ry nodes with the
distance to the nearest inner nodes smaller than /ij or hr,. In spite of the
obvious fact that the grid in the plane is equidistant in x.^ and x, both, the
grid iJj^ — ijj-j^ + 7;j in the domain G is non-equidistant near the boundary.
This case will be the subject of special investigations in Chapter 4.
The approach we have described above depends for its success on
replacing the domain G of the argument x by the grid cD^j, that is, by a
finite-dimensional set of points x^ belonging to the domain G, The grid
functions y(x{) will be quite applicable in place of the functions v.(x) of the
continuous argument a; £ G with x^ being a node of the grid Gj^ = {^i}'
Also, we may attempt the grid function y(xi^ in vector form. This is due
to the fact that by enumerating the nodes in some order x^^x^^... xj^
the values of the grid function at those nodes arrange themselves as the
components of the vector
^ = (j/i >•••,?/,;,••• J/w).
If the domain G of forming the grid is finite, then so is the dimension
A^ of the vector Y. In the case of an infinite domain G the grid consists of
an infinite number of nodes and thereby Y becomes a,n infinite-dimensional
vector. Within the framework of this book the sets of grids w^^ depending
on the step h as on the parameter will be given special investigation, so
that the relevant grid functions yii{x) will depend on the parameter h or on

54
Basic Concepts of the Theory of Difference Schemes
Figure 3.
the number A^ in the case of an equidistant grid. In dealing with various
non-equidistant grids w;j we mean by step h the vector h = {h^, /i,,.. . , hpj)
with components /ij, /ij,... , h,^.
The same observation remains valid with multidimensional domains
G. Here x = (x^ , ■ ■ ■ , x^) and, on the same grounds, h — (/ij, h^,... , h^)
"p-
if the grid oj^ is equidistant in each of the arguments a'j,.
Throughout the entire chapter, the functions u{x) of the continuous
argument a; G G are the elements of some functional space Hq. The space
Hh comprises all of the grid functions Vhix), providing a possibility to
replace within the framework of the finite difference method the space Hq
by the space Hh of grid functions yh{x). Recall that although the fixed
notation || • || is usually adopted, there is a wide variety of possible choices
of the functional form of || • ||.
In a common setting the set {Hh] of spaces of grid functions depending
on the parameter h corresponds to the set of grids {oJh}, making it possible
to introduce in the vector space Hh the norm || • ||, , which is a grid analog
of the norm || • |L of the initial space Hq. We give below two norms in the

Difference approximation of elementary differential operators 55
space Hh for the grids Oj^ = {x^ = ih} on the segment 0 < x- < 1. These
norms are widely used in many applications to numerical analysis (here the
subscript h ol y^ is omitted):
(i) the grid analog of the norm on the space C:
||j/||c = max |y(a;)| or || y ||c = max | j/J ;
xiiOh Q<i<N
B) the grid analog of the norm on the space L'j'-
\\y\\= ( E y]h\ or ||j/|| =
In what follows we will use, as a rule, several norms associated with
inner products in the space Hh (the grid analogs of the L2 and H^j^-norms
are available in Chapter 1).
Given a solution u[x) of the original continuous problem, u G TJq, and
a solution y^ of the appropriate approximate (difference) problem, y^ G Hh,
the main goal of the possible theory of approximate methods is the accurate
account of the proximity between j/;j and u. It is worth noting here that in
a common setting the vectors yh and u belong to different spaces. In this
context, two interesting possibilities reduce to the following ones:
• the grid function yh defined at the nodes W;j(G') should be extended
(for instance, via the linear interpolation) to all of the remaining
points X of the domain G. As a final result we get a function y{x^ h)
of the continuous argument x ^ G for which the difference y(x', h) —
u{x) belongs to the space Hq. The proximity of yh to u is well-
characterized by the number || y{x, h) — u{x) ||q, where || ■ ||q is the
norm of the spa.ce Hq]
• the space Hq is mapped onto the spa.ce Hh, thereby putting every
function u{x) £ Hq in correspondence with a grid function Uh{x),
X ^ ojh, so that Uh ~ Vh u G Hh, where Vh is a linear operator
from Hq into Hh- It is possible to establish this correspondence in
a number of different ways by approval of different operators Vh-
If u{x) is a continuous function, we might accept tih{^) = u{x) for
X G W;j. Sometimes Uh{Xi) is determined at a node x^ G ujh as the
integral mean value of u(x) over some neighborhood of this node (for
instance, of diameter 0(h)). In the sequel we will aJways assume
that u{x) is a continuous function and keep Uh{xi) = u(a;J for all
Xf G LOh unless otherwise is explicitly stated.

56 Basic Concepts of the Theory of Difference Schemes
For a difference function u^, we turn to the difference y^ — u^, which
gives a vector of the space Hh ■ The nearness of j/;, to u is well-characterized
by the number || j/^j — tt^j ||, , where || ■ ||, is the norm of the space Hh- It
seems natural to require that the norm || • ||, should approximate the norm
II • IIq in the following sense: for any vector u £ Hq
li_m \\uh\\h = \\u\\Q-
This condition is known as the condition of concordance of the norms
on the spaces Hh and Hq.
Under the second approach mentioned above we proceed to the
accurate account of the errors of difference methods in the space of grid
functions. In the most cases the spaces involved appear to be finite-dimensional.
As we will see later, it is possible to present the principal aspects of the
theory of difference schemes with further treatment of Hh as an abstract
vector space of arbitrary dimension.
After preliminary discussions of the simplest examples illustrating
some ways of producing grids and, thereby, of forming the spaces Hh of
grid functions we concentrate primarily on the problem of the difference
approximation of differential operators.
2. The difference approximation of elementary differential operators. Let
a linear operator L assign the values to a function v = v{x). By replacing
the derivatives built into Lv by their difference counterparts we derive the
difference expression Lh f/j, which is a linear combination of the values of
the grid function Vh on some set of grid nodes known as a pattern:
LhVh{x)= J2 M^'Ovhi^)
{LhVh)i= E ^h{xi,x-)vh{Xj),
Xj €Patt(x^)
where Ah{x,^) are the coefficients, /i is a grid spacing and Patt[x) is a
pattern at a point x. Any replacement of Lv by Lh Vh is called the
approximation of a differential operator by a difference operator or
the difference approximation of an operator L.
The way this approach is used in practice is connected with
preliminary studies of difference approximations of the operator L locally, that is,
at an arbitrary fixed point of the space. If v{x) is a continuous function,
then Vh{x) = v{x). Before giving further constructions of a difference
approximation of the operator L, it is necessary to choose a pattern of proper

Difference approximation of elementary differential operators 57
form, that is, to compose the set of nodes adjacent to x at which the values
of the grid function v{x) are aimed at approximating the operator L.
In this section we consider several examples of difference
approximations for elementary differential operators.
dv
Example 1 Lv = ~r~- Let us fix some point x on the Ox-axis by capturing
,dx
the neighboring points x — h and x + h, where h > 0, and try to approximate
Lv. Also, it will be sensible to introduce the following expressions:
A) Ltv.'-i^^±^p^^v^,
B) L-..^M^^fc^..,.
Expressions A) and B) are called the right difference derivative
and the left difference derivative and are denoted by v^. and v^,
respectively. The difference expressions Lf^ and L^ are defined at two points. In
other words, these are based on the two-point patterns x^ x + h and x — h,
X. Moreover, any linear combination of expressions A)"B) such as
C) Li"K = <Tv, + {l~a)v,,
where cr is a real number, can be adopted as the difference approximation
of the derivative dv/dx. In particular, for a = 0.5 we get the central
two-sided difference derivative
1 v(x + h) - v(x - h)
D) v° = 7. (v,. + vJ = ~ '~~~^ .
It turns out that there is an uncountable set of difference expressions
approximating Lv = v' and this is something one might expect. The following
question is of significant importance: what is the error of one or another
difference approximation and how does the difference ip{x) = Lh v[x)— Lv[x)
behave at a point a; as /i —>■ 0? The quantity il}{x) = Lh v[x) — Lv[x) refers
to the error of the difference approximation to Lv at a point x. We
next develop v{x) in the series by Taylor's formula
v{x ±h) = v{x) ± h v'{x) + — v"{x) + Oih"^),
assuming v[x) to be a sufficiently smooth function in some neighborhood
[x — h^jX -\- ho) of the point x and h < h^, where the number /Iq is kept

58 Basic Concepts of the Theory of Difference Schemes
fixed. Substituting the preceding series into A), B), and D) yields
V{x + h) — v{x) I ^ III \ rM-L'i\
v^
v'{x) + -v"{x) + 0{h'),
h "■ ' ' 2
/.-x v{x)-v{x — K) .. , h 11, , „,,9.
E) % = ^ ' ^ '- = v'{x)--v"{x) + 0{h'),
v[x + h) — v{x — h) ,. . r^/u2\
2h
thereby justifying that
v'{x) + 0{h^),
■0 = v^ - v'{x) = 0{h) ,
ip = v,,- v'{x) = 0{h) ,
yj = Vo -v'ix) = 0(/j2).
Let ]/ be a class of sufficiently smooth functions v ^ V defined in a
neighborhood PaU[x,ho) of a point x containing for h < h^ the pattern
Patt(x, h) of a difference operator Lh- We say that Lh approximates the
differential operator L with order m > 0 at a point x if
iP{x) = Lhv{x)-Lv{x) = 0{N").
Thus, the right and left difference derivatives generate approximations of
order 1 to Lv ~ v', while the central difference derivative approximates to
the second order the same.
(fv
Example 2 Lv = v" = -r-;j. In order to construct a difference approxi-
Illation of the second derivative, it is necessary to rely on the three-point
pattern {x — h, x, x + h). In that ca.se we have
v(x + h)-2v(x) + v(x-h)
F) Lh V = p .
Observe that the right difference derivative at a point x is identical
with the left difference derivative at the point x + /i, that is, the relation
v^.[x) = Vg[x + h) occurs, permitting us to rewrite F) a.s
G)
vJx) - vJx) If/ , \ / xl / X

Difference approximation of elementary differential operators 59
With the aid of the expansion of the function v{x) in Taylor's series it is
not difficult to show that the order of approximation is equal to 2, meaning
Vg^. — v"[x) = 0[h'^) by virtue of the expansion
(8) y^^:=y"+^^y(^) + 0{h^).
Example 3 Lv — v'^'^'>. We now deal with the five-point pattern consisting
of the points {x — 2h, x — h,x,x + h,x + 2h) and accept LhV = v^^^^. By
applying formula F) to Vg^ we derive the expression for v^^g^, which will
be needed in the sequel:
LhV = Vg^g^ = ^ [vg^{x + h)- 2vs^{x) + Vf^^{x-h)\
= —^ [v{x + 2h) - 4 v{x -I- /i) -I- 6 v{x) - 4 v{x - h) + v{x - 2/i)] .
It is straightforward to verify that Lh provides an approximation of order
2 to L, so that v^^.^.^ — i^'-'*) = ^i;'-'^) -|- 0[h'^). Indeed, this can be done
using the expansion in Taylor's series
v(x + akh) =v(x)+y ; -^-FO/i*, (T = ±1,
^—' s ! dx^
s = l
for k ■= 1/2 and exploiting the obvious fact that the sum v[x+kh)+v[x — kh)
contains only even powers.
The expansion of the approximation error ip = Lh v — Lv in powers
of h is aimed at achieving the order of approximation as high as possible.
Indeed, we might have
V,. -^" = Y2 "^'^ + ^(^'^ = V2 "^^'^ + ^(^'^ •
whence it follows that Lv — v" is approximated to fourth order by the
operator
^h ^ — ^Sx ■" Y^ ^3-xSx
on the pattern [x — 2/i, x — h, a;, x + h^x + 2h),
In principle one can continue further the process of raising the order
of approximation further and achieve any order in the class of sufficiently
smooth functions v ^V. During this process the pattern, that is, the total

60 Basic Concepts of the Theory of Difference Schemes
number of the nodes will increase. However, the way of achieving a higher-
order difference approximation we have described above is of little practical
significance, since the quality of resulting operators becomes worse in the
sense of computational volume, the conditions of existence for the inverse
operator, stability, etc. Later in this chapter we will survey some devices
that can be used to obtain higher orders.
In the sequel an auxiliary lemma is needed.
Lemma The formulae are valid:
v{x + h) — 2 v{x) + v{x — h)
T2
(9) ^^... = ^^^""^ -\->-"^- ->^v"{0. ^ = ^+eh, \e\<i,
ifv eCB)[a;-/i,a; + /i],
A0) y^^=y"{^)+'l-yW{^)^ ^ = ^ + ^^/, _ | ^ J < 1 ,
if i; e C'-'*)[x' — h^x + h]. Here C^^^a^b] stands for the class of functions
with the kth continuous derivative on the segment a < x < b.
Proof We have occasion to use Taylor's formula with the remainder term
in integral form
A1) v{x) = v{a) + {x^a)v'{a)+ ■■■ + ("^ ~ ")' v^.'') + R^^^{x) ,
rl
where
A2) R^^,(x) = ^J (x - ^Y v'^^+'^iO d^
a
1
= ^"^ ~^?"^' / A - ■')" y^"^'^ («+ s (* -«)) ds,
Applying Lagrange's formula to the last integral reveals the remainder term
Rr+i{x) to be

Difference approximation of elementary differential operators 61
where £, is the mean value of x on the segment [a, e]:
1
e = a + e(x-a), 0<^<1, (l-sYds=^—.
' J ^ r+1
0
Upon substituting x -\- h for x and x for a. into formula A1) we find that
1
A3) v{x + /i) = v{x) + h v'{x) + h^ I (i^-s) v"{x + sh) ds ,
0
A4) v{x + h) = vix) + h v'{x) + — v"{x) + — v"'{x)
1
+ — I {1 - sf v'^'^\x + sh) ds
0
for r = 1 and r — 3, respectively. Replacing here h by —h and then s by
— s gives for later use the new formulae
0
A5) v{x -h) = v{x)-hv'{x) + h^ I {l + s)v"{x + sh) ds,
-1
A6) i;(a; - h) = i;(a;) - /jw'(x') + — v"{x) - — v'"(x)
0
+ ^ /(l + sKt;D)(^ + s/l)rfs.
Adding A3) and A5), placing the term 2v{x) on the left and then dividing
the resulting expression by /i^, we finally get
1
vix + h) + v{x - h) - 2 vix) f , , ,,, ,, ,
Vs^ = -^ ~i ~ — = / 92is)v"{x + sh) ds,
-1
where
r 1 + s for - 1 < s < 0 ,
^^^ ~ I 1 -s for 0 < s < 1,

62 Basic Concepts of the Theory of Difference Schemes
Since fi'2(s) > 0, Lagrange's formula provides support for the representation
1
v^^ = v"{x + 9h) I g^{s) ds = v"{x + eh) = v"{Q, -1<^<1,
-1
where cj is the middle point of the segment [x — h, x-\-h\. The second formula
A0) can be derived in the same manner as before.
Below we follow these procedures in a step-by-step fashion: collect A4)
and A6), carry 2v{x) over to the left-hand side and divide the resulting
expression by /i^. Simple algebra gives
1
,2
v"{x)+^ J g,{s)v^''\x + sh)ds.
where
r(l + sK for -l<s<0, [ , , , 1
^''(^^=1A-.)^ for 0<.<1, y ^*(^)^^=2-
-1
Because g^{s) > 0 and v^'^"){x) is continuous, applying Lagrange's formula
yields
v,^=v"{x) + ^^v(''\x + 9h), \e\<l.
Remark 1 By exactly the same reasoning as before, we can establish
A7) .,^=."(x) + ^.W(x) + ^.(«)@, ^ = x + 0h, \0\<l,
if v{x) belongs to the class C^^")[x — h^x + h].
Remark 2 A similar result is still valid for the derivative
A8) Vg^g^ = -^ [v{x + 2h) -4v{x + h) + &v{x)
-Av{x-h) + v{x - 2/i)] = i;(^)@ ,
where ^ = x + 9h, \9\ < 2, is the middle point of the segment [a; — 2/i, x + 2h]
and V G C^^'[x — 2/i, x + 2h], This expression will be proved if we succeed
in showing that
2
_ 1
" / g{s) v^'^^x + sh) ds,
6
-2

Difference approximation of elementary differential operators 63
9is) = <
' 8A +0.5 s) for - 2 < s < -1
8(l + 0.5s)^-4(l + sK for -l<s<0,
8(l-0.5s)^-4(l-sK for 0<s<l,
t 8(l-0.5s)^ for 1 < s < 2.
The reader is invited to do it on his/her own,
dv d'^v
Example 4 Lv = -7- -tt^, v = ^(^,^). Let (x t) be a fixed point in
ot ox^
the plane (x/i) and let /i > 0 and r > 0 be two arbitrary numbers taken
as steps in the sequel, A difference approximation LhT of the operator L is
connected with a proper choice of the pattern.
We begin by placing approximations of the simplest type for which
the pattern consists of the four points (Fig, 4,a), so that Lhr is certainly
expressed by
A9) ^i,)^^v{x^ + r)^-v{x^t)
v{x + h,t) - 2v{x^t) + v{x - h,t)
A suitably chosen symbolism may symplify the form of writing various
difference expressions and makes our exposition more transparent. This is
acceptable if we agree to consider
V = v[x,t) , V = v{x^t + t) , ii ■= v{x^t — t) .
Within these notations, the difference derivative with respect to t becomes
v[x,t + t) — v{x^t) V — V
B0)
r
By virtue of relations G) and B0) we write down A9) in the form
A9') l["^'y = vt - y,, ■
In the preceding constructions of L\J we have taken the value of Vg^, at the
moment t, that is, on the lower layer. On the pattern depicted in Fig. 4.b it

64
Basic Concepts of the Theory of Difference Schemes
{X,t+T)
{x - h,t + t) {x,t + T) {x + h,t +
{x - h,t) (x^t) {x + h,t)
{x,t)
b
{x - h,t+ t) {x,t+T) {x + h,t+ t)
II II II
II <■ <i
{x-h,t) {x,t) {x + h,t)
Figure 4.
is reasonable to place v^.^ at the moment t + t^ that is, on the upper layer,
so that
B1)
r(l)
By a linear combination of A9') and B1) we get a one-parameter family
of difference operators on the six-point pattern shown in Fig. 4,c for cr ^ 0
and a ^ I:
B2) 4'^« = ^'^*-('^%.. + (l-'^)%.).
In the estimation of the order of approximation we substitute into the

Difference approximation of elementary differential operators 65
formulae for L\^ v, L)J v and L\J v the following expressions;
+ 0(r2)
dv(xf) T d'^v(xt)
y, — ^ ' ' A ^^ '
* St 2 dt'^
"^XX
9.(x-,t + r/2) ,
dx' ^12 dx^ -^^{n )
= 5^;2 2 5i^5^ + ^('^ +"^'
_g^.(x,t + r/2) r a^.(x,t + r/2) _
"^-- 5^^ +2 5^?^5^ + ^('^ +"^-
The outcomes of such manipulations are:
1) 4> = ^-%^ + o(,.' + .)
meaning, ^(°) = l|°^^ i; - Lv{x,t) = 0{h'^ + r);
2) 4V. = ^^^(^-^^!^%^
= Lv{x,t + T) = Oih"^ + t),
meaning, ^^^^ = l[^^ v - Lv{x,t + r) = 0{h^ + r);
3) 4-% = M^i4±iZ^ - ^!!%1±^
= Li;(x-,< + r/2)=:0(/i2 + r2),
meaning, •0^°-^) = 4°^^^ i' " Lv{x,t + r/2) = 0(/i^ + r^) ,
All this enables us to conclude that the operator Lj'^ provides the
approximation of order 2 with respect to h for any cr, the approximation of order
1 with respect to r for cr = 0 and cr = 1, and the approximation of order 2
with respect to r for cr = 0.5,

66 Basic Concepts of the Theory of Difference Schemes
Example 5 Lv= -^r-:^— -;:-^ , In that case the values of a grid function at
ot-' ox-'
three instants of time t — T^t and t + t are aimed at selecting the difference
operator Lhr- The five-point pattern plotted in Fig. 5,a,b,c is the best
possible for our purposes.
{x - h^t + t) (x^t + r) {x + h^t + t) (x^t + r)
• a • a
{X,t)
{x,t — t)
ix,t)
[x — h,t — t) (xJ—t) (x + h,t — t)
{x -h,t)
(x^t + T)
{x - h^t + t) (x^t + r) {x + h^t + t)
(x^t) {x + h,t)
<1
(x-
A
•
1
-hj)
1
<
1 (
{x,t)
< <
1 •
{x + h,t)
{x,t -t)
[x — h^t — T] {x,t — T) [x + h^t — r)
Figure 5.
One of the possible approximations on the pattern 5.c with using the
value Vgj. on the middle layer t is of the form
B3)
where
Vui^J)
v(x^t + t) — 2v{x^t) + v{x^t — t)

Difference approximation of elementary differential operators
67
In a similar manner one can write down on the pattern 5.a the operator
B4) LhrV = Vtt~Vg,^.
On the other hand, a two-parameter family of difference operators
B5) -^i?'"''' V = '"it - (c^i %:t- + A - cTj - crj i;j;^, + a,_ Vg^)
is based on the nine-point pattern 5.d, whence B3) follows for a^ = a^ = 0
and B4) follows for cr, = 0 and a^ — 1. We have occasion to use the
asymptotic formulae
dp
+ 0(r'),
d^v{x,t)
0{h')
thereby justifying that operator B3) provides an approximation of 0{h''^ -\-
r^). Operator B5) has the same order of approximation for <j^ — <j^ = cr,
where <j is an arbitrary number.
We note in passing that a key role of the parameters cTj and a^ just
as the parameter a in the previous example is connected not only with the
approximation order, but also with stability of the appropriate difference
scheme. This important property will appear in subsequent discussions in
Chapter 5, Section 1,
Example G. L v = v". An irregular pattern (a non-equidistant grid).
Granted two numbers /j_ > 0 and /i_|. > 0, other ideas are connected with
the three-point pattern [x — /j_,x',a; -|- hj^). When /«_ ^ /j_|. , any such
pattern is said to be irregular, since the grid including this pattern turns
out to be non-equidistant. Being concerned with new members
){x) — v{x — h-)
hi
v{x + /i_|_) — v{x) /i_
we refer to the operator L/j with the values
v{x + /i_|.) — v{x) v{x) — v{x — h-)
B6)
Lh V
For h- = h^ = h the preceding is identical with expression G) (see Example
2). Plain calculations of the local approximation error at a point x show
that
i){x) - Lh v{x) - Lv{x).

68 Basic Concepts of the Theory of Difference Schemes
Having stipulated the condition /ij_ < 2?i, the well-established expansions
of a sufficiently smooth function v[x) in a neighborhood of the node x such
as
v(x + h+) = v{x) + h+ v'{x) + -^ v"{x) + -^ v"'{x) + 0{hX) ,
v{x - /j_) = v{x) - /i_ v'{x) + ~ v"{x) - ~=- v"'{x) + 0{ht)
2 ' ' 6
lead to the following ones:
., = v'{x) + ^ v"{x) + ^ v"'{x) + 0{hl),
L, V = '-^^ = v"{x) + ^^i—^ v"'{x) + 0{h^).
With these relations in view, we derive the usefull expressions for ip{x)
B7) '^^LhV-Lv= ^^Z.^" v'" + 0{n'^) = 0{n).
o
Thus, operator B6) provides the local approximation of order 1 on any
irregular pattern with /j_ ^ h^,
3. The error of approximation on a grid. So far we have considered the
local difference approximation; meaning the approximation at a point. Just
in this sense we spoke about the order of approximation in the preceding
section. Usually some estimates of the difference approximation order on
the whole grid are needed in various constructions.
In preparation for this, let oj^ be a grid in a domain G of the Euclidean
space {x = (Xj,. . . , a; )}, TJ/j be a vector space of grid functions defined on
the grid w^^ and let the space Hq comprise all of the smooth functions v[x)^
whose norms are defined by || • ||q and || • ||. , respectively. In the sequel
we take for granted the following assumptions:
A) there exists an operator Vh such that 'Pf^u = u^ ^ H^ for any
B) the norms || • ||q and || • ||^ are concordant, that is,
|lim^ IIP.mII, = |hllo,
where | h \ stands for the norm of a vector h.

Difference approximation of elementary differential operators 69
The next step is to introduce an operator L in the space Hq and an
operator L/j carrying a grid function v^^ into a grid function Lk Vj^ on the
grid u)^ {Lh- Hh ^ Hh).
A grid function
where V}^ = VhV, {Lv)i^ = Th{Lv) and v is an arbitrary function (vector,
element) of the space TJq, is called the error of approximation of the
operator L by the difference operator Lji,
If ll'i^/ill; —* 0 as /i —>■ 0, we say that the difference operator Lh
approximates the differential operator L.
Likewise, the difference operator Lh approximates the differential
operator L with order m > 0 if
B8) \\^K\\h = \\LhVh- {Lv)h\\h = 0{\hr)
or II Lh 'Vh~{Lv)f^ ||. < M I /i I™, where M is a positive constant independent
of|/i|.
Remark 1 We give below several examples of projectors Vh onto the set
of grid functions;
A) for a continuous function v{x)^ Vh = Vh v{x) = v{x)^ x G w^^;
B)
x + h 1
^h = '^hV = JT / v{t) (it = ~ / v{x + Sh) ds
x-h -I
if v[x) is a summable function, etc.
Remark 2 The length \h\= (/j^+.. •+/i2)i/2 of a vector/i = (/ij,... ,hp)
with components /ij, ,. ■ , h may be taken as the quantity involved in the
above definitions. Observe that approximations with respect to /i„, a =
1,2,,,, ,p, may be different in order. If so, instead of B8) we might have
\\LhVh-{Lv),\\^<M E /i™», m„>0.
a = l
Choosing among the numbers nij,,,, , m. the minimal one and denoting it
by m, we get estimate B8).

70 Basic Concepts of the Theory of Difference Schemes
Remark 3 If the grid u.-;j is non-equidistant, that is, h = (/ij , . .. , /j^y),
where A^ is the total number of nodes, it is sensible to deal with h =
maxi<j<Ar /ij or the mean square value \h\.
Several examples add interest and help in understanding the outlined
theory.
Example 1. The difference approximation on a non-equidistant
grid. In working in the space Hq = C'^'*)[0, 1] comprising all the functions
defined on the segment 0 < x < 1 we refer to the operator L with the values
Lv- —
and take on this segment an arbitrary non-equidistant grid
Qf^ = {x^^ I = 0,1,.., , A^, x'o = 0, x^ = 1},
On account of Example 6 in the preceding section the difference operator
defined at the node x^ on the irregular three-point pattern [x^_^,x^,x^_^^)^
is associated with the operator Lv. With this, we rewrite Lh v as
with more compact notations
In Section 1,2 we have found the local approximation error taking now
the form
•0, = {Lu v\ - {Lv\ = h±l_h y^' + o{h^) , i=l/2,...,N~l.
From here it is easily seen that the operator L^ is of order 1 with respect
to the grid norm of the space C:
\\th\\ = max I ^,-I = 0(/i) , h= max /;,,.,
" '"-^ i<i<N-i ' ' ^ i<i<N ''

Difference approximation of elementary differential operators 71
The same is still valid in the grid norm of the space L2'-
N-l
1/2
Oih)
We claim that in an alternative norm known as the negative norm
1\ — i / Z \
ll^ll(-.l)
W-1 / i \ 2
8=1 ^k=l
the local approximation error ip is of order 2 and behaves like
B9) \\^p\\ -Oih'), /i= max /i,.
Indeed, rewriting ip as
and taking into account that ?;'" = v'.'' +0(/ij_|_j), we establish the
decomposition
h'^,v"' -h'^v'"
+ '0„: = tPi + A
ipi
6h,
with Tp* = 0[h''^) in any suitable norm. The main term tpi in the established
0
decomposition 1|}^ =tp. +tp* is of "divergent form", thereby justifying that
s ■ = J2^k^k--J2
' hl,,v'l'-hlv''' li:?..v'''~-Pv'"
^k + l "k + i '"k "k
8+1 i+1 1 1
k = l
k^l
6
Since \Si\< Mh'^, we find that
/N-i \ 1/2
-2
Hii(-n= E ^-^n =o(/'')>
which in combination with the relations
ll^ll(_i)<IUII(-.i) + N*ll(-.i), ll^*ll(-.i) = 0(/i^)

72 Basic Concepts of the Theory of Difference Schemes
implies that || if) |L_jn < Mh"^. This means that the approximation error is
of order 2 in the negative norm || • ||/_,y Observe that the norm || • |L_jn
is concordant with the norm \\u\\q = [Jq dxi^j^ ti(^) rf^) ] , so that
I|waII(_i)^ Ihllo as /i^O.
Another conclusion can be drawn from this example. Of course, the
study of the local approximation is unsufRcient for determination of the
order of the difference approximation and proper evaluation of the quality
of a difference operator.
In the estimation of the approximation error the well-founded choice
of the norm depends on the structure of an operator and needs investigation
in every particular case. A precise relationship between an operator and a
norm in the process of searching the error of approximation will be
established in the general case in Section 4, Its concretization for the example
of interest leads naturally to the appearance of the negative norm || • ||/_,y
A similar obstacle occurs in trying to construct difference approximations
of the operator Lu = (ku')', where k(x) is a piecewise continuous function
(see Chapter 3).
Where searching a solution u[x,t) to a nonstationary equation (for
instance, to the heat conduction equation), it is not unreasonable to
separate the variable t (the time). Some consensus of opinion is that the sought
function u[x, t) as a function of the argument x is an element of the space
Hq. Let u)f^ be a grid in a domain G of the space {x = (xj,. . . , x )] and let
LOj be a grid on the segment 0 < i < ig. The grid function y[x, t) = yhj{x, t)
is defined on the grid LOf^^ = uif^xui^ = {(x, t), x (zui/^, y (^ui^] and gives as
a function of the argument x ^ ui/^ a vector of the space Hh with the norm
II • ||. . In dealing with y(x,t) on the grid uji^^ many scientists generally
prefer either
C0) ||y||,^=max||y(i)l|.
or one of the alternative norms
C1) ||ylL= E r\\y{t)\l, ||y||,,= (E r\\y(t)\l')
J/2
t^LUr ^t^LOr
Let LhT Vhr be a difference approximation of the operator Lu, where
u = u[x, t). The operator L^j assigns really the values to the grid functions
Vf^^(x,t) defined on the grid tJ/j.^. If v(x,t) as a function of the argument x
belongs to the space Hq, then Vf^[x,t) = Vhv{x,t) G Hh for any t G [0,io]'
For any continuous in t function v(x,t) and all t ^uj^, we put Vh^(x,t) =

Difference approximation of elementary differential operators 73
Vf^{x,t), thereby specifying Vf^^ on the grid ujf^^ and reducing the error of
approximation to
i'hT{x,i) = ^hTVhr{x,t) -{Lv)^^(x,t), {x,t) eui,^^ .
We say that Lhr approximates L with order m > 0 in x and with
order n > 0 in i if in the class of sufficiently smooth functions v(x,t) one
of the estimates
U,ri^,t)\l^=.0{\hr + r"), ||^..||,, <M (I/J r + r")
is valid with a positive constant M independent of | /i | and r both.
Example 2
dv d'^v
Lv = -^ - -^ , 0 < X < 1 , 0 <t <to, LhrV := V^ - Vg^ .
The difference operator Lhr defined at all inner nodes of the grid
^Ar = { i^t .^j). ^^ = ^h, ij = J'T, 0 < J < TV, 0 < i < io , io = iol'T }
complements subsequent studies and proves to be useful in many aspects.
If v(x,t) possesses two derivatives with respect to t and four derivatives
with respect to x (ii G C^) that are continuous in the rectangle {0 < x < 1,
0 <t <to}, then
i'hri^,*) = LhrV^^ -iLv)hT = 0(/l^ +r)
at each of the inner nodes of the grid uji^^. This is also consistent with
the results of Section 1.2. Hence, Lhr approximates L to second order
in X and first order in t in either of the norms C0) and C1), where
Ei=i 'fpf h) . etc. Thus, in the
case of interest the approximation on the grid follows from the local
approximation.
So far we have studied the error of the difference approximation on the
functions v from certain classes V. In particular, in the preceding examples
the class of sufficiently smooth functions stands for V.
In what follows it is supposed that v = u is a solution of a differential
equation, say
Lu = u" = -f .

74 Basic Concepts of the Theory of Difference Schemes
The meaning of LhV ~ Vg^ as the difference approximation of the operator
L V ^ v" reveals that
Lh V - v" + ^ (/4) + ^ v('\x + eh) , -1<6<1.
We substitute here v = u, u" — —f, «('') = —/" and accept /" = 0,
implying that f^^^ = 0 for all k > 2. Then L^u = u" = Lu, that is,
the approximation error happens to be from the class of solutions to the
equation Lu = —f, where / is a linear function identically equal to zero:
i/' = 0. In that case the approximmation is said to be exact. But if/" ^ 0,
the accuracy will be improvable once we attempt the operator in the form
Lhv = Lhv+ — f" ,
thereby providing the approximation ip = LhU — Lu = 0{h'^).
Thus, the accurate account of the error of the difference approximation
on a solution of the differential equation helps raise the order of
approximation.
4. The statement of a difference problem. In the preceding sections we were
interested in approximate substitutions of difference operators for
differential ones. However, many problems of mathematical physics involve not
only differential equations, but also the supplementary conditions
(boundary and initial) which guide a proper choice of a unique solution from the
collection of possible solutions.
The complete posing of a difference problem necessitates specifying the
difference analogs of those conditions in addition to the approximation of
the governing differential equation. The set of difference equations
approximating the differential equation in hand and the supplementary boundary
and initial conditions constitute what is called a difference scheme. In
order to clarify the essence of the matter, we give below several examples.
Example 1. The Cauchy problem for an ordinary differential
equation:
C2) w' = f(x), x>0, m@) = Wo .
We proceed, as usual, on the simplest uniform grid tj^^ = {x^ = ih, i =
1,2,.,.} and put the difference problem
or = ^i , J = U,1,...; yo = «o .
Vo = "o . "

Difference approximation of elementary differential operators 75
in correspondence with problem C2), Here the right-hand side (/j, can be
specified in a number of different ways, for instance, by the formulae
^i = fi^i), <Pi =
f(Xi) +f{Xi^,)
2
provided the condition f^ — /^ = 0(h) holds.
Thus, given y^ — Ug, the solutions can be found from the recurrence
relation y^_^_^ = y^ +hip^, i = 0,1,2. ...
Au :=0 , t > 0 , w@) = Wq ,
Example 2. The Cauchy problem for a system of first-order
differential equations:
du
dt
where A. — (ciik) i^ ^ square n X 7i-matrix and u = (uj , . ., , w„) is an n-
dimensional vector. As we will see later, it seems reasonable to introduce
the grid uj^ = {tj = JT, j = 0, 1, 2,, . . } with step r. One of the possibilities
is Euler's difference scheme
yj+^ _ yj
■Ay^ = 0, i = 0,1,2,.,., y^ ^{yi ,yi ,... ,y,
T
The difference problem under consideration will be completely posed if the
subsidiary information is available on the vector y^ for j = 0 with the
initial condition y^ — Uq- The value y-''^^ is successively calculated by the
explicit formula
y^+^ — y^ — tA y-' .
This is one way of gaining experience with Euler's scheme that can be
accepted and used in theory and practice.
Example 3. The boundary-value problem:
C3) u"(x) = -f(x), 0<x<l, w@)=/ii, w(l)=/i2.
We take once again the equidistant grid
CD;,= {X^ = th,i = 0,l,... ,N, hN ^1}
and set up on it the difference problem
C3h) Vs^ = '-<P or j^ = -<Pi ,
J = 1,2,... ,yV - 1 , yo = y"i . yN~l^2-
Thus emerged the system of algebraic equations with a tridiagonal matrix.
Because of this form, the elimination method may be useful (see Chapter 1,
Section 1).

76 Basic Concepts of the Theory of Difference Schemes
Example 4. The first boundary-value problem for the heat
conduction equation:
du d u
C4) Lu= —-^ = f(x,t), 0<x<l, 0<t<t,,
w@,i) =/ii(i) , u(l,t) = ij.^(t) , u(x, 0) = Ug(x).
Choosing the equidistant grid uij^^ = {(x,- = ih, tj = jr), i = 0, 1, . . . , Ni,
j = 0, 1,. . . , A''2} and the simplest four-point pattern from Example 4 in
Section 1.2, we are now in a position to set up the difference problem
which admits the
C4J
index
y, -
r
1 < i
form
7
-Vi
<Ni
Vt =
7
Vi-
-1,
Uxx
-1 "
+ ^
■2y/
/l2
0 <
!
+ y!+i
j < iV2
+
—
^/
1,
Here the right-hand side ip can be defined in a number of different ways:
V-=f(xi,tj), Vi = f{xi,tj + i/2), etc.
The difference problem C4|j) illustrates the implementation of the so-called
explicit scheme in which the values of the solution on the upper layer
1/^+^ are expressed through the values on the current layer by the explicit
formulae
yj+l. = yJ -^. T{yl^ +^0 •
Common practice involves also the implicit scheme
yt=ysx+'P' j/(.-c,0) = u.o(a;), x^Lo^,
y{{},t) = ti,{t), j/(l,i) =/i2@, iecj, ,
which finds a wide range of applications. In order to have for later use the
values y = y^"*"^ on the [j -\- l)th layer, we must solve by the elimination
method the system of algebraic equations with a tridiagonal matrix
7,^+1 . . 7li
ylt' = F'^ F^ = ^ + <p\

Difference approximation of elementary differential operators 77
y^Y - B + ^) y/+^ + y.VV = -h' f/, o<t<N,
So far we have considered the first kind boundary conditions
approximated exactly on grids. In the case of the third kind boundary conditions
the question of their approximation needs investigation. In the next section
we will say a little more about this.
5. Convergence and accuracy of schemes. While solving a problem by
an approximate method the accuracy which is provided by this method
should be properly evaluated and predicted before proceeding to further
constructions. In this regard, the question of convergence and accuracy of
difference schemes arises naturally.
For convenience in analysis, we look for in a domain G with the
boundary r a solution to the linear differential equation
C5) Lw = /(a;), x^G,
subject to additional (boundary or initial) conditions
C6) lu = 1-l{x) , x G r,
where f(x) and ij,(x) are given functions (the input data of the problem)
and / is a linear differential operator, under the agreement that a solution
of problem C5)"C6) exists and i.s unique.
Within the framework of the present chapter the domain G -|- F of
continuous variation of an argument (point) is replaced by some discrete
set of points (nodes) x^ known as a grid.
Let h he a vector parameter related to the distribution density of the
nodes of the grid and let tj^ and 7^, be the sets of its inner and boundary
nodes. With these ingredients, the difference problem
C7) Lh yf^ = (ff^, X eLOh< hi Vh = Xh for x e j^ ,
where iPh{x) and X^i^) are given grid functions, is put in correspondence
with problem C5)-C6). Here the operators Lh and l^ assign the value.s to
grid functions defined for x ^ ui^ — ^h ~^ lh- ^ solution y^ of problem C7)
is a grid function of the nodes of the grid tj;^. Varying h and thereby
composing different grids ui^, we constitute the set of solutions {y^} depending
on the parameter h. In this context, the family of difference problems C7)

78 Basic Concepts of the Theory of Difference Schemes
corresponding to different values of the parameter h is of great interest and
refers to a difference scheme.
The main goal of any approximate method is to solve an original
(continuous) problem with a prescribed accuracy e > 0 in a finite number
of operations. In order to clarify whether it is possible in principle to
approximate a solution u of problem C5)"C6) by a solution j/;^ of problem
C7) with any prescribed accuracy e > 0 depending on the step h{e), we
follow established practice. This is concerned with further comparison of
Hi^ with u(x) in the space of grid functions i"J/j. Let w^ be a value of the
function u(x) on the grid tUf^, so that Uf^ £ Hh- The error Zf^ = Hf^ — U)^ of
the difference scheme C7) needs more a detailed exploration for a complete
and rigorous treatment.
The condition for Zf^ can be derived upon substituting y^j = z^j + M;^
into C7). Through such an analysis the problem of the same type arises as
problem C7):
C8) LhZ^=^^, X euj^, l^z^ = v^, X ejh,
where ipi^ and ly/^ are residuals equal to ■i/);^ = ifj^ — Lh Uf^ and Vf^ — Xh~^h ^h-
The right-hand sides ipj^ and Vf^ of problem C8) are called the error of
approximation of equation C5) by the difference equation C7) and the error
of approximation of condition C6) by the difference condition /^^ y/^ = Xf^ on
a solution of problem C5)"C6) or, briefly, ^^^ is the error of approximation
for the equation Lh J/^ = (/J;^ on a solution u[x) to equation C5) and Vf^
is the error of approximation for the condition /^ y^ = \ on a solution of
problem C5)"C6).
The accurate account of the error Z/^ of a scheme as well as of the
errors of approximations i)f^ and Vf^ is carried out in suitable norms || • |L. -.,
II • |L„ s and II ■ |L„ . on the space of grid functions.
We say that a solution of the difference problem C7) converges to a
solution of problem C5)"C6) (scheme C7) is said to be convergent) if
lUJI(i^) = l|yA-«JI(U) —^0 as |/i|^0
or
IUJI(U) = II/'(IM)II, where p{\h\)^Q as \h\^Q.
The difference scheme C7) is said to be convergent with the rate 0{\ h |")
or is of the 7ith accuracy order (of accuracy 0{\ h |")) if for all sufficiently
small I /i I < /iq the inequality
IUJI(u) = l|y.-«JI(u)<^w|/jr

Difference approximation of elementary differential operators 79
holds with constant M > 0 independent of | /i | and n > 0 both.
We say the difference scheme C7) is of the nth approximation order
if
ll'A;J|B,) = 0(|M"), Ik. 11C,)= 0A M"),
Denoting by fj^ and (Lu)i^ the values of f(x) and Lu(x) on the grid lvi^
and taking into account that (/ — Lu))^ = 0, we can rewrite iph ^s
^h ^ Wh- Lh Mft) - (A - (iw)ft )
= {Vh - fh) + {{Lu)^ - LhU^)
thus rearranging the approximation error -i/)^ of a scheme as a sum of the
error of approximation -i/jj^ ^ = i^^ — /^ of the right-hand side and the error
B)
of approximation tp^ = {Lu)i^ — Lh w^ of the differential operator.
Since ipf^ is the error of approximation in the class of solutions to
a differential equation, the condition ||'!/'a||(-2 i = C'd^l") holds true if
neither ip\ ' nor ip\ is of order n. An example in Section 1.3 confirms this
statement.
We now should raise the question; how does the accuracy order of
a scheme depend on the approximation order on a solution? Because the
error z^ = y^ —w^ solves problem C8) with the right-hand side i/'ft (and Vf^),
the link between the order of accuracy and the order of approximation is
stipulated by the character of dependence of the difference problem solution
upon the right-hand side. Let Zf^ depend on -i/j^j and Vf^ continuously and
uniformly in h. In other words, if a scheme is stable, its order of accuracy
coincides with the order of approximation,
A rigorous definition of stability of a difference scheme will be
formulated in the next section. The improvement of the approximation order for
a difference scheme on a solution of a differential equation will be of great
importance since the scientists wish the order to be as high as possible.
6. The attainable order of approximation of a difference scheme. As we
have stated in Section 2,3, the order of approximation to a differential
operator on a solution of a differential equation can be made higher without
enlarging a pattern. For convenience in analysis, we take into consideration
only two different difference schemes which will be associated with problem
C2) for u' = f[x) and u{Q) = u^. Our starting point is the difference
scheme of the form
yj; = ^ ^ yo = "o •

80 Basic Concepts of the Theory of Difference Schemes
At the next stage we look for the residual if) = u^ — </? on a solution u = u[x)
to the equation u' = J{x). The traditional way of covering this is to develop
Taylor's series
u{x + /i) = u{x) + hu'+'^u"+^ u'" + Oih^)
with the forthcoming substitutions u' ~ f, u" ~ f, ... and further
reference to expansions
^P =^ u'+ y^ il" + ^ u'" - ^ + 0{h')
= f(x)-<pix)+'^u"+^u"' + Oih')
= fix) - ^{x) + ^ fix) + ^ u'" + 0{e) .
Upon substituting ^p — f{x) + ^ h f or ip^ = (f + ^ h f^). we obtain the
scheme of order 2 on the solution u = u(x), thus demonstrating that the
residual ip behaves like 0(h'^).
By virtue of the relations
u"' = f"=f,^ + Oih^)
and
we find that
i'=^f+'^L-~f,.+Oih')-^,
thereby clarifying that the scheme y^. — </? with the right-hand side
<P = J + 2 U~- -^ hx
is of order 3 on the solution ii. = u{x) (■0 = 0{h^)).
For the boundary-value problem
u" - qu~ -f{x) , 0 < X < 1 ,
w@) = 0 , w(l) = 0 , q = const ,

Difference approximation of elementary differential operators
81
we focus our attention on the three-point difference scheme
Vxx - dy = ~-^p{x) , x = ih, f = 1,2, ...TV - 1 , y^ = y^ = 0 .
We will show that its order of approximation on a solution u = u{x) can be
made higher without enlarging a pattern under a proper choice of d and </?.
For a solution u = u(x) of the original problem, the residual ■(/) =
w^j. ~ du + if will be involved in the expansion
,D)
12
Oih')
nth regard to the equation u" = q u — f{x), permitting us to deduce that
thus causing the behaviour like 0(h'^) for rf = g and ip = f. Substitution
of u^'^^ = q"u — /" = q {q u — f) — fg^ + 0{h^) into the formula for ip yields
^ - / ~ Y2 '^■^^^ "^ '^^^
d- q
qH/
~12~
Oih')
A higher-order approximation ip = 0{h'^) can be achieved on the solution
u{x) of the initial equation by merely setting
12
^ = /+Y^D. + ?/)
We now turn to the question of approximations of boundary and initial
conditions on a solution of the original problem. This question is intimately
connected with the statement of a difference problem.
7. Approximations of boundary and initial conditions. From Section 2.5
it seems clear that the accuracy of a scheme depends on the order of
approximation on a solution of the original problem. By an approximation we
mean not only the equation itself, but also the supplementary (boundary
and initial) conditions.
In this section we give several examples of raising the order of
approximation for boundary and initial conditions without upgrading a pattern.

82 Basic Concepts of the Theory of Difference Schemes
Example 1. The third boundary-value problem, for an ordinary
second-order differential equation:
-j-^ — qu = —j{x) , q = const , 0 < a; < 1 ,
C9)
—~ = (Tti(O) -/ii, ti(l) = /i, .
On the equidistant grid tUf^ = {x,- = ih, 0 < i < N} we may attempt the
difference equation in the form
D0) ygj; - g y = -^
with (/5j = /(Kj), where f(x) is a continuous function.
At the point x = I the boundary condition is satisfied exactly:
D1) y(l) = Vn = 1^2-
Furthermore, we replace the first derivative w'@) by the first difference
derivative y^ q = (y^ — j/o)//i and impose the boundary condition at the
point X — 0 such as
D2) y^._ 0 = <T yo - Ml or l,^ y ~ t-i^ ,
where an operator //j is defined on the two-point pattern @, h). Substituting
here y = z -\- u, where u is a solution of problem B9), we establish for the
error z the condition
where the error of approximation v^ for the boundary condition on a
solution is equal to v^ ~ fi^ -\- u^. q — au^.
Developing u{x) about the node x = 0 in Taylor's series
«i = «o + /i«o + y <+ 0(/i^),
we find that
D3) «,,o = < + |< + 0(/j2),
V, = [fi, + u'{0) - a «@)] + ^h u"{0) + 0(/i2)
= lhu"{0) + Oih'^),

Difference approximation of elementary differential operators 83
since /ij + w'@) — a u{0) — 0. This provides support for the view that
Vi ~ 0(h). It is necessary to make some changes in condition D2) to
achieve the order of approximation OQi^). This is due to the fact that
u{x) is just the solution of the original problem C9). From the governing
differential equation the value w"@) can be expressed by
D4) «"@) = g«@)-/@).
Substitution of D4) into D3) yields
D5) «,,o ~kh{q•a(O) - /(O)) = «'@) + 0(/r),
thereby justifying that the left-hand side of D5) approximates to second
order the derivative u'[x) at the point x = 0 on the solution to the equation
u" - qu = -/.
From here and D2) it follows that the approximation of the boundary
condition
D6) y^^fi-'^Vo-p-i, a = a+\hq, /ij = Mi + ^/i/(O) ,
is of order 2 on a solution of problem C9).
It is worth emphasizing here that we have succeeded in raising the
order of approximation without enlarging the total number of grid nodes
which will be needed in this connection for approximating the boundary
condition.
Example 2. The boundary-value problem for the heat conduction
equation:
du d'^u ., , , „
^= ^ + ./(.^,i), 0<x<l, Q<t<t,,
D7) u(x,0) = Mo(^-),
du{{),t
dx
= auiO,t)~-fi,it), u{l,t) = fi,(t)
On the grid ui^.^ arising from Section 2.1 it is simple to follow the explicit
scheme of accuracy Olh"^ + r):
D8) yt = yg^ + f, y{x,o) = Ua{x), y{i,t) :=/j^d),

84 Basic Concepts of the Theory of Difference Schemes
where i^ = ip^ = f(xf,tj). In giving the difference approximation of the
same order for the boundary condition at the point a- = 0 we have occasion
to use
^duiO,t) h dhi{0,t) ,
By having recourse to the heat conduction equation for x = 0 we establish
a precise relationship
d^u@,t) _ duiO,t) _^^^ ^^
which implies that
,„ , h fdu@,t) ,,„ A du@,t) ^,,
2
-^r"-^("'^V = "^
The expression on the left-hand side of this equality approximates the
derivative du/dx for x = 0 to 0A1^). Replacing |y| __ by the appropriate
difference derivative
u@,t + T) -u{0,t)
T
we impose the difference boundary condition at the point x = 0:
D9) ya,.,o = I ^ %,o + c^ % -/*i . /«i = Ml +-1/1/@,^),
whose approximation on a solution of problem D7) is of accuracy 0(h + T'^).
In the case of the implicit scheme y^ = y^^ -|- (f the following condition is
acceptable to be an alternative:
E0) y^,o~^hy^Q + ay„-~fM^, ft, = ji, + ^ h f{0,t) .
Example 3. The second-order hyperbolic equation:
d u d u
E1) 5^" 5^ + -^'^'''^^' 0<^-<l. 0<t<to,
u{0,t) = u^{t) , u{l, t) — U2{t) ,
/ \ / \ du{0,t) _
w(x,0) = Wo(x), —^— = Uo{x) .

Difference approximation of elementary differential operators 85
Evidently, a special attention in approximating problem E1) is being
paid to the difference form of writing the initial condition for the derivative
du/dt. On any equidistant in x and t grid ujj^^ with steps h and r (see
Section 2.1) the simplest approximation u^{x,Q) = Uq{x) gives the error of
approximation 0{t). Plain calculations show that
du{x,Q) T d'^x{x,Q) ^
From the governing differential equation it follows that
(92w(x,0) __ (92w(x,0)
(fua
This is due to the fact that
9'w(.T,0) d'^Ugix)
dx^ " "ix-^
+ /(x,0) = L«o(x) + /(x,0)^
In this line,
u,ix,0) - i r (L u„ + fix, 0)) = ^^^ + 0(r2).
Therefore, the difference initial condition j/;(a;,0) = ito{x), where
Wo(x) = u(x) + ^ t{L Wq + f{x, 0)) ,
approximates to second order in r the condition du[x, 0)/dt = Ug(x) on the
solution of problem D1).
In this case the condition u(x,0) = Ug(x) and the boundary
conditions are approximated exactly. For instance, one of the schemes arising in
Section 1.2 is good enough for the difference approximation of the initial
equation. No doubt, we preassumed not only the existence and continuity
of the derivatives involved in the equation on the boundary of the domain
in view (at x = 0 or t = <J), but also the existence and boundedness of
the third derivatives of a solution for raising the order of approximation of
boundary and initial conditions.

86 Basic Concepts of the Theory of Difference Schemes
Example 4. The three-layer difference scheme for the heat
conduction equation. A special attention is being paid to the first boundary-
value problem
E2) = +/(x,i), 0<x<l, Q<t<t
0 I
dt di
u({),t) = Uj^(t) , u(l,t) = U2(t) , u(x,0) = Ug(x) .
The usual practice in numerical analysis of the heat conduction
equation E2) is connected with three-layer schemes. The values y^~^{x), y^{x)
and y^'^^(x) of a grid function on the three time layers i,_i, t, and i,-_|_j
are aimed at constructing such schemes.
The three-layer symmetric scheme on the equidistant grid uij^^ with
steps h and r, being the most familiar one, comes first:
E3) y'^'-y' ' ^ A iay'+' + (l ~ 2 a) y^ +ay^-') + ^\
/ r
y° = ^oi^i) , yl = ul , y^ = u^ ,
where A y = y^^, cr is a real parameter and (p^ = /(.t^, i).
Since the central difference derivative in t approximates to the
second order in r fyL ^ and Aw = |^ -|- 0{h'^), scheme E3) approximates
equation E2) with accuracy 0(/i^ -|- r^). From what has been said above
it is clear that problem E3) is overdetermined as it were. Applications
of the three-layer scheme concerned necessitate imposing one more initial
condition, for instance, by doing this on the first layer. But under such a
condition the approximation 0{t'^ -\- /i^) should remain unchanged. There
seem to be at least two ways of determining y{x, r). One way of proceeding
is to approve at the first step the two-layer scheme of accuracy OIt"^ -\- h"^)
in specifying y{x, r):
yl ^ yO 1
0
= ^A(yi+y°) + ^' .
r 2
Under the second approach the value of y(x,T) arranges itself as a sum
y(x, r) = Ug(x) + Tii{x) and /i is so chosen as to obtain the error y{x, t) —
u(x,t), not exceeding 0(r^-|-/i^). Substitution of the value ^1 _ , arising
from the differential equation
du
~dt
L Uo + f(x, 0), L Ua
d^u
t-o ' ' dx'-^

Difference approximation of elementary differential operators 87
into the formula
u(x, t) — Ug(x) = r
du
i = 0
Oir^^
yields fi = Lu^ + f(x, 0), so that
y(x,T) = u,(x) + T {u'^(x) + f(x, 0)) .
2.3 STABILITY OF A DIFFERENCE SCHEME
1. Examples of stable and unstable difference schemes. As we have shown
in the preceding section, the introduction of a difference scheme permits one
to reduce the solution of a problem associated with a differential equation to
a system of linear algebraic equations. In this situation the right-hand sides
of equations, boundary and initial conditions, all of which we will call in the
sequel the input data, are specified with a certain error. This procedure
causes rounding errors to be inevitable in the process of numerical solution
of a system. Because of this, it seems natural to require that small errors
in specifying the input data should not increase during the process of the
execution and result in wrong reasoning. The schemes in which initial errors
grow during the course of calculations turn out to be unstable and, from
the viewpoint of possible applications, cannot find response.
We will not attempt to encompass a wide variety of situations, but
instead look in more detail at several exhaustive examples before formulating
the definition of stability of a difference scheme with respect to the input
data, the concept of which we have intuitively developed earlier.
Example 1. A stable scheme. Let
A)
-aw, x > 0 , w@)
a > 0
It is straightforward to verify that the function u(x) — Ug exp { — ax}
gives the exact solution of problem A). This solution does not increase with
increasing x: | u(x) | < | Wq I f°'" ^ > 0, so that u(x) continuously depends
on «(,. An excellent start in this direction is to approximate problem A) on
the equidistant grid u)/^ = {x^ = ih, i = 0, 1, ...} by the difference problem
B)
Vi -Vi-i
h
■aVi
Vo
i=l,2.

88 Basic Concepts of the Theory of Difference Schemes
which can be rewritten as
y, = s yi_i , s = , i = i, 2,... , yo = "o .
i + an
yielding y^ = s^y^ .
We regard a point x to be fixed and take a sequence of steps h so
that X would always belong the set of grid nodes: x ~ i^h. The attached
number i^ may be made arbitrarily large once we will refine the grid in any
convenient way, that is, letting h —^ 0. The value of y at this point becomes
y(^) = Via ~ s'° Vo ■
Since | s | < 1 for a > 0 and any h, we thus have
I y(s) I < IH^'" I yo I < I y@) I
for any h.
The last inequality implies that the solution of the difference problem
B) continuously depends on the input data. In such cases we say that a
difference scheme is stable with respect to the input data.
Example 2. An unstable scheme. For problem A) we rely on the
scheme
C)
a -^ ^ A - cr) -^—^ h a yi = 0 .
% = «o . % = Wo . i = 1,2,... ,
where cr > 1 is a numerical parameter. Observe that this scheme is a three-
point one, since the difference equation is of order 2, so there is some reason
to assign the value y-^ in addition to the value y^. The approximation order
of scheme C) is no less than 1 regardless of the choice of the parameter a.
With Uq = A — ah)ug, a proper evaluation of the deviation gives w — u(h) =
0{h'^). We look for particular solutions to the difference equation C) in
the form y^ = s\ Substituting y, = s* into C) gives the quadratic equation
related to an unknown s:
D) (a-l)s'^-Ba-l + ah)s + a = 0,
which possesses two distinct roots
2a-l + ah± s/l + 2B(T- l)ah + a'^h'^
2(^-1)

stability of a clifFerence scheme 89
It is well-known that the general solution to equation C) is of the form
E) y,^As\+Bsl.
Putting J = 0 and f = 1 and using the assigned values y^ = u^ and y^ = u^,
it is easy to calculate the constants A and B:
It is clear that s-^s^ > 1, since a > a — \ > Q. We claim that Sj < 1
for any value of ah. Indeed, for a > \
2 (<T - 1) - BG - 1 + ah - n/1 + 2B<t- \)ah+a'^h'^)
Let us stress also that Sj > 1 for any value of ah on account of the bound
Sj Sj > 1 in view.
From the well-established representation y^ — As'^ + B s' it is easily
seen that y^ —^ooasi—^coifA^/^O, But we can always select j/j = Ug
so that the condition yi = 0 holds true for the choice u^ ~ u^s^. The
rounding errors arise inevitably in developing the solution s^, thus causing
the instability of the indicated type. In this scheme the solution increases
along with increasing x^ = ih if h is kept fixed. Successive grid refinement,
that is, successive refinement of h, leads to the growth of errors at a fixed
point X = j'o h, since ig = x/h increases along with decreasing h. A small
change of input data results in an enormous change in the solution of the
problem at any fixed point x as fe ^ 0.
We quote below the results of computations for problem C) with
?/(, = 1 and j/j = Sj, where Sj is the smallest root to the quadratic equation
D). Once supplemented with those initial conditions, the exact solution of
problem C) takes the form j/j = jBs^ (A = 0). Because of rounding errors,
the first summand emerged in formula E). This member increases along
with increasing i, thus causing abnormal termination in computational
procedures.
Modern computers allow the implementation of model problems. We
have carried out the calculations for several variants:
1) G = 1.1, ah^Om, Sj= 11.11, 52=0.99;
2) G = 1.1, a/i = 0.1, Sj = 12.09, Sj = 0.91;
3) G = 2, a/j = 0.01, Sj = 2.02, .s^ = 0.99;
4) G = 2, ah = 0.1, Sj = 2.17, s^ = 0.92.

90
Basic Concepts of the Theory of Difference Schemes
Table 1 contains the values j/j at several nodes of the grid uif^ for the four
variants. In the final line of this table the numbers of grid nodes connected
with abnormal termination are indicated by "infinitiy".
Table 1
i
6
7
11
12
13
14
15
16
20
25
32
Variant 1
Vi
0.952
0.942
0.900
0.832
0.174
-7.05
-8.72-101
-9.77-102
-1.49-10^^
-2.52-10^2
CO
i
7
8
10
11
12
13
14
15
20
25
30
Variant 2
Vi
0.567
0.516
0.420
0.306
-0.641
-1.17-10^
-1.45-10^
-1.76-10^
-4.54-10*
-1.17-101''
CO
i
32
33
37
38
39
40
50
60
80
90
100
Variant 3
Vi
0.724
0.703
0.260
-0.196
-1.11
-2.95
-4.10-10^
-4.64-10«
-5.92-10^2
-6.69-lOi'^
CO
i
28
29
32
33
34
40
50
60
80
90
92
Variant 4
Vi
8.97-10-2
7.87-10-2
3.78-10-3
-7.41-10-2
-0.237
-3.17-10^
-7.84-lO''
-1.94-10*
-1.19-101^^
-2.94-101*
CO
2. The Cauchy problem for a system of differential equations of first order.
Stability condition for Euler's scheme. We illustrate those ideas with
concern of the Cauchy problem for the system of differential equations of
first order
F)
du
dt
Au = 0.
t > 0,
u@)
where u = {u^^\ u^'^\ . . . , «(")) is the vector of unknowns, Uq = (wj; , Wq i
. .. , Wo ) is an n-dimensional given vector and A = (cj,-) is a symmetric
positive definite nxn-matrix. In what follows a vector space H„ is equipped
with the inner product (u,v) = X^"_j u^''^v^*) and associated norm ||u|| =
y(u,u). Under these structures, a linear self-adjoint operator A: Hn i—>
i?„, A = A* > 0, is associated with the matrix A of interest.
We denote by {Aj,,^^,} the system of eigenvalues and orthonormal
eigenvectors of the operator A:
Aik = Ki
k <,k I
1,2,

stability of a difference scheme 91
so that
0<A, <A, < ... <A„, (ik,is) = h. = [l' I'^l'
With these entries, we look for a solution of problem F) in the form
n
s = l
Substitution of this expression into equation F) yields
s = l
which implies that
-^ +X^ak = 0, afe(O = at@)exp{-A^i}, k = l,2,...,n,
providing the same grounds for the series
G) u(i)= E afe@)exp{-A,O6.
k = l
Under the initial condition
u@) = uo= Ea,@)^,,
k = l
we arrive at the relations
n
«.@) = (uo,4), l|uo|P= E «'@).
fc=l
From G) it follows that
n n
II uW II' = E E "^-@) "^@) e^^P ^-^k t} exp {-A, t} D , ^J
Jt = l 5-1
n n
= J2 a^@) exp {-2 A, i} < exp {-2 A, t} ^ al{0)
k=l k=l
=:exp{-2AJ^}||uo||^

92 Basic Concepts of the Theory of Difference Schemes
which implies that
(8) ||u(i)||<exp{-Aji}||uJ|
From such reasoning it seems clear that the solution of the Cauchy problem
decreases along with increasing t:
(9) ||u(i)||<||uj| for t>0.
Of special interest is Euler's scheme for the Cauchy problem
A0) li+lZll + Ayj = 0, i = 0,l,..., yo = Uo,
T
where y ■ = y(i,) and tj = jr. We seek its solution as a sum
A1) yj = E «fc9i4.
k = l
SO that yo = Uq = Efc=i '^k ik ^■^'^ '^k — i'^o )ik)- We note in passing that
expression A1) satisfies equation A0) only if
'?fc-l
k=i
or, what amounts to the same,
T
It follows from the foregoing that g^. = 1 — rAj,.
By rearranging the norm as
n
\\yj\\'=E \<ii?al
k = l
it is easy to derive the chain of the relations
n
II Yi IP < max I ,1 \'Eal= max | qi \' \\ y„ f < \\ y„ |p,
I" fc = i I"
which are valid under the conditions maxj; | g^, | < 1 and || y,' |P > || yo IP if
minfc I g^, I > 1. Being an alternative of (9), the inequality
J2a,['^ + X,)qU,=0
A2) l|y/ll<l|yo

stability of a difference scheme 93
expresses the stability of the problem A0) solution.
The condition max^ | g^. | < 1 is fulfilled if — 1 < g^. = 1 — rXf. < 1,
that is, rAj, < 2 for all fc = 1, 2, .. . , n. For the latter to hold, it is sufficient
that
A3) r<|,
where A = maxj, A^, = A„ .
By a rather similar argument we take Yq = c ^,j, leaving us with Uj, = 0
for fc = 1, 2,. .. , ri — 1, a„ = c and the relations
Yj =cqiL =qiyo, II Vj II = I in K II Yo II > II Yo II.
since |g„| = rA„ — 1 = rA — 1 > 1 ifrA > 2. For instance, when
rA = 2 + p, p > 0, we draw the conclusions that | g„ | = 1+p and
I ?n I ~ A +-P)'' ^ oo as J ^ CO. In that case scheme A0) becomes
unstable and makes the method so inefficient as to be unusable.
In mastering the difficulties involved, we impose the initial condition
Yq = c^j, thereby providing
Yj =?/yo. II Yj II = l?iK llyoll.
where gj = 1 — rAj.
It may happen that r > 2/A, but r Aj < 2 and | g^ | < 1, which assures
us of the validity of the estimate
A4) l|y,ll<|gil^'yo<l|yol|.
Because of rounding errors, y^ is determined with some error e still subject
to the approved decomposition
n
e = E £^4. Sfc = (e,^i),
k = l
which is not surprising. No matter how the value e,j is chosen, there always
exists Jo such that |e„||9„|''° > Moo, where M^ is computer infinity,
meaning that for j = jg abnormal termination occurs because \q„\ > 1.
Thus, scheme A0) turns out to be unstable with respect to rounding errors
under any initial condition in the case where r A > 2.
For the solution y ■ of problem A0), the requirement of having an
estimate similar to inequality (8) necessitates imposing one more condition
1 — rAj > rA,j — 1 or, what amounts to the same,
2
T < — .
Aj + A„

94 Basic Concepts of the Theory of Difference Schemes
Assuming this to be the case, we deduce in light of the obvious relations
1 — r Aj < exp { — AJ r} and | 9i | > | 9^ | for k > 1 that
II Yi II < max I g,K' || y^ || < A - t\)^' \\ y„ || < exp {-X, tj} \\ y„ \\ .
We have here one of many examples reinforcing the view that the
general ideas of stability are sensible. Consider a system of two equations
du dv
\- au -\- bv — \), -— -\- hu -\- av = \)
dt dt
nth the matrix A
K l), where a = |(A + .5) and 6 = i (A - ,5). It
is straightforward to verify that its eigenvalues and eigenvectors are equal
to
With these entries, scheme A0) reduces to
^^l+l-lIl + ay^+bzj^O,
A5) ^ J = 0,1,2,...,
-j+i
- + byj + azj = 0 .
Observe that the vector {j/qi -^o) coincides with the first eigenvector ^j if we
agree to consider
A6) J/o = 75, ^o = -75.
After scrutinising the available information we initiate the review of final
results of calculations for problem A5)-A6) with the following values of
parameters:
1) .5 = 1, A = 2, T= I:
2) E = 10, A =400:
A
6
"=A
In both cases | 1 — rE | < 1 and, therefore, for a solution of problem
A5)-A6) estimate A4) is valid, but r > 2/A. Because of rounding errors,
the computational process is unstable: for large j the growth of its solution
causes abnormal termination in the computer realization of the algorithm.
The final results of calculations are tabulated in Table 2 with the
values of the functions y- and z.- for several j. For variants 1 and 2 abnormal
terminations have occurred at the 105th and 46th time steps, respectively.

stability of a difference scheme
95
Table 2
Variant 1
J
8
9
10
17
18
19
20
25
26
27*)
28
30
50
68
69
70
105
Vj
-5.52
2.76
-1.38
1.08
-5.46
2.82
-1.60
8.18
-1.63
3.26
-6.51
-2.60
-2.73
-7.16
1.43
-2.86
CO
10^3
10-3
10-3
lo-'^
io-«
10-'^
10-'^
10-'^
10-^
10-'^
10-5
10-"
10^
10^
10*
10*
^J
5.52
-2.76
1.38
-1.08
5.33
-2.57
1.09
8.10
-1.62
10-3
10-3
10-3
lo-'^
10-'^
10-^'^
10-'^
10-'^
10-5
Variant 2
j
8
8
10
13
14
15
16
17
18
19
20
21*)
22
38
39
40
46
yi
0.227
0.193
0.164
0.101
8.49 10-2
7.56-10-2
4.69-10-2
0.127
-0.326
1.89
-9.23
4.63 10^
-2.32-102
-3.53-10*3
1.77-10'''
-8.83-10'"
CO
'j
0.227
-0.193
-0.164
-0.101
-8.61-10-2
-6.96-10-2
-7.65-10-2
2.16-10-2
-0.415
1.81
-9.30
*) Once started with this number j we have Zj = y,- .
Observe that in variant 1 the accuracy in specifying the first
eigenvector ^j is of significant importance. For the values
A7)
Vo = 1
-1
abnormal termination occurred at the 197th time step. At the beginning
the variables j/j and Z; continue down to some quantities of order 10-'® for
j K, 65-70 and then begin to grow. Unlike this tendency, the initial data
A7) did not change essentially the results of computations for variant 2:
abnormal termination occurred for j = 47.
If for E = 1 and A = 2 we could take the value r = 2.1/A, for which
the condition t < ^ is slightly violated (r is exceeded by 5%), the same
still holds at the 760-790th time steps.
The preceding examples provide enough reason to conclude that the
concept of stability with respect to the input data is identical with the
concept of continuous dependence of the solution of a difference problem upon

96 Basic Concepts of the Theory of Difference Schemes
the input data. The two major aims, improved accuracy and better
stability, have to be balanced against the need to avoid expending unnecessary
computational effort. In the next few sections the advanced theory in such
matters will be developed and presented in full details.
3. On the concept of well-posedness for a difference problem. There is
another matter which is one of some interest. In conformity with statements
of problems of mathematical physics, it is fairly common to call a problem
well-posed if the following conditions are satisfied:
A) the problem is uniquely solvable for any input data from some class;
B) a solution of the problem continuously depends on the input data.
Being concerned with a solution y^^ and input data ipf^ of a difference
problem depending on the grid step h as the parameter, we introduce the
concept of well-posedness for a difference problem in a similar manner.
Varying h we are in possession of two sequences of solutions {j/^} and
input data {</5;j}, thus causing not only a single difference problem, but
also a family of problems depending on the parameter h. The concept of
well-posedness is introduced for a family of difference problems (schemes)
as I/i I ^0.
We say that a difference problem (scheme) is well-posed if for all
sufficiently small \h\ < h^
A) a solution y^^ of the difference problem exists and is unique for all
input data i^f^ from some family;
B) the solution y^^ continuously depends on ipf^ and this dependence is
uniform in h.
More precisely, the second condition implies that there exists a
constant M > 0 independent of h such as for sufficiently small \h\ < h^ the
inequality
A8) \\yh-yh\i,)<M\\'Ph-^AB,)
holds, where y^^ is the solution of the problem with the input data (pf^, and
II • |L s and II ■ IL^ -, are suitable norms on the set of grid functions given
on the grid ujf^.
The property of the continuous dependence of the solution for a
difference problem on the input data is expressed by inequality A8) and is
treated as the stability of the scheme with respect to the input data or
simply stability (see Section 4.2)
4. Stability, approximation and convergence. We now examine a
continuous problem of the type
A9) Lu = f(x) for xgG, Iu — ijl{x) for x £ F,

stability of a difference scheme 97
which is approximated on a grid lJ/^ = tUf^ + J/^ by a difference problem
B0) Lh iJh = fh foi' « e ^ft , ihVh^ K foi' »■ e ih ■
The statement of the problem for the error Zy^ = y^^ — Uf^, where Uf^ is
the projection of the solution u of problem (f9) onto the grid lo^^, is
B1) Lh Zh = i^h for 3: ^LOh, l^z^-Vh for a; G 7^ ,
where ip^ and z/;^ are the errors of approximation to the governing equation
and the additional condition. Instead of B1) we may write down formally
'Lh Zh='Ph-
If the operator Lh is linear and the difference scheme is well-posed, we are
led by A8) to
B2)
Zh\,,,<MUh\^^)
IUJI(,,)<M(||^J|B,) + lkJIC,))-
With these relations established, we conclude that if the scheme is stable
and approximates the original problem, then it is convergent. In other
words, "convergence follows from approximation and stability" and the
order of accuracy and the rate of convergence are connected with the order
of approximation.
Everything just said means that in establishing convergence and in
determining the order of accuracy of a scheme it is necessary to evaluate
the error of approximation, discover stability and then derive estimates of
the form B2) known as a priori estimates.
Being the elements of different spaces, the solution Zh and the right-
hand side tpf^ of the difference problem should be evaluated in different
norms.
The examples of suitable norms are available in the preceding
sections in which the errors of solutions and approximations on the grid LOh
are investigated in the usual way. Unfortunately, it is impossible to get
estimates of the form B2) directly for stability of concrete difference
problems. Auxiliary mathematical tools and techniques such as the summation
by parts formulae, the difference Green formulae and elementary analogs of
embedding theorems are aimed at deriving various estimates for solutions of
difference analogs of boundary-value problems associated with an ordinary
second-order differential equation. In this particular case we will encounter
typical situations related to stability, approximation and accuracy of
difference schemes. Later we will elaborate on this for rather complicated
problems. At the same time, subsequent discussions of them will involve
issues which are different from those occurred before.

98 Basic Concepts of the Theory of Difference Schemes
2.4 MATHEMATICAL APPARATUS IN THE THEORY
OF DIFFERENCE SCHEMES
1. Some difference formulae. In the sequel, when dealing with various
difference expressions, we shall need the formulae for difference differentiating
of a product, for summation by parts and difference Green's formulae. In
this section we derive these formulae within the framework similar to the
appropriate apparatus of the differential calculus. Similar expressions were
obtained in Section 2 of Chapter 1 in studying second-order difference
operators, but there other notations have been used. It performs no difficulty
to establish a relationship between formulae from Section 2 of Chapter 1
and those of the present section.
1) Formulae for difference differentiating of a product. As
known from the differential calculus, the formula
(uv)' = il'v + uv'
holds for differentiating the product of two functions u[x) and v{x).
In Section 1 we have already introduced two types of difference
derivatives for grid functions: the left and the right ones, which correspond to
different formulae for difference differentiating of a product
A) (w^),-!; = ^x^ ~^ W^"*" ■'^.r = ; v'^'^^' -f U V^ ,
B) {UV)_ = UgV + U^'^^^V^ = Wj.?j(~^) +UVj, .
Here we retain the notations
/(±i) = fix ± h), /,. = ~^, f, = ~ .
In this context, we draw the reader's attention to the fact that these
formulae involve the index shift. Let us prove, for instance, the first equality.
With this aim, we write it in the index form
h h h '
which assures us of the validity of the equality.
2) The summation by parts form/ulae. We recall the integration
by parts formula
i i
IJ — / u V dx .
0

Mathematical apparatus in the theory of difference schemes 99
As in the previous case similar formulae of two types
C) (U, V^) = Uj^ Vj^ - Wo TJj - (Uj , V] ,
D) (W,%) = M^fTJAf^l - Wo^O - K-.^)
are valid for grid functions keeping the notations
N-l N N-1
E) iu,v)= J2 ^i^ih, (u,v]= J2 '^^i^i^h [u,v)= E "i^i^-
!=1 2=1 i-0
Let us prove, for instance, C). First, applying formula A) yields
N-l N-l N-l
2 = 1 !=1 2=1
N
= iuv)N - (W^)l -E('"«^J-'^
2 = 2
= (w^)Af - "l ''l - E '"S,i ^^ '* + («5 ^)l ^ ■
Second, it is obvious that h(ugv)-. = Wj iij — Wq i"! ■ Substitution of the
preceding into F) along with E) yields C).
In what follows we will frequently employ a non-equidistant grid
denoting it by LVj^ in contrast to an equidistant introduced in Section 1. On
any such grid the formulae of the inner product and the difference
summation by parts formulae are somewhat different:
N-l N-l
(w, v)^ - Y^ Ui Vi fi, , (w, v)=z Y^ u. v^ hi^^ ,
!=1 2=1
G)
N
2 = 1
where H^ = ^{h^ + h^.^). The notations for the difference derivative on a
non-equidistant grid

100 Basic Concepts of the Theory of Difference Schemes
and the inner product on a non-equidistant grid ( , )* allow a simpler writing
of the ensuing formulae.
To prove G), a .simple observation that
"i- = -r- V,;, v^ = — v^
may be useful. Indeed, substituting this expression into the inner product
N-y
(w, v^)^ = (w, v^), where (tt, w) = Y. ''-h ^i /^,+i ,
i-V
and following the proof of identity C), we arrive at G) as desired.
3) The first Green formula. The equality
i 1
/ u (kv )' dx = — ku'v' dx -\- kx
is usually called the first Green formula.
For grid functions an analog of the Green formula can be obtained by
the summation by parts formulae. This can be done by substituting
u= z, V = ay^
into C). The outcome of this is the first difference Green formula
(8) (^, (aVs),;) = -{uiJs , %] + azy^ |^ - Cj y^, q z^ .
If Zg = Zj^ = 0, the last two terms in (8) vanish and the first Green formula
appears in simplified form:
(8') {z, Ay) - ~{ay,. , Zj,], Ay - {ay^)^ .
In particular, for z = y the preceding becomes
(8") (Ay,y) = -(a,(y,,J], y„ ^ y^ = 0.
In the case of a non-equidistant grid we obtain a similar result:
(z, {ay^.)i)^ = -{ay.^ , z^] for z^ = 2^ = 0 .

Mathematical apparatus in the theory of difference schemes 101
4) The second Green formula. In the integral calculus the
following formula
u (kv'y dx — V (ku'y dx — k {uv' — 'yw')|o
0 0
is known as the second Green formula. Inserting u = y and v = aZj in C)
yields
(9) (y. {az^.).^.) = -{ay^ , z^] + ayz^\^ - a, y^ ^,, „ .
Substracting (9) from (8) we derive the difference analog of the second
Green formula
A0) (z, (ayj,)^.) - (y, (azj,),.) = a^ (zy^ - z^ y)N - a, (y^ z ~ z^ y)^ .
In just the same way as was done on a non-equidistant grid, we find that
A1) (z, (ayg)^)^ - (y, (azg)^,)^ = a^ {zy^ - z^ y)^ - a, (y^ z - z^ y)^ .
If y and z vanish at the points x = 0 and x — 1, the appropriate terms
in the expressions being substituted into C) are equal to zero, thereby
reducing to
A0') (Ay,z) = {y,Az), Ay = {ays,)^,
A1') (Ay, z), = (y, Az)*, Ay = {ay^)^ .
These formulae confirm that the operator A is self-adjoint.
5) The Cauchy-Bunyakovskii inequality and the e-inequaliiy.
In the sequel we shall need the well-known Cauchy-Bunyakovskii inequality
l(«-^)l<ll«ll-IIHI,
where (, ) is the inner product in a vector space and j|w|| = \/{u,u) is
the associated norm. To be more specific, (, ) designates one of the inner
products we have introduced above.
In what follows we will frequently employ the inequality
2 ^'
\ah\ < ea'^ + -—

102 Basic Concepts of the Theory of Difference Schemes
where e > 0 is an arbitrary number, wliich is sometimes referred to as the
e-inequality, permitting us to establish
A2) \iu,v)\<\\u\\-\\v\\<e\\u\\^+j-^\\v\\\
2. The search for eigenfunctions and eigenvalues in the example of the
simplest difference problem. The method of separation of variables being
involved in the apparatus of mathematical physics applies equelly well to
difference problems. Employing this method enables one to split up an
original problem with several independent variables into a serie.s of more
simpler problems with a smaller number of variables. As a rule, in this
situation eigenvalue problems with respect to separate coordinates do arise.
Difference problems can be solved in a quite similar manner.
In this section we consider the problem of searching for eigenvalues of
the simplest difference operator.
Results we are going to obtain here will be needed in the sequel because
applying the method of separation of variables leads to problems just of the
same type. In the next chapters we give various examples of employing this
method for the discovery of stability and convergence of concrete difference
schemes.
Before giving further motivations, we would like to recall the basic
aspects concerned with the elementary problem of determining eigenfunctions
and eigenvalues for the differential equation
A3) w"(x) +Aw(x) = 0, 0 < X </, w@) = w(/) = 0.
The nontrivial solutions of this problem, that is, the eigenfunctions Uf, and
the appropriate eigenvalues A^, are expressed by
I.
12 . kiTx
2. The eigenfunctions Uf, constitute what is called an orthonormal
system:
r / N / ( 0, k^m,
j «fc(a;) «,„(»;) dx = Ej.,„ , where 6,^^ =
0 [ I, k = m.
«fc(^) = V 7 sin —;-, Aj, = -j^, k = l,2,
3. If f{x) is twice differentiable and satisfies the nonhomogeneous
boundary conditions /(O) = /(/) = 0, it arranges itself into a uniformly
convergent series
oo
/(») = E /fc«fc('-c)
fc=l

Mathematical apparatus in the theory of difference schemes 103
1
where f^ — J f{x) Uj.{x) dx; in so doing
0
1 oo
\\f\\' = f p(x)dx=Y: II-
0 k = l
We put the difference problem
A4)
j/jj; + Ay =1 0 X ~ ih, 0 < i < N, h ~ --
y@) = y(/) = 0, yix) t 0 ,
of searching for nontrivial solutions being the eigenfunctions and eigenvalues
of problem A4) in correspondence with the differential problem A3). For
this, the index form of A4) is
A5) %+j-2(l-i/i2A)y,+yi_j=0, i = 1, 2,... , TV - 1 .
We look for a solution of problem A4) in the form
j/(a;) = sin cxx
with a to be determined. Along these lines,
j/f-l-i + Vi-i = sin aix + /i) + sin aix — /i) = 2 sin ax cos ah .
Substituting the expression just established into A5) yields
2 sin ax cos ah = 2 A — i h}X) sin ax .
Since only nontrivial solutions are those to be found, that is, sin ax ^ 0, it
follows from the foregoing that
1 — I h}\ = cos ah
A 2 4 . 2 a/i
^ = 7^ A - <=°s a/i) = ^ sm — .
The value of the parameter a is chosen in such a way that the function
j/(a;) — sin ax satisfies the boundary conditions of problem A4):
y@) = y{l) = 0.

104 Basic Concepts of the Theory of Difference Schemes
Observe, that the boundary condition at the point a- = 0 is automatically
fulfilled for any a. At the point x = I we have
sin a/ = 0 ,
giving a = ttj. = kir/l, k = 1,2, ... , N — 1.
Thus, we have obtained the eigenfunctions and eigenvalues of problem
A4). A brief survey of their properties is presented below.
1.
A6) y(*')(x) = sm -p , A, = — sm^ ^^ , k = 1,2,... , N - 1.
2. The eigenvalues A^, are enumerated in increasing order and for the
whole collection {A^,} the estimates holds:
/. X „ , 4 . , 7r/i , , 4 . , TTh(N - 1)
A7) 0 < A^ = ^ sin' ^ < ^2 < ■ " < ^^-1 = F 21
4 2 '^^ 4
In particular, it follows from A7) that all the eigenvalues of problem A4)
are positive.
3. Eigenfunctions of problem A4) y^''\ y^'"\ corresponding to distinct
eigenvalues, are orthogonal in the sense of the inner product E):
A8) (y(*'),y('")) = 0, k^m.
To prove this fact, we make use of the second Green formula for the
homogeneous boundary conditions A0'):
0 = ii/^, y^™)) - {t/'K y'^?) = (A„ - A,) (j/^), y('")).
In light of our supposition, y^''-* and y'-™-' are eigenfunctions corresponding
to distinct eigenvalues, that is, A^, ^ A^, the orthogonality of y'-*'-' and y'-™-'
follows from the preceding equality:
(yW,y('»)) = 0.
4. The norm of an eigenfunction y^^\x) is ||y(*')|| = yip. Recall
that the norm is understood in the sense of the inner product E):
Af-l
i=l

Mathematical apparatus in ttie ttieory of difference sctiemes 105
We perform simple algebraic calculations and write only the final
results
N-l N-1 ,
A9) ||y(fc)|p=^(y(^-)K))\=^/.sin2"'"'
/
s=l s=l
h / 2iTkx,
= E 9 ^-'^"^
,2V /
Using g^. = exp {?' ^2±ll^ and taking into account that g[. = exp {i ^y^ x^}
and g^ = 1, we find that
N-l N-l N __ 1
y hcos~x,=Rey hql = Keh'^ ^ = Re/j illit = „/,,
"^ I /l^ ^i g,. - 1 Oj. - 1
s = l snnl ^* ^*
Substitution of this value into A9) yields
II (^)||2 hjN-l) h hN I
\\y'>\\ =—^—+2 = ^ = 2-
Thus, the set of grid functions
B0) fiC'^x) = ^J^ y('\x), fc=:l,2,...,yV~l,
constitutes an orthogonal and normed system in the sense of the inner
product (A«(*'),A«('"^)=*i™.
5. Let a function f(x) with the values /^ = /^ = 0 be given on a grid
Ljf^. Then, obviously, it can be represented as a sum of the eigenfunctions
of problem A4):
Af-l
B1) f{x)= E hf^^'K^)
k = l
where the coefficients are defined by the relations ff. =■ (/(x),/i(*'-'(x)).
What is more, in that case
Af-l
Ik
B2) ||/|p= E /^
k=l
Indeed,
N-l /N-l N-l \
i=l \k=l k=l J
( N-l \ Af-1 Af-1
= E A-/«.M^'V'"\i = E /./™(A^^'^A'('"^) = E/^
\k .m^X J k .m=l k~l
ince {fi^^\fi^^^) = S^.^ as established before.

106 Basic Concepts of the Theory of Difference Schemes
3. Eigenvalue problems with boundary conditions of the second and third
kinds. We now turn to the second eigenvalue problem
B3) u" + \u = 0, 0<x<l, u'{0) = u'{l) = 0, uix)^0.
With the relations w^.o = u'{0)+^hu"{0) + O{h'^) = -^^h\u{0) + O{h'^) and
u^ 0 + ■|/iAw@) = 0(h'^) in view, we approximate the boundary conditions
with accuracy 0(h'^), leading to the second boundary-value problem on
eigenvalues:
ysx + ^y{^) = o, x=.ih, ?'=: 1,2,... ,yv~ 1, h = jq,
B4)
Vr-fi + k '^•^^0 = 0, ■~yx,N + h ^'^^vn = 0 •
In such a setting it is required to find the values of the parameter A
such that these homogeneous equations have nontrivial solutions y{x) ^ 0.
In contrast to the first boundary-value problem, here the parameter A enters
not only the governing equation, but also the boundary conditions. The
introduction of new sensible notations
B5) Av - \ -Vj.^ for x ~ ih, 0 < i < N , ,
~r V.-. for X = I,
h
is connected with setting problem B4) in the operator form
B6) Ay = Ay ,
where the operator A acts in the space H = Q comprising all the functions
y(x) defined on the grid lvj^ = {x^ — ih, z =; 0, 1,. .. , A''}.
We introduce in that space the inner product
N-l
[y, ^] = E ?/; '"z ^ + t (Vo Vo + ?/Af '"Af)
2 = 1
and show that the operator A is self-adjoint and nonnegative. By definition,
this means that
[Ay,v\ ~ \y,Av\,
[Ay, y] > 0 for any y e H .

Mathematical apparatus in the theory of difference schemes 107
Indeed,
[Ay, v\ = {-y^^ '^'^^ ~2\1l ^° ^''''° "^ ~h '"^ ^*'^/ '
Making use of the second Green formula A0), we get
[Ay, v] = (y, -Vg^) + {vy^ - yv^)^ - {vy^ - yv^)]^ + (-Vg y^^ + ^N Vx,n)
- {y, -^xx) + i-Vo ^xfi + vn Vx,n) = [y> M ,
that is, A = A*. The first Green formula (8) gives for z = y
[Ay, y] - (y, -ysx) + {-Vo Vx.o + Vn Vs.n)
= ivx^-yx]" (yyx)N + {yyx)o + (-% y^.o + yNys,N)
= [yx^yx] > 0.
Problem B6) can be solved by appeal to the general theory (see
Section 1). Let us determine the eigenvalues A^, and eigenfunctions /ij;(x) of
problem B4) accepting a solution of problem B4) in the form
y = i__i[x) = A cos ax, A^ Q .
Upon substituting /-<(x) into equation B4) we get
4 . , a/i
We now require the boundary conditions to be satisfied at the points
X — Q and x = /. The condition cos a/i — 1 + i h'^X = 0 is automatically
fulfilled at the point x = 0. The case where x = / is not simple to follow
and needs investigation:
(l — i h'^X) cos a/ — cos a(/ — /i) = 0
or
cos ah cos al — cos al cos ah + sin al sin ah = sin al sin a/i = 0 ,
implying that al = irk, k = 0,1,2,... , iV. Thus, we have determined the
eigenvalues
4 o 7rA;/i
B7) Ao = 0, ^A-=/^sm2^^, fc=],2,...,yV,

108 Basic Concepts of the Theory of Difference Schemes
and the orthoiiormal eigenfunctions {/ij,(a;)}, for which
1, k = m,
where
Wk ) Mm] — hm 1 ri 7 /
'0, k ^ m,
1 I ttN
B8)
I^N = \ 1 COS
; ; ^-1 7
COS —^, COS ——- = > h COS
/ ' ^^^ V / / '
/ii^^ycos-p, fc=l,2,... ,yV~l.
The normalizing multiplier yl^ can be recovered from the condition
ir Ti9 r T ^9 7T fC X 7T tC 3^
lil^kW = [Mi .MiJ = ^i [cos -p, COS —j-
where the sum
_ 7. „ _ 7. „ s
s=l
is calculated by analogy with the preceding section. The orthonormaUty of
the eigenfunctions {/i^,} follows from the general theory, since A — A* > 0
and all the eigenvalues are simple.
Any grid function f(x) defined on the grid uif^ arranges itself into a
series of {yUj,(x)}:
N
k = 0
N
under the inner product structure [/,/] = J2 fk-
k = 0
The statement of the third boundary-value problem on
eigenvalues is
B9) w" + Aw(x) = 0, 0<x</,
u'@) = cTj w@), cTj > 0 ,
— «'(/) = (Tj u{l) , (Tj > 0 , u{x) ^ 0 .

Mathematical apparatus in the theory of difference schemes 109
As can readily be observed, the difference scheme of second-order
approximation acquires the form
C0) yg^. + \y=0, x = ih , i = 1, 2,... , TV - 1 ,
-(ys,Af+ '^2J/Af) +l^'^J/w = 0, y(x)^0,
yielding
M^
0 - cTj Wq -I- A /lA ztg = M^ -f i /iM^' -I- 0(/i^) - a^Ua + ^hXug = 0(h'
It will be sensible to introduce the operator A with the values
-I
C1) Ay = < -Vs:^ for x = ih, 0 < i < N ,
1
^ Q-^ivx + c^zy) foi' X = I,
by means of which problem C0) admits an alternative form of writing
C2) Ai.t = Xii.
The domain of operator C1) coincides with the entire space H, the domain
of operator B5), A + A* > 0 and, what is more,
[Ay, y] = (fe , yj ] + cTj y^ + (T2y%,
it being understood that yi > 0 if either at least one of the coefficients a^
and (Tj becomes nonzero or a^ + a^ > 0, but Cj > 0 and cTj > 0. In that
case [Ay, y] = 0 only for y(x) = 0,
Unlike the preceding two problems, we come to nothing in trying to
derive the explicit formula for a. The parameter a should be recovered
from the equation
C3) (sin^ ah — Ji^a^ a^) tgal = h (cj -|- cr^) sin ah ,
but the eigenvalues are expressed, as usual, by the formulae
C4) ^'^ = ^'^'''^' ^ = 0,1,2,.,, ,iV.
The eigenfunctions are determined within a constant by
C5) /ij;(x) = cos ttj; {l~~x)+ ^.^ J ^ sin a^ {I ~~ x).

110 Basic Concepts of the Theory of Difference Schemes
4. Difference analogs of the embedding theorems. In the estimation of
various properties of difference schemes such as stability, convergence, etc. we
shall need yet inequalities corresponding to the simplest Sobolev
embedding theorems. In this respect the appropriate re.sults have been obtained
with the following lemmas.
Lemma 1 For any grid function y(x) defined on the grid
'^h ~ { ^i — *'^> 0 < J < A'', Sq = 0, a;„ = 1 }
and vanisfiing at the points x = 0 and x — I the inequality holds:
C6) l|y||^<il|yjl,
where \\y\\^ = max^^g^^^ | y(a;) | and ||y_]| = (y_, y-Y'"^.
Proof The function y(x) given on the grid w^^ can be expressed in the form
of the identity
C7) i/(x) = {l-x)y''{x) + xi/(x).
With the assigned values j/@) = j/(l) = 0, one can write down
yHx)=[j: y^.ix')h or y^x) = [ E vA^') h
\x' = h J \x' —x-\-h
Upon substituting these equalities into C7) we find that
y\^) = A-^) f E hy^[x')\ ^x[ -t hv^(x')] .
\x-' = h J \x' = x-\-h J
Let us estimate the sums on the right-hand side of the preceding relation
with the aid of the Cauchy-Bunyakovskii inequality:
X s: 11
y\x)<{\~x)Y,hY,y%x')h^x ^ h ^ yl{x')h
x' — h x' = h x' =x-\-h x' = x-\-h
1
= x{l~x)Y^ ylix')h = xil-x)\\y^]\'\
x'—h
where y? = (j/-.)^' The maximum of the expression x (l — x) on the segment
[0, 1] is attained at the point a; = I and equals ^. Therefore,
and, consequently,

Mathematical apparatus in the theory of difference schemes 111
Remark 1 Lemma 1 remains valid on an arbitrary non-equidistant grid
Remark 2 In the sequel we will also use an inequality of the type C6) for
segments of arbitrary length /. Such an inequality can be derived from C6)
by merely substituting x' = Ix. Then x' varies on the interval @, /) and
y^' "=" y^/ ^' h' = hi.
By inserting y_ = y_, I and h = h' /1 in B4) we arrive at the chain of the
relations
\\y,]f=f:(y.')'i'i-^h' = i f:(y,')'h' = i\\y,,]f.
i=l i=l
Consequently, on any interval of length /
C8) \y(x)\<\\y\\^<'^Vl\\y.]\.
Remark 3 Inequalities C6) and C8) are valid for all the functions
vanishing at both ends of the interval in view. Being concerned with the function
y(x) vanishing only on the boundary, one can derive another inequality
C9) \\v\l,<^i\\y,]\-
Inequalities C6), C8) and C9) fail to be true, in general, for arbitrary
functions. However, it is plain to show that in this case the inequalities of
alternative forms occur:
l|y||^<2(/||yjr+y;), \\y\\l<2{l\\y^]f + yl).
Lemma 2 For any function y(x) defined on an arbitrary grid
^h = {'-Ci, 2^0 = 0, a;„ = /}
and vanisfiing at the points x = 0 and x = / the inequality holds:
D0) \\y\f<j\\y,]\\ yo = y. = o.
Indeed, it is easy to check that II y II < /||y|L. Substituting this inequality
into C6) yields D0). In the case of an equidistant grid estimate D0) can
be improved and the reader is invited to do it on his/her own.

112 Basic Concepts of the Theory of Difference Schemes
Lemma 3 Any function y(x) defined on the equidistant grid
'^h — {^i = *'^! J = 0, 1, . . ., iV, Xq = 0, a;„ = /}
and vanishing at the points x = 0 and x = / admits the estimates
D1) ^l|y,]r<l|yir<^l|y,]r
Proof We have occasion to use the expansion of y(x) with respect to the
eigenfunctions of problem A4):
A'-l ^ N-1
y(«)= E c,/i(^)(x), c, = (y(x), /i(*)(x)), l|y||'= E c^
k=l k=i
By the first Green formula (8) we thus have
D2) (-Ay, y) = ||yJ|^
where Ay = j/_^, || y_]f = A, (j/_) ] .
By definition, A jj.^''^ = —A^./i^*') and, therefore,
N-i
-Ay= E c,X,f,('\x).
k = l
Let us substitute this expression into D2) and take into account that { /jS''' }
is an orthonormal system. As a final result we obtain
\\yJ' = ~-(^y,y)^'Y: Kcl
k = l
We deduce from here that Aj || y |p < \\y^]f < A^_j || y |p, where
4.2"'^ 4 9 7r/i
The next goal of our studies is to construct a lower bound for Aj with
respect to a = irh / B1);
2 / ■ \ 2
TT / sm a ^
^' = ¥
Since h < 0.5/, the quantity a is varying on the half-interval @, 7r/4],
It is easy to check that the minimum of the function for a £ @, tt/4] is
attained at the point a = 7r/4, that is, Aj(/i) attains its minimum in the
case where h — 0.5/. This serves to motivate that Aj > 8//^. Taking into
account also that A^f_ < 4//i^, we come to D1) and finish the proof of
the lemma.

Mathematical apparatus in ttie ttieory of difference schemes 113
5. The method of energy inequalities. One of the general and very effective
ways of constructing a priori estimates is the method of energy inequalities.
We bring several examples illustrating how to use this method in deriving
a priori estimates in difference problems and find, for instance, the rate of
convergence of a difference scheme on the basis of these estimates.
In this section we consider the simplest model problem
D3) w"(x) + /(x) = 0, 0<x<l, w@) = w(I) = 0.
Example 1 Assuming that an equidistant grid Cof^ is given on the segment
[0, 1], we now consider the difference approximation of problem D3)
D4) y_^ + /(x)=0, kGcj;,, y„ = y^=:0.
Multipling equation D4) by hy and summing up the resulting equality over
the grid nodes of tj;j, we eventually get
N-l N-1
D5) E {y,J.y^h+ E fi^Mh = o.
1=1 8=1
We now rewrite D5) in terms of inner products, whose use permits us to
reduce it to the following one:
D6) (%,,y) + (/, y) = 0,
Via transform of the first summand in D6) by the Green difference formula
(8") we find that
D7) -{y,,y,] + {f,y) = 0 or ||yJP = (/, y).
The estimation of the inner product (/, y) can be done using the Cauchy-
BunyakowskiT inequality A2): | (/, y)| < ||/|| ■ ||y||. By Lemma 3,
lly||<l|y,]|/V8.
Putting these together with D7) we deduce in agreement with Lemma 1
that
l|y,]|<ll/ll/V8.
We finally get one possible a priori estimate for the solution of problem
D4):
D8) ||y|L< 11/11/D72).

114 Basic Concepts of the Theory of Difference Schemes
This inequality will be aimed at estimating the rate of convergence of
scheme D4). We write beforehand the equation for the accuracy of scheme
D4): z = y — w, where u is a solution of problem D3) and y is a solution
of the difference problem D4). Upon substituting y = z + w into D4) it is
plain to set up the problem for z:
D9) z_^. + ^{x) = 0, x&uj^, z,=z^ = 0.
Here ipi^) = W- + /(x) is the approximation error of scheme D4). For all
sufficiently smooth functions u(x) it is well-known that il){x) is a quantity
of order 0{h'^)^ thus causing the same type of the problem for the function
z{x) as occured for the function y{x). Because of this fact, estimate D8) is
still valid for z{x):
E0) II z 11^ < II ^11 7D^/2).
However, ^ = 0{h'^) and, consequently, || z H^,, = || y — u ||^, < M /i^, where
M is a positive constant independent of step /i, In agreement with the above
definitions (see Section 1) estimate E0) provides the uniform convergence
of a solution of the difference problem D4) to a solution of the differential
problem D3) with the rate Oih"^).
So far we have established an estimate for the rate of convergence in
a very simple problem. It is possible to obtain a similar result for this
problem by means of several other methods that might be even much more
simpler. However, the indisputable merit of the well-developed method
of energy inequalities is its universal applicability: it can be translated
without essential changes to the multidimensional case, the case of variable
coefficients, difference schemes for parabolic and hyperbolic equations and
other situations.
Let us show, for example, that this method leads without any
difficulties to the desired result for the case of a non-equidistant grid.
Example 2 Let a non-equidistant grid ui/^ be given on the segment [0, 1].
On this grid problem D3) can be approximated as follows:
E1) y_.,j.-F/(x) = 0, xetof^, y(,=y_^ = 0,
(for the notations see Section 1.3, Example 1).
For problem E1) one can derive an a priori estimate of the same type
as estimate D8) for problem D4). However, in this case such an estimate
fails to provide with a quite reliable idea on the speed of convergence of

Mathematical apparatus in the theory of difference schemes 115
scheme E1). We have shown in Section 1.3 that the local approximation
error ip = m_. + / of scheme E1) is a quantity of order O(h^) and
E2) U\\^<Mh^^^.
Estimate E2) indicates an effective reduction of the speed of
convergence of Scheme E1) on the non-equidistant grid tj/, in comparison with
scheme D4) on the equidistant grid. However, as we have stated above, if
the error of approximation is evaluated not in the grid norm of the spaces
C or Lj! but in a specially constructed negative norm || ■ ILi-,, then the
error of approximation on any non-equidistant grid will be of the same order
0{h'^). Namely, the negative norm
\V\
l(-i) -
N-l /N-l
i=l \ k=i
1/2
= 0(h')
is good enough for our purposes. What has been said above imphes that in
the further derivation of a priori estimates for problem E1) the right-hand
side should be evaluated in the negative norm || • |L .y
Let us obtain this a priori estimate by multiplying equation E1) by
j/j/ij and summing over all grid nodes ofilii^. In terms of the inner products
the resulting expression can be written as
E3) (y,,,y)* + (/, y)* = 0.
Via transform of the first summand in E3) by the Green difference formula
(8) it is not difficult to establish the relation
E4) (?/,,yJ = (/,?/)*.
As we will see later, it will be sensible to deal with the fimction ri(x) specified
by the relations
E5) V,i = f^, i = l,2,...,yV-l, ,y„ = 0.
vAf-l
While solving problem E5) it reduces to —'ri(x^) = J2k=i fk^k-
The inner product on the right-hand side of E4) is modified by the
summation by parts formula G) into
(/, y)* = ('?£., y)* = -(?y, yJ ■

116 Basic Concepts of the Theory of Difference Schemes
On account of the Cauchy-Bunyakovskii inequality,
l(/, y)J= l(^, y,)l<lhll ■ l|yj|.
Substituting this estimate into E4) and eliminating ||y-]| from both sides
of the resulting inequality, we arrive at the relations
N-: /N~i
8 = 1 V k=i
1/2
I/Il(-1)-
l|yjl<lhll =
By Lemma 1,
lly|lc<ll2/JI/2
and, consequently, we derive the relation
E6) l|y|lc<lll/ll(-i).
thereby justifying the desired estimate. Further development is connected
with the accurate account of the error z = y — w, where y is a solution of
problem E1) and u is a solution of the original differential problem D3).
Upon substituting y = z + u into E1) the problem arises for the error z:
E7) z_^ + ?/) = 0 , X edi^, Zg= z^ = 0.
Applying estimate E6) to problem E7) yields
IUIIc<|ll'All(_i).
However, we have stated in Section 1.3 that || V'Ilc_i-| "£ M K^^ where h =
maxi<j<Af/ij. Therefore, scheme E1) on an arbitrary non-equidistant grid
Cjf^ converges in the space C with the rate 0{h'^).
2.5 DIFFERENCE SCHEMES AS OPERATOR EQUATIONS.
GENERAL FORMULATIONS
Difference schemes for the simplest differential equations have been
considered in preceding sections, the basic topics in the theory of difference
Schemes have been introduced for them as well as all the tricks and turns
available for investigating stability and convergence of such schemes have
been demonstrated with a great success.
In this section a unified interpretation of difference equations as
operator equations in an abstract space is carried out and, after this, the
corresponding definitions of approximation, stability and convergence are
presented. This approach is quite applicable in mathematical physics for
stationary problems.

Difference schemes as operator equations. General formulations 117
1, Difference schemes as operator equations. After replacing differential
equations by difference equations on a certain grid loj^ we obtain a system of
linear algebraic equations that can be written in matrix form. The outcome
of this is
A)
iiY = '^ ,
where il is a square A'' X A''-matrix, Y = (j/j, j/ji ■ • ■ j Vn) '^ ^^'^ vector of
unknowns and $ = (i^j, ip^^ . . ., ip^) is & known right-hand side including
the right-hand sides of boundary conditions. With every matrix il one
can associate a linear operator A acting from R^ into i?^. With this
correspondence in view, equation A) takes the form
B)
^y = ^:
where the unknown vector y is .sought, while the right-hand side ip G R^^ is
a given vector. The operator A maps onto itself the space of grid functions
defined on loj^ and satisfying the homogeneous boundary conditions. Several
examples can add interest and aid in understanding.
Example 1. The first boundary-value problem. Given on the segment
[0, 1] an equidistant grid uij^ = {x^ =: i h , i = 0, I, . . ., N , h = I / N } ,
let us look for a solution of the first boundary-value problem
C) Ay= y^^ =-/(x), 0<x = ih<l
Vo
C')
p ivi-i -'^yi + Vi+i) = -/i
i = 1,2,... ,yv-i,
; Wo
As the first step towards the solution of this problem, we form the vector
^ = (j/i) ^2' ■ ■ ■' Vn-i)' ™c^king it possible to rewrite equation C) in the
form A) with the (TV - 1) x (TV - l)-matrix
U =
/ 2 -1 0 0 ... 0 0\
-1 2 -1 0 ... 0 0
0-12-1 ... 0 0
\ 0 0 0 0 ... -1 2/

118 Basic Concepts of the Theory of Difference Schemes
The right-hand side vector $ = ((p^, (p^, ... , <Pj^_i) includes the right-hand
sides of the boundary conditions C)
u u
9i=h, ^ = 2, ...,yv-2, ^j = /^ + ^, ^^_^ = /^_i + ^,
so that ipi may differ from /, only at the grid nodes adjacent to the boundary
points, that is, for i = 1 and j = A'' — 1.
The matrix il specifies an operator A = —A, carrying a grid function
y{xi), that is, a vector of the (A'' — l)-dimensional space into a vector of
the same space (into the grid function (—A j/)j). The operator A coincides
with the operator A on all grid functions vanishing at the boundary nodes
(for i = 0 and ? = TV), so that (A y), = (A y)^ for i = 2, 3, ..., TV - 2 and
D) (Ay)j = -^ , (AyL_, = -^ .
Let 0;j be the set of grid functions defined at the inner nodes of the
grid tj;j. The set so constructed is certainly linear. Once equipped with the
inner product (y, v) = "^Zi-i Vi ""i ^ ^'^'^ associated norm \\y\\ = \/{y, y),
the space 0;^ becomes a normed vector space. The above operator A is
linear and maps 0;^ into itself, meaning that the domain and range of the
operator A coincide with the entire space 0;^.
0
Let 0^ be the space of all grid functions defined for all nodes of the
grid LOj^ and vanishing at the boundary nodes, that is, for x £ 7;^. Then
0
the operator A may be treated as an operator from 0 into 0;^. Obviously,
A = Aj^, y = yi^ and ip = ip^^ depend solely on the grid step h. Just for
this reason the subject of subsequent discussions is a family of equations
depending on the parameter h rather than the single equation B). A family
of such equations constitutes what is called an operator-difference scheme
(see Section 1.2).
While solving the operator equations B) we establish the basic
properties of the operator A such as .self-adjointness, positive definiteness, the
lower bound of the operator and its norm and more. The operator A
constructed in Example 1 will be frequently encountered in the sequel. Before
stating the main results, will be sensible to list its basic properties.
The operator A is self-adjoint, that is, {Ay, v) = (y, Av) for any
y, -y e 0;^. As a matter of fact, {Ay, v) — {—Ay, v). Making use of the
second Green formula (Section 3) and taking into account that A coincides
with A on the set of grid functions vanishing at the boundary nodes, we
establish the relation (Ay, v) = (y, Av), implying that A = A*.

Difference schemes as operator equations. General formulations 119
The operator A is positive definite, that is, (Ay, y) > 8|| y|p. This is
a consequence of Lemma 3 in Section 3.4.
The norm of the operator A is equal to
E) IMII=^cos^^ < -.
Indeed, the norm of a self-adjoint positive operator in a finite-dimensional
space 0;j is equal to its greatest eigenvalue: \\A\\ ~ A^_.. In that case, in
complete agreement with the results of Section 4.2, we might have
A^_i = S2 cos —
and, therefore, formula E) holds true. In addition,
{Ay,y)<\\A\\ ■ \\yf .
The next quantity we will introduce is an operator with the values
F) Ay = —(ay^)^ + dy , 0 < Cj < a < Cj, 0 < rf < Cj .
Due to the second Green formula it is self-adjoint. In turn, the first Green
formula assures us of the validity of the relation
G) iAy,y) = ia,yl] + {dy,y),
where y'i = (j/-)^. This implies that
(8) {Ay,y)>c, A, y^] = cj| yjp > 8cJ| y |p ,
meaning that A is positive definite. It seems clear from formulae G) and
(8) that its norm satisfies the estimate || yl || < 4 Cj //i^ -|- Cj.
Remark Here the operator A = Af^ depends only on the grid step h playing
for the moment the role of parameter. When ui/^ = {.-?;, G [0, 1], J =
0, I, .. ., N, Xg = 0, a;„ = 1 } is a non-equidistant grid, its step h^ =
Xj — Xj^j itself becomes a grid function or an A''-dimensional vector h =
(/ij, /ij, . .., /i„). The operator A admits the form A = —A, where

120
Basic Concepts of the Theory of Difference Schemes
In this case we write, as before, A = Af^ with further reference that /i is a
vector of the dimension A'', that is, an element of the space 0^ consisting
of all functions defined on the grid
c^+ = {x,. e(o, i],i = i,2,... ,yv, x„ = i}.
Example 2. The third boundary-value problem. Given the same
grid w^j as in Example 1, we now consider the difference boundary-value
problem of the third kind
(9)
0 < X = ih < I,
-Vg N = '^2^^ -A'2
Let 0;j be a set of all functions defined on the grid w^j = { a;,- = j/i, i
0, 1, ..., N }. We begin by specifying the operator A by the relations
/ 1
(j/.r,o-'^1 J/o) = A y,
(Ay),
0.5/1
A y = y^.^.
1
-q:^ (y:.,iv + '^2y«) = A+y, i=N
?: = 1,2,... ,yv- 1,
By merely setting A = —A problem (9) is recast as
A0) Ay^^,
where
r 1
/ij , J = 0 ,
j = 1,2,... ,yv-i.
^i
1
oH'h
/,.
The hnear operator A maps 0;^ onto 0;^. To make our exposition more
transparent, it will be convenient to introduce the inner product
iV-l
[y> v]= E yiVih+\h (ya v^ -^ y„ v^)
i = \
and the associated norm | [y] | = \/[y, y\- The operator A is self-adjoint,
that is, [y, Av] = [v, Ay], where
[y, Av\ = -(y, A ?;) - - /j (yo A" ?; -F y„ A+ ?;)

Difference schemes as operator equations. General formulations 121
and (y, ui) = J2i=i Vi ^i ^- Making use of the Green formula from Section
3 and substituting the relevant expressions for A~ v and A"*" v, we find that
-(y> A?;) = (-Ay, v) - {y^^v^ - y_^^, v^) + {y^ v^^^ - y„ v^^^),
-~{).bh{y^h- v + ij^h_+ v) = -yoiv^^o - en ^o) + y^ (\., Af + (^2 ^n) ■
Since y^ o = '^iVo + 0.5/iA~ y and y_j^ = —a^ y„ — 0.5/i A"*" y, we arrive
at the chain of the relations
[y, Av] = -{Ay, v) -0.5/i(?;o A" y + v^ A+ y) = [Ay, v]
as required.
We are going to show that if Cj > Cj > 0 and cr^ > Cj > 0, then the
operator A is positive definite:
A1) [Ay,y] > ^ IMP.
1 + Cj
With this aim, we follow the .same procedures as we did in the proof of
Lemma 1 in Section 3. Namely, those ideas are connected with attempting
the function y"^(x) in the form
y'{^)= (yo+ t y,{^')A =yl+2y, t y,{x')h+( t y,{^')h)
\ x'— h J -p'~h \x' — h J
and applying then the ^-inequality. The outcome of this is
y2(x)<(l+£)y2+(^l+i) ( E^J/,(^')/i) ■
By the Cauchy-BunyakovskiT inequality,
E y,{^')h] <At y^(^')/i) <^(i.yl].
■■'■=.h J \x' —h J
implying that
i
(x)<(l + £)y2+(^l+l)x(l,y2].

122 Basic Concepts of the Theory of Difference Schemes
Likewise, one can prove another inequality
By the same token,
Putting £ = Cj and using the identity
A2) [Ay,y] = {l,yl] + a,yl^a,yl,
we establish relation A1).
It is plain to deduce for the norm of the operator A the estimate
4
A3) \\M\ ^ Ti A + 2 "^2^) ! where Cj = max (cTj, cTj) .
Indeed,
2
(ygJ'^ Ti(y' + j/'_i)
A'4] < ^IMP,
where | [y] p = J2i:=i^ Vi ^ + k^ivl + vl,)- By virtue of the relations
[Ay,y\<\\A\\ ■ \[y\\\
'^iyo+'^2yl = li^h a, yl + -ha,yl) < ^ | [y] ^
we get from A2) the desired estimate A3),

Difference schemes as operator equations. General formulations 123
Example 3. Non-self-adjoint operators. Let Coj^ = {x^ — ih , i =
0,l,...,yV,/i=l/yV}bea grid on the segment 0 < x < 1 and let the
difference operators
A4) A- y=y^, A+ y = -y.,
0
mapping the set 0;^ of grid functions defined on Cof^ and vanishing for i = 0
and i = N into 0;^, be such that
,^-.^iyilh, i = 1 ,
(A+y), = |
%_i//i, i = yv~i.
These formulae show that the operators A and A"*" can be treated under
such an approach as operators from 0;, into 0;^.
Let (y, v) = X^^_j Hi v^ h be the inner product in Clf^ and Clf^. It seems
clear that the operators A~ and A"*" are mutually adjoint:
A5) ( A" y, -y ) = (J/, A"*" -y ) for any y, -y £ 0;^ .
Indeed, by the summation by parts formula we have —(y, v^) = (i/_, v) if
y = -y = 0 for j = 0 and i = N. This just implies the fact that A" and A"*"
are operators adjoint to each other. We note in passing that
A~ + A+ = -( y^. - y J = -/i A y ,
where A j/ = y_ , that is, A~ + A"*" = —/i A. Therefore, the operators A~
and A"*" are positive definite:
(A-y, y) = (A+y, y) = ^ (-Ay, j/) = ^ II yJP > 4 ft || y |p.
The last inequality holds true on account of Lemma 3 in Section 3.4.
Later we will deal with operators from 0;^ into 0^ of the .structure
A6)
1 1
h ^ h

124 Basic Concepts of the Theory of Difference Schemes
They are mutually adjoint, that is, (R^ y, v) = (y, R^v) and
The useful relation
\\R,y\\' = h-' Y.yl^<'^^~'\\y\\'
i = l
provides the validity of the following estimates:
A7) ll^lli<Ti, 11^112 <
2 2
Since R^ = R*
,2 ^ 1 ^^' .., . . 1 ^ .2 . 1
iR,R,y,y) = \\R,y\\':^j^ E <. ^ < ^^ E <,'^ = ^ II yJP,
yielding
A8) \\R2y\\^<^iRy,y), R = R, + R,.
0
The difference operator R^y = p- j/o = ^{R^ — R2) y, y G 0;^ , is
obviously slcew-symmetric due to the relationships
-^3 - 2 (-^1 ~ -^2 ) " 2 (-^2 - -^1) = --^3
and, consequently, (R^y, J/) = 0. Its norm satisfies the estimate
11^311 < ^(ll«JI + ll^.ll) < ^.
It is not difficult to refine the estimate for || -R3 || with the aid of relations
1 ^-1 1 ^-1 ll?y||2
llyxlP = r^ Eiyi+r-y.-.)'h < ^ E (yf+,+ yf_J/^ < ^ ,
giving II iZ3 II <l//i2.
Non-self-adjoint difference operators appear, for example, in the
approximation of second-order elliptic operators with the first derivatives.
The operator Lu = u"(x) + bu'(x) , x £ [0, 1] , 6 = const, is approximated

Difference schemes as operator equations. General formulations 125
by the difference operators Aj y = y^^ + ^Vx for ^ > 0 or Aj y = j/- , + 6 j/j.
for 6 < 0 with y £ 0;^.
^ 0
Let A y be an operator from 0;^ into 0;^ coinciding with A y for y G 0;^.
The operators A^ = — Aj and A2 = — Aj, acting from 0^ into Clj^, are
positive definite for any h. Indeed,
{A,y,y) = {^y^^,y)-b(y,.,y) = (l + Lhb)\\y^]\',
A9)
(^y, y) = (l-|/z6)||yJP = (l + i/j|6|)||y^,]|2.
We conclude from here that
iA,y,y)>8il+'^h\b\)\\yf
for a = 1, 2.
Be re-ordering A^ - A + bA'^ and A^ = A+ \b\A~, where A = -A,
Ay =: y^^ and || A^*" || < 2 //i, || yi || < 4 //i^, we are led due to the triangle
inequality for the norms to
|MJ|<|M|| + 6||A+|| < A(l+i/j6), 6>0,
B0)
IM2ll<IMIi + 6||A-|| < ^(l+iMH). 6<0.
We note in passing that if we approximate the operator Lu — u" + b u' by
the expression A y = y^^ + 6 y- for 6 > 0, then the member \ — ^hb arises
in place of 1 + |- /i 6 m A9) and thereby the operator —A will be positive
definite only for /i < 2/6.
If w'( X ) is approximated by the central difference derivative u for
any sign 6, we have the operator A^y — —Aj y, where A^y = y^ + by .
This operator A^y = —y^ — h b R^ y is of second-approximation order and
satisfies the relations
(Ay, y)~{ --y^x. y) - ^ M ^3 y. y) = 11 j/j ] P
and
\\A,\\<\\A\\ + h\b\\\R,\\<^{l+h\b\).
Although the complete theory could be recast in the general case, we
confine ourselves to simplest examples. In subsequent chapters the
difference operators approximating elliptic operators (in particular, the Laplace

126 Basic Concepts of the Theory of Difference Schemes
operator) in rectangular domains will be studied by means of similar
methods. If the initial differential operator is self-adjoint and positive definite,
one should construct the difference operator also to possess these
properties in the grid space. This can be achieved, for example, by employing the
balance method (the integro-interpolation method from Chapter 3) or the
variational method in designing difference schemes.
We learn from the examples under consideration that the difference
equations can be treated as operator equations with operators in a finite-
dimensional normed vector space. A feature of such operators is that they
map the entire space into itself as further developments occur.
We proceed to the description of the theory of difference schemes
treated as operator equations, the meaning of which we have discussed
above.
2. Stability of a difference scheme. Let two normed vector spaces .8> and
B)
Bf^ be given with parameter h being a vector of some normed space with
the norm | /i | > 0. In dealing with a linear operator Aj^ with the domain
V(Ai^ ) = Bj^ ' and range TZ(A)^ ) C Bf^ ' we consider the equation
B1) A,y, = ^,, y. esf\ ne^P.
where ipj^ is a given vector. Varying the parameter h, we obtain the set
of solutions { j/;j } to equation B1). We call the operator equation B1)
depending on the parameter h a difference scheme.
Let II • |L , and || ■ \\,^ , be the norms on the spaces BJ^ ' and Bf^ ,
respectively. Scheme B1) is said to be correct or problem B1) is said to be
well-posed if for all sufficiently small \h\ < hg
A) a solution j/;^ of B1) exists and is unique for all ifj^ £ Bf^ ' (scheme
B1) is uniquely solvable),
B) this solution continuously depends on ipj^ and this dependence is
uniform in h (scheme B1) is stable). In other words, there exists a
positive constant M independent of h and (pj^ such that a solution
B)
to equation B1) admits for any ipj^ G Bf^ the estimate
B2) ||yJ|(^^)<M||^J|(,,^).
The meaning of the solvability of scheme B1) is that there exists an inverse
operator A'^ such that
B3) Vh = Al' ^, .

Difference schemes as operator equations. General formulations 127
Stability of the scheme is to be understood as the property that the inverse
A'j^^ acting from BJi into Bj^ is uniformly bounded in h:
B4) IMft Ml < M where M > 0 does not depend on h .
Combination of B3) and B4) gives estimate B2):
l|j//JI(U)<IM;;M|-iinllB.)<^ll^'>llB.)-
That is to say, the meaning of stability of scheme B1) is that a solution B1)
depends continuously on the right-hand side and this dependence is uniform
in the parameter h. This implies that a small change of the right-hand side
results in a small change of the solution. If the scheme is solvable and
stable, it is correct. Note that the uniqueness of the scheme B1) solution
is a consequence of its solvability and stability and, hence, we might get
rid of the uniqueness requirement in condition A). Indeed, assume to the
contrary that there were two solutions to equation B1), say i/j^ and y, j^ j/;^.
By the linearity property of the operator Aj^, their difference z^ = y, -- y^
should satisfy the homogeneous equation
^4a z^ = A^iy^-- y,^ ) = A^y^- A,, y^ = ^^^ - ^^^ = 0 .
Because scheme B1) is stable, inequality B2) holds true and, therefore,
IUJI(U) = 11^/."% 11A.) ^^11 011B.) = °'
implying that y,^ = y^^.
To prove the stability of B1), we need an a priori estimate of the form
B2). A derivation of some a priori estimates for the operator equation B1)
will be carried out in Section 4. A difference scheme A^ y^ = ip^ is said
to be ill-posed if at least one of the conditions (l)-B) we have imposed
above is violated.
Suppose that a solution y^^ of problem B1) exists for any ^pf^ (z Bji , so
that y^'" = Af^ ^pf^.. Since Bj^ ' and Bj^ are finite-dimensional spaces, the
inverse operator A^^ from Bji ' into Bj^ ' is bounded and its norm equals
II AT II = M;j, where M^^ is a positive constant depending on the parameter
h. If the Scheme is stable, there exists a constant M > 0 independent of h
such that M;j < M for all \h\ < h^. The meaning of instability of scheme
B1) and, hence, of its ill-posedness is that M;j ^ oo as | /i | ^ 0, it being
understood that the above constant M does not exist. For an ill-posed
problem only an estimate of the form
B5) ||y||(j^)<Mj|^J|B,)

128 Basic Concepts of the Theory of Difference Schemes
may be true.
Here the meaning of weak stability is that condition B5) should be
valid. In other words, this asserts that there exists a domain for \h\, for
example, 0 < /i« < |/i | < /iq, where B2) is satisfied with constant M,
depending on h^.
The definition of well-posedness and ill-posedness of a scheme is closely
connected with the selection rules for norms || • |L -, and || ■ \\,^ ,. It may
happen that for some choices of these norms estimate B5) is fulfilled, while
for the others estimate B2) is true. It is worth noting here that for the
0
scheme Ay = —y~ = (^ for y G 0;^, as stated in Section 3, estimate B2) is
still valid in the norms
l|j/JI(U) = 11% He = i<^^X^_i \yhi^i)\' M =1,
1/2
11^11B,0
Af-l /Af-1
. i=l \ k~i
3. Convergence and approximation. Let B^^' and B^'^' be normed vector
spaces with norms || • |L> and || • ||,2-., respectively. One assumes, in
addition, that
A) there exist linear operators V[ from B'--^^ into Bj^ and Vf^ from B^'^')
B)
into Bj^ ' known as projectors such that
V'-p u = u^, e B^P if ueB^^^ and T'f ^ / = ^ G 4'^ if/G-B^');
B) the conditions of concordance of norms are satisfied:
..... II ' 1 M||n ^ = II W lin-i > 1™ II ^i J MO ^ - II J \\('>\
B6) .limJlT'l^^^ll^^^^^llHln. ,!™ Jl n V H^.. = 11/I
Given a vector yy^ of the space Bj^ , we study the convergence of { j/;^ }
as I /i I ^ 0 to a fixed element u from B^'^>.
1) A sequence { y^^ }, where y^^ (z Bj^ , is said to be convergent to an element
wGS(^)if
B7) ^limj|y,-7'[^)«||(,^^ = 0.
2) A sequence { J/^ } is said to be convergent to Uj^ G B^^' with the rate
0( I /i I" ), n > 0, or { j/;j } approximates u with accuracy 0( | /i |" ) if for
all sufficiently small \h\ < hg the estimate is valid:
B8) \\y,~u,\\^^^^<M\hr, n,=vl^'Ki,
where M > 0 is a constant independent of h.

Difference schemes as operator equations. General formulations 129
Remark 1 Convergence conditions B7) imply that
B9) ^limj|yj|(,^^ = ||«||(,^.
Indeed, with the aid of the triangle inequality and the relation Uj^ — 'P^.'u
one can write down
11% 11A,^) = WiVh " Wft) + M;JI(u)
<ll%-«/JI(U) + ll«/JI(U)
<
A),
ii%-"ji(u) + iin'«ii(i,)
Passing to the limit as | /i | ^ 0 and taking into account B7) and B6), we
obtain
|limj|yj|(i,)<ll«ll(i).
A similar reasoning yields
II « 11A) < |[™JIJ/JIA,.)'
giving B9).
Remark 2 The sequence { j/;j } can converge only to a single element
u e B'^^y On the contrary, let there exist two limit elements w, w £ B'^^\
u ^ ii ({ y^ } converges to each of them), that is.
We are going to show that u ^ u by considering the difference
"ft " U;, = ( W;, - y J + ( y;, - U J
and appealing to the triangle inequality
II "ft " ^h 11A,) < II yft " "ft ll(u) + II Vh " Uft ll(u) ■
Passing to the limit as | /i | ^ 0 and taking into account the convergence of
y^ to both u and u and the norm concordance condition B6), we deduce
that II u — u |L> = 0, implying that w = u.

130 Basic Concepts of the Theory of Difference Schemes
Let iiy^ be a solution of problem B1). We say that
1) scheme B1) converges if there exists an element u £ B^^^ such that the
limit relation B7) occurs;
2) the Scheme is of accuracy 0( | /i P) if there exists an element u ^ B'^^^
such that for | /i | < /iq relation B8) occurs.
Let us define the notion of the approximation error on an element
u e B'^^K To this end, we must write down the equation for the difference
■^h ~ Vh '" ''^h- Substitution of y^ = z;^ + UJ^ into B1) gives
C0) A,,Zh = iPh, V'ft = ^ft " ^ft Wft, V*;, e4^\
In this context, we call the right-hand side ipj^ = ipii{u) depending on the
choice of an element u from B^^^ the error of cipproximation on the element
u e B^^^ for scheme B1). Obviously, il))^{u) is the residual emerging as a
result of replacing j/;^ by the element W;^ G Vj^ u in B1).
We say that
1) scheme B1) generates an approximation on an element u G B^^' if
C1) ^ lim^ II ^^(u ) 11B,) = I lim^ || ^n - A^h 11B,.) = 0 '
2) scheme B1) is of the nth approximation order on an element u G B^^' if
for sufficiently small \h\ < hg
C2) ||^,(«)||B,)<M|M" or ||^ft(«)||B,) = 0(|M"),
where M is a positive constant independent of /i, h > 0.
Our next goal is to establish direct links between stability,
approximation on an element u G B^^^ and convergence to u for scheme B1). If
scheme B1) is correct, then problem C0) for Z)^ is well-posed. Because of
this fact, its solution obeys the estimate
C3) ' ||zJ|(i^)<M||^J|B,).
In this respect, we obtain the profound result with the following
theorem.
Theorem 1 If scheme B1) is correct and generates an approximation on
an element u G B^^\ then it converges. More precisely, a solution y^^
of problem B1) converges to this element u G S*-^^ as |/i | ^ 0 and, in
addition, the order of accuracy of scheme B1) coincides with the order

Difference schemes as operator equations. General formulations 131
of approximation, that is, approximation and correctness (stability) of a
scheme imply its convergence.
Until now we spoke about convergence of a scheme and approximation
on a fixed element u of the space B'--^\ However, if u belongs to the domain
of a linear operator A from B'^^^ into B^'^\ then Au = f, f (z B^'^\ Hence,
u can be adopted as a solution to the equation
C4) Au = f, u^B^^\ feB<~'^\
and, therefore, one can speak about the approximation of this equation by
a difference scheme. The only reason we did not introduce equation C4) for
it is that no restriction on the operator A was imposed in the definitions.
Everywhere we have dealt only with an element u £ B^^'.
However, if w is a solution to some equation like C4), then one can
speak, as usual, about the approximation of equation C4) by scheme B1) on
a solution of equation C4), about the convergence to a solution of equation
C4), etc.
Once started with the notion of approximation to an element / from
.8"-' by a set { fii} from {.8^ }, we can speak about the approximation of
/ by the elements ip)^ as well as about the approximation of an operator A
by A^:
1) ip)^ approximates / with order h if
C5) ||^,--pfVl|B,) = 0(|M");
2) an operator A)^ approximates an operator A with order n if for any
C6) \\A,u,^vP{Au)\\^^^^ = \\A,{vl;'\t)^vl;'\Au)\\^^^^
-o(|M").
Obviously, if conditions C5) and C6) are satisfied, then scheme B1)
is of the nth order of approximation on the solution u to equation C4),
With the relation V)^'{f — Au) = Q in view,
^hiu) = 'Ph - A^u^
= {^k-VPf)^{A,{vt'\)^VP{Au)),
lh^/„(«) 11B,.) <ll^''."n''VllB,) + iM/,.(n^'^«)-n^'^(^«) 11B.)
<M|/jr,

132 Basic Concepts of the Theory of Difference Schemes
provided conditions C5)-C6) hold. From such reasoning it seems clear that
the converse assertion, in general, fails to be true, that is, relations C2) do
not imply conditions C5) and C6).
Let us stress once again that in order to estimate the order of accuracy
of a Scheme, it is necessary to estimate its order of accuracy only on a
solution of the original problem.
So far we have always preassumed that the operator Aj^ is linear
(scheme B1) is linear). If Aj^ is nonlinear (scheme B1) is nonlinear), the
preceding arguments need minor changes only in the concept of stability.
In dealing with a nonlinear scheme
B1*) A,y, = ^,, ^h^^i'^l
where yj^ is a solution and y^ is a solution with the right-hand side (p)^ G
B\ , Scheme B1*) is said to be stable if there are positive constants /Iq > 0
and M > 0 independent of the parameter h and disregarding to the choice
of (fij^ and (fij^ such that for \h\ < h^ the inequality
B2*) ||%-yJ|(i^)<M||^,-^J|B,)
B)
is satisfied for any fh^fh G B\ '. We note in passing that for a linear
Scheme with i^^ = 0 and y^ ::= 0 this implies B2), All of the above
definitions of approximation and convergence remain valid. Theorem 1 is also
true. However, its proof follows another reasoning: instead of C0) it is
more cinvenient to write down
where ^^^ is the approximation error (residual) on the element u G &--^^.
Denote by y^^ a solution of equation B1*), y^^ = u^^ and make use of the
stability condition B2*). We obtain in this direction estimate C3)
lUII(U) = ll%-«JI(U) <^ II ^ft 11B.) = ^ ll^ft-^ft«JIB.)'
thereby completing the proof of Theorem 1 for a nonlinear scheme as well.
4. Some a priori estimates. We now consider several simplest a priori
estimates for a solution to equation B1), the form of which depends on the
subsidiary information on the operator of a scheme. These estimates are
typical for difference elliptic problems.
For simplicity of writing we will omit the subscript h if this does not
cause an ambiguity. The equation we must solve is of the form
C7) Ay = ^,

Difference schemes as operator equations. General formulations 133
where A is a linear bounded operator defined in a real Hilbert space H, (p
is given and y is sought in the space H.
We will assume that problem C7) is solvable for any right-hand sides
f (^ H: there exists an operator A""^ with the domain V{A~^ ) = H. All
the constants below are supposed to be independent of h. In what follows
the space H is equipped with an inner product ( , ) and associated norm
II .T II = s/i X, x). The writing A = A* > 0 means that A is a self-adjoint
positive operator. Set
^J
-, = v/(^-V, ^), lli/IL =x/(Aj/, J/), A^A*>0.
An a priori estimate depends on the nature of the subsidiary information
on the operator A.
Let _ff be a finite-dimensional space.
1) Consider first the simplest case when A is a non-self-adjoint positive
definite operator:
{38)A>6E, 6>0 or {Ay,y)>6\\y\f for any yeH,
where E is the identity operator. Then the inverse A~'^ is bounded in norm
by the constant 1/6:
C9) \\A-'\\<].
Indeed,
0<{Ay,y)^6\\y\\':={A-Kx,x)~-6\\A-'x\\'
< ||A-^x|| ■||.t||-E||A-^t||2
= 5||A-^x||('^||x||-||A-^i-||V x = Ay,
implying that 11 A ^ x 11 < E ^ 11 x 11 or 11 A ^ 11 < i5 ^. Since y = A ^ i^ and
II J/ II ^ II ^"^ II • II 'i^ 111 *'he solution to equation C7) admits the estimate
D0) l|j/|l<7lkl| for A>6E, 6>0.
0
2) The precise estimate appears to be useful:
||j/iL=||^|L^. for A = A*>0.

134 Basic Concepts of the Theory of Difference Schemes
To make sure of it, we take the inner product of C7) and y = A~^(p:
{Ay, y) = {^, y) = {f, A'V)
or ||J/||^ = 11^11^-..
3) If A > 0, then
D1) ||j/|U„<||^IU-i, A, = ^iA + A*).
Furthermore, taking the inner product of C7) and y we obtain
D2) {Ay,y) = {^,y), A = A, + A, ,
where Ag = t: {A + A* ) is the symmetric part and Ag = t: {A — A* ) is
the skew-symmetric part of the operator A. This provides the sufficient
background for the relation (Ay,y) = (Agy,y) due to the fact that
( Aj J/, J/) = 0. As far as Ag > 0, the inverse operator A~^ exists,
because the space H is finite-dimensional*^ and, therefore, by the generalized
Cauchy-Bunyakovskii inequality one can write down
k- -II^IUo
i^,y) = {A;'^,A,y)<\\^\
Substitution of this inequality into D2) leads to the relations
iA,y,y) = \\y\\l<\\yh-\\^h--
^0
from which the desired estimate D1) immediately follows.
4)l{ A>6 E with E > 0, then
D3) \\y\\. < ^ll^ll
1
76
To prove this fact, we make use of estimate D1) and the inequality
which is a consequence of
A,>6E and || ^ jj^ _, = ( A'V, ^) < IM^Ml ' II ^ iP < 7 II ^ I
*'If H is an infinite-dimensional space, one should require instead of /i > 0 that
A> 6 E with 6 > 0.

Difference schemes as operator equations. General formulations 135
5) Assuming A to be a non-self-adjoint operator subject to the inequality
0 0 0
A > J A with 7 > 0 and A* = A > 0, we derive for a solution to equation
C7) the estimate
D4) ||y||o < ^IIHI = _..
The identity {Ay,y) = (ip,y) yields the chain of the relations
7{Ay,y)<{Ay,y) = {^,y)<\\^\\o ■ \\y\\o
A A
thereby providing the validity of D4),
6) For a solution to equation C7) we thus have
D5) \\Ay\\ < i||^||
7
under the following conditions;
A* = A>yA, A = A*>0, AA = AA, y>0.
It suffices to show that ||Aj/|| > 7||Aj/|| and to involve in further reasoning
0 0
equation C7), giving ||Aj/|| = \\<p\\. The conditions A>jA and AA =
0
A A together imply the chain of the relations
\\Ay\\^ = {Ay,Ay) = {A{A'/'y),{A'/'y))
>j{^A{A'/'y),{A'/'y))=j{AAy,y)
= j{A{A'/'y),iA'/'y))>j'{A{A'/\j),{A'/'y))
= 7'\\Ay\\',
0
meaning || Ay|| > 7||Ay||. Here we exploit the fact that the operators A
0 0
and A-^/^ as well as the operators A and A^^"^ commute with each other.

136 Basic Concepts of the Theory of Difference Schemes
5. Negative norms. In a priori estimates D1) and D2) we were dealing with
the negative norm || ip |L-i. Just for this reason the question of computing
negative norms for the simplest operators will be of great importance. First
of all we quote a result of simple observations:
D6) IklU-. < ^ll^lll-i if /i>7/i,
0 0
where A = A* > 0 and A = A* > 0. This follows from the equivalence of
the operator inequalities
D7) A>7A and \-^>-iA-^.
Our first goal is to prove the equivalence between the inequalities Q > 0
and L*Q L > 0, where Q, L: H <-^ H and L"^ exists. Indeed,
{L*QLy,y) = {QLy,Ly)={Qv,v),
0 0
where v — Ly and y = L~^v. Accepting A>7AorE = A — 7A>0 and
0
setting L — L* — A~^''^, we obtain C - j E > 0, where
C'= A-^f^AA-^f^ = C* > 0.
Since the operator C~^'^ = {C~^'''^)* > 0 does exist, the inequality C —
7 J? > 0 is equivalent to
C-^f^{C-jE)C-^f^ = E-jC-^ = E~jA^f^A-^A^f^ >0,
0 0
By merely setting L = L* = A"^!"^ we get A~^ — ^f A~^ > 0, what means
that inequalities D7) are equivalent as required.
Example 4 Consider the first boundary-value problem
Aj/= (aj/ ),, =-(^(xi), Xi-ih, i = 1, 2, , .,, A^ - 1 ,
D8)
/zA^ = l,j/o =J/n = 0, a,->Cj>0, f = 1,2, .,., A,
on the uniform grid ujf^ = {x^ — ih , i = 0, 1, 2, . . ., N , h = 1 / N ]. One
0
assumes, as usual, that H/^ = Q/^ is the space of all grid functions defined

Difference schemes as operator equations. General formulations 137
on LOf^ and vanishing for i = 0 and i = N under the inner product structure
As a first step towards the solution of this problem, the operator A is
defined by the relation Ay = —Ay, where Ay = Ay for j/ G fi, so that
D9) (A J/),: = -( a J/J,,, i = l,2,...,N-l, {Ay), = {Ay)^=0
and the vector if is taken to be if = @,f^, f^, ■■■, ^m^u 0)- The next
step is to recast the problem in view as an operator equation
Ay- f.
It is easy to see from Green's formulae
(j/, (aj/J,) = -(a, (j/j2j
and
(^> (aj/j).;) = (j/> (a^s)^),
0
where y , v ^ VL, that the operator A is self-adjoint and positive definite,
that is, A > E J?, E = 8 Cj. Recall that we have established in Section 3
from Chapter 2 that
{Ay,y) = {a,{y^f]>c,{l,{y^f]>8c,\\yf.
This implies that the inverse A~^ exists and {A~^y = A~^ > 0. More
exactly,
E0) ^^E<A-^<]e.
Let us show that the negative norm ||iic|L-i of operator D9) is repre-
sentable by
^i, / f^ u \2/A'i,\-l
3=1 ' ^i=l ' ^ ^i=l
3-1
E2) 5, = ^/i^, , i = 2,3,...,N, S,=0
k = l

138
Basic Concepts of the Theory of Difference Schemes
All this enables us to derive for this norm the estimate
E3)
1 '-i
k-l
For its proof the right-hand side of equation D8) is expressed in the
form (fi = 5'j, i, i = 1,2,.., , A'' —1, where Si is specified by conditions E2).
The equation {ay.)^ +ip — {ay^ + S)^ =0 implies that O; j/_ i+S^ = c =
const for all i = 1, 2,.. . , A^. Whence y^ — j/j-_j = h[c~ Si) / a^. Summing
up over f = 1,2,... , A'' and taking into account that y^ = j/„, we deduce
that
--^Si
2=1
a.
N
E
On the other hand,
W^Wl^, = (A-'p, p) = (y, p) = (y, S,) = ~{S, y^]
= — y^' 1 hv^ . Si.
Substituting y^ . = (^ c — Si) / cii yields
i = l ' i = l '■
N
h
h S.;
N
E^s,- E^ E
as required. Estimate E3) is simple to follow:
1=1 '
iV-l
c
i = l
1 = 1
1
C,
E
k=i
hfk
Example 5 Consider now the third boundary-value problem (9). As in
Example 2 of Section 1 it will be convenient to introduce the space Hf^ = fi;,
of the dimension A'-|-l consisting of all grid functions defined on the uniform
grid W;j = { Xi = i h, i = 0,1/2, ..., N, h = l/JV } under the inner product

Difference schemes as operator equations. General formulations 139
structure [ J/, t;] = ^.^1 Vi Vih + ^ h {yoV^ + y^ v^). Then problem (9)
reduces to an operator equation of the form
Ay = f,
where y and (p are the vectors of the dimension A'' + 1:
y = { J/o) yi ^ • • • ) J/n-1 . 1/n } ^ 'P= \ 77TT /^i. /i. ■ ■ ■ > /n-1 > TTTT /^2
, 0.5 h
and the operator A acts in accordance with the rule
1
0.5/z
E4) {Ay),= {
0.5 h
1
(j/^r.o-cTiJ/o), i = 0,
l<i<N -I
0.5 h ' -^.^
The negative norm of operator E4) is expressed by the collection of formu-
E5) II^IP ^=[A-V,^]
N 1 / N 1
i = l
s = l
1 + 1+1
E6) 51 = 1/1^0, 5, = i/z^o + E /z^j^ , 1 = 2,3,
/t = i
Af-l
S'^+l = H ( ^0 + ^N ) + E /* ^/t
/t=l
and admits the estimate
E7)
fWl-^ < \\\hp,+ t n]\' + ~[i,pr.
k = l
Af
In establishing these relations the grid should be enlarged by recording
two "artifical" points x_j = —h and x^^^ = I -\- h and assigning the values
y( x_j ) = j/_j =0 and j/( x^^j ) = j/„^i = 0. All this enables us to impose
the boundary condition in (9) for i = 0 as follows:
(^1 -yo)-aoiyo -y.i)
= fo
aa = ha^,

140
Basic Concepts of the Theory of Difference Schemes
In just the same way as before, we write down the boundary condition in
(9) for i = N:
a^+i ( J/n+1 - J/n ) - ( J/n - J/n-J
^A
«N+i = ha^
^N = 2 ^~ ■
In such a setting problem (9) turns out to be equivalent to the first
boundary-value problem
E8) ^(aj/_)^ , = ^,, i = 0,l,...,N
y.i = j/n+, = 0:
where (p^ = ipi, i = 1,2,.. . , N - 1, <p„ = ^fo, 'Pn = ^'Pn, «; = !>
i = 1,2, . . . , N and a^ = ha^, a^^j = ha2- If J/ is a solution of problem
(9), then
N-\
A V, p]=[y,f]=\h{yaipa + y^ip^)+ Y. Vifih
N-l N
= E yiPih + h{ya^a + yNV'N)= T. ViPih
i=l s=0
N
= Y.{A-'p\p.h.
s = 0
By applying successively formulae Ei)-E2) to problem E8) and estimate
E3) we arrive at E5) and E7).
Corollary If u^ > Cj > 0, then operator E4) admits the estimate
2
h (o„
2
E9)
ll/ll.^. <
k = l
1
c,
6. Operator equations of divergent type. We now deal with operators of
the special structure known as divergent or conservative operators:
F0)
A = T*ST.
where T, S and T* are linear bounded operators. Operators of this type
are frequently encountered in this book in the approximation of differential
operators having the form Lu = div ( A; grad u). Let H he a vector space

Difference schemes as operator equations. General formulations 141
with the inner product {y,v) and associated norm \\y\\ — \/{y,y) and let
H^ be a vector space with the inner product (y, v] and associated norm
II J/II = \/{y, y]- Let, in addition, T be an operator from H into H^, S he
an operator from H^ into H^ and T* be an operator from H^ into H. Then
the operator A really acts from H into H: A: H <-^ H. The operators T
and T* are mutually adjoint in the following sense;
{Ty,v] = {y,T*v) for all yeH,veH,.
Before going further, we give below several examples illustrating how
to construct factorized operators F0) for the simplest difference schemes.
Example 6 Of our initial concern is the first boundary-value problem
F1) (aj/_),, = ^^, 0<x = ih<l,
a > Cj > 0, l/o = 0) ?/n = 0 ■
In this case H ^ Q/^ i,s the space of all grid functions defined at the inner
nodes of the uniform grid w^j = { x^ = i h , i = 0,1,2,... , A'', h N = 1},
that is, for 0 < i < N and H^ = Q^ is the space of all grid functions defined
at the nodes of the uniform grid w^"J" = {x^ = ih,i=l,2,...,N,hN —
1 }. The operator A: H <-^ H equals Ay = —Ay, where Ay takes the form
(Aj/); = (aj/_),_,, i = 2,5,...,N-2,
02 ( J/2 - J/l ) - fll J/l
(Aj/), =
(Aj/)„_i =
/l2
''^N J/n-1 ~" "^N-l I J/n-1 ~ J/n-2 j
The inner products in the spaces H and H^ are defined by
A'-l
i=l
N
{y,v]=Y. Vi^ih for j/i, Vi e H, .
Under these structures, we specify the operators Ty, T*v and Sv by the
relations
(Tj/), = ^^^^ = j/,_,. i = 2,3, ...,jV^1,

142
Basic Concepts of the Theory of Difference Schemes
so that V = Ty e Hj if y e H;
{T*v\ =
i+i
= 1,2 N ^l
so that T*v e H if V e H,. Finally, {Sv)i = a^ t;, for i = 1,2,... ,N,
that is, Sv Q H^ifv Q H^ and {Sv, v] > Cj||t;|p. From such reasoning it
seems clear that
(STy)^ = Giy.^., i = 2,3, ..., N -1, {ST y)^ - a^yj h ,
(STy)^ = -a^ j/„_, / h,
{T*STy\=~{ay.).., i=l,2,...,N, for % = j/„ = 0 ,
which serve to motivate that the operator of problem F1) can be factorized
in the form F0) so as to have instead of C7)
F2)
T*STy = ip.
The fact that the operators T and T* are mutually adjoint is a corollary to
the summation by parts formula: (j/, v^ ) = —(t;, y_], so that
iy,T*v)=-iv,Ty].
Example 7 The third boundary-value problem comes next:
F3) (ay^)^ = -if, Q< x = ih < 1, a > Cj > 0 ,
CTj > Cj > 0 , cr^ > Cj > 0 .
In this case the operator A is of the form (see Example 2 in Section 1):
F4) (Aj/), = J "(aj/J,, z = l,2,.,.,Af-l,
1
( «~ J/« w + '^2 ?Av ) ) i = N .
0.5/z
The operator F4) can be made of the same type as the preceding operator
F0) if we impose another grid
— I Xq, X-^j2
-1/2)
., x^
-1/2' ■"^nS J ^1-1/2 — {''' 2>'^-

Difference schemes as operator equations. General formulations 143
and work in the space H^ of all grid functions defined on the grid w^ under
the inner product structure {y, v] = ^^^^ ^J/i-i/2 ^i-i/2 + J/N ^w + J/o ^o and
the norm \\y]\ = \/{ y, y]- As before, H is the space of all grid functions
defined on the uniform grid uif^ = {x; = i h , i = 0, 1, . . ., N ,h = \ / N ]
with the inner product (j/, v) = ^^ = 1 J/i Vi h+ ^ h [y^ Vg + y^ v^) and the
associated norm \\y\\ = \/{ y, y). We specify the operators T: H <-^ H^
and T*: H^i-^ H as follows:
(Ti/)o = J/o, (Ty\-i,2 = , , z'=l,2,.,.,Af, {Ty),=^y,
Vi - Vi-
h
1/2 ''O frp* . __ "n "jV-i/a
iT*y)o = -^WTT^> iT*v),^~-
0.5 h ' ^ ^" 0.5 h '
{T*v), = ^'''^"'~'J'-"\ i=l,2,...,Af-l, j/ei/, v^H,.
This provides support for the view that A = T*ST, where the operator
S: H^ H^ i?j is defined by the relations
( 5 J/)o = cTj j/o , E'j/)i_i/2 = a, J/i_i/2, i<i<N,
{Sy)^ ^a^y^ .
Obviously, 5 is a self-adjoint operator in H^ and {Sy, y] > Cj|[j/]p with
constant Cj = min ( a,, cTj , Cr^) . Let us show that the operators T and T*
are mutually adjoint in the following sense: {T*v, y) = [v, T y] for y ^ H,
V (^ Hj. Indeed,
N-l
{T*v,y) = - E {'+1/2-Vi_i/2)yi~-Vo {^1/2-^o)
i=l
N
= E Vi-1/2 ( % - J/i-1 ) + J/o ^0 - J/n t'w
! = 1
We are now in a position to derive some a priory estimates for the equation
Ay = T*STy = (f. Under the natural premise S > c^E, c^^ > 0, we obtain

144 Basic Concepts of the Theory of Difference Schemes
{Ay,y) = {T*STy,y) = {STy,Ty]>c,\\Ty]\\
meaning A > jAg, where Ag = T*T and 7 = Cj. The operator Ag is
self-adjoint:
iA,y,z) = {T*Ty, z) = {y, T*Tz) = {y, A,z).
Therefore, estimate D4) holds true and takes the form
F5) \\Ty]\ < l||(r)-V-]|.
It is worth bearing in mind here that the inverses T~^ and ( T* )~^ do exist.
Indeed,
\\y\\l^ = {A,y,y)^\\Ty]\\
\\p\\l.. = iT-\T*r'f,f) = \\ir)-'f]\'-
Estimate F5) can be simplified in the case when the right-hand side (p of
equation C7) is of the special form p = T*rj and A = T*ST. The inner
product of C7) and y discovers the relationships
(r5Tj/, j/) = (T*77, j/) = (Tj/, 77],
Putting these together with the inequalities
{T*STy,y)>c,\\Ty]\\ ( T j/, 77] < || Tj/] | • |h?]|,
we derive the estimate 11 T j/ ] | < Cj "^ ^ 11 ^/ ] | •
We have restricted ourselves to the simplest examples demonstrating
how the a priori estimates that can be obtained through such an analysis
for the operator equation Ay — p apply equally well to important problems
arising in theory and practice.

Homogeneous Difference Schemes
This chapter presents the theory of homogeneous difference schemes for the
solution of equations with variable coefficients
Lu + f{x) = 0 , Lu — — I k(x) — — g(x) u .
A special attention is being paid to various forms of homogeneous
difference schemes and their approximation and convergence for discontinuous
coefficients k, q and / as well as for non-equidistant grids. Not much is
known in such cases. Later in this chapter we will survey some devices that
can be used to obtain simpler forms and higher orders.
3.1 HOMOGENEOUS SCHEMES FOR SECOND-ORDER EQUATIONS
WITH VARIABLE COEFFICIENTS
1. Introduction. Modern computers permit implementation of highly
accurate difference schemes. Just for this reason, it is unreasonable to develop
difference methods and create high quality software for solving particular
problems. An actual problem consists of constructing difference schemes
capable of describing classes of problems that are determined by a given type
145

146 Homogeneous Difference Schemes
of the governing differential equation, a class of bounda.ry and initial
conditions as well as of forming a functional space comprising the coefficients
of the differential equation. Naturally, such universal difference schemes
should possess the convergence and stability properties on any sequence of
grids and for any original problem from the given class. The requirement
of universality of computational algorithms for solving a class of problems
necessitates imposing the notion of homogeneous difference scheme. By a
homogeneous difference scheme we mean one whose form is
independent of a concrete problem from this class and the choice of a grid. At
all grid nodes the difference equations talce the same form for any problem
from this class. The coefficients of a homogeneous difference scheme are
treated as functionals of the coefficients of the differential equation.
For instance, of great interest are "through" or "continuous"
execution schemes available for solving the diffusion equation with discontinuous
diffusion coefficients by means of the same formulae (software). No
selection of points or lines of discontinuities of the coefficients applies here.
This means that the scheme remains unchanged in a neighborhood of
discontinuities and the computations at all grid nodes can be carried out by
the same formulae without concern of discontinuity or continuity of the
diffusion coefficient.
Homogeneous "through" execution schemes are quite applicable in the
cases where the diffusion coefficient is found a,8 an a.pproximate solution
of other equations. For instance, such schemes are aimed at solving the
equations of gas dynamics in a heat conducting gas when the diffusion
coefficient depends on the density and has discontinuities on the shock
waves.
In the theory of difference schemes with a priinary family of schemes
the coefficients of a homogeneous difference scheme are expressed through
the coefficients of the initial differential equation by means of the so-called
pattern functionals; the arbitrariness in the choice of these functionals
is limited by the requirements of approximation, solvability, etc. There
are various ways of taking care of these restrictions. The availability of a
primary family of homogeneous difference schemes is ensured by a family
of admissible pattern functionals known in advance.
Let us clarify the subject of investigation in a more simpler situation
through the use of difference operators acting on functions of only one
variable x^ = i h, i = 0, ±1, .... One way of proceeding is connected with
the following two steps. A difference operator is defined beforehand on
an integral pattern, that is, on a set of the type OTq = {—m^, —m^ +
1, . . ., —1, 0, 1, ... , TO,}, where m^, 771^ are positive integers. The next
step is the transition to the real grid uif^ = {x^ = ih, i = 0, ±1, . . .}

Homogeneous schemes for second-order equations 147
with step h. Let k{s) be a vector function defined for —m^ < s < m^ and
called the coefficient pattern. In the sequel we are dealing with pattern
functionals
A'^[Hs)], ^m,<j<m,, B"[~k{s)],
which usually depend on the parameter h and are defined for the vector
functions k[s), s G [-""ij, tti^]' By a linear with respect to a grid function
J/'' homogeneous difference scheine is meant (L^ J/''). = 0, where
D'^/).= E A'^[kix, + sh)]y'^{x, + mh) + B''[kix, + sh)].
m = — mi
Omitting the subscript i one can rewrite the preceding as
'2
4*)/= V Al[k{x + sh)]y''{x + mh) + B''[k{x + sh)]
m = — mi
The principal question in the theory of homogeneous difference
schemes is connected with further design of admissible schemes within a primary
family for solving a class of typical problems as wide as possible and
choosing the most efficient ones (in accuracy, volume of computations, etc.).
3.2 CONSERVATIVE SCHEMES
1. An example of the scheme which is divergent in the case of discontinuous
coefficients. We now consider problem A) of Section 2.1 with g = 0 and
/ = 0 incorporated:
A) (A;w')' = 0, 0<x<l, w@) = l, mA) = 0.
As one might expect, the derivative {ku')' should be replaced by ku" + k'u'.
As a first step towards the construction of a second-order approximation,
it will be sensible to carry out the forthcoming substitutions
1,1 ^ I, _ 2-1-1 ~" 2-1 „,/ ., ... _ "j-l-i ~ "■
U ~ Uo =
2h ' ^ 2h
Within these notations, a reasonable form of the difference scheme is
/z2 2h 2h
/r)\ L yi + 1 '^ ill ~ i)l-\ "-j-l-l "-j-l fi-l-1 i)l-\ _ r^
0<i<N, j/o = 1- J/iv = 0,

148 Homogeneous Difference Schemes
Reducing scheme B) to the form D) from Section 2.i we find that
C) a, = ki , hi = ki + ^ , d^. = ^. = 0 ,
whence it follows that scheme B) belongs to family D) from the preceding
section.
Conditions E) and F) from Section 2.1 hold true, since on segments,
where the function k(x) is smooth enough, the relations occur:
a, = A;. - i /i A;; + 0{h^) , b^ = k^^\h k[ + 0{h^),
K«i + ^i) = h - bi - a, = \{k^^, - k^_,) = hk[ + 0{h^) ,
so that fl; > 0 and b^ > 0 for sufficiently small h.
We are going to show that scheme B) is divergent even in the class of
step coefficients
{ k^ , t, < X < I ,
with ^ being an irrational number such that ^ = x^ + Oh, x„ = nh, 0 <
e < 1,
The exact solution of problem A), D) subject to the continuity
conditions is of the form
ri-ftoX, 0<x<^, fto = (x+A-x)^-! ,
E) u[x) = <
L/?o(l-x), i<x<l, Pa = xao, x:^kjk^.
We proceed to solve the difference problem B), D) in the usual way. Since
a^ = hi = fcj for 0 < i < 72 and O; = 6^ = fcj for n -|- 1 < i < A^, equation B)
reduces to j/^_j — 2 j/; -|- ?/;_,_j = 0 for i 7^ n and i ^ n -|- 1, Whence it follows
that
f 1 — a X-; , 0 < X' < x„ ,
F) y^ = J/(a;i)= / / 1
The coefficients a and /? can be most readily recovered from the appropriate
equations with i = n and i = n + 1 incorporated:
&n [/?(!- ^n+i) - (l-"a;„)] +a„a/z = 0,
G)

Conservative schemes 149
From C) and D) we find that a„ = ^ {5k^ — k^), a„_,_j = \{k^ + SA;^),
6^ = i (Sfcj + fcj) and 6„_|_j = ^{5k^ — k^). Solving equations G) with respect
to a and /? and recalling the expressions x„ = ^ — 6h and x„_,_j = ^ + {l — 6)h,
we determine the values ascribed to the parameters
P = fj, a ,
(8)
1
/i =
/i+(l-/i)e+/i(A-(?-(l-(?)/i) '
3 + X , , 5>ir — 1
A,
5 — X
The passage to the limit as h ^ 0 yields
lim a = a„ , lim /? = /?„,
where
(9) fto = (/i+A "/i),^) , Pa^tJ'^o-
Via the linear interpolation we extend the functions F) on the whole
segment 0 < X < 1. Under such an approach we have at our disposal a new
function y{x, h), x Q [0, 1], which coincides with y^ at the grid nodes x^ ~ ih
and possesses the limiting function as /z ^ 0:
fl — iSoX, 0<.'!;<(J,
A0) u{x) = hm y{x, h) = (
h^o I fj^{^l -- x), ^ <x <l .
Comparison of u(x) with the exact solution u{x) specified by E), where the
coefficients a^, Pg are determined by formula (9), shows that u(x) = u{x)
for (Xg = ag and /?o = Pg. But it is possible only if x = 1 or k^ = k^.
This provides support for the view that, as h ^ 0, solution F) of the
difference problem-B), E) approaches the function u{x) other than the
exact solution u{x) of problem A) in the case k^ ^ fcj- Due to this fact
scheme B) is divergent.
Before stating the main results, it will be sensible to clarify a
physical sense of the function u{x), which solves problem A) subject to the
conditions [u] = 0 and [k u'] = —cig [n — x) k^ = q at the point x = ^.
Here q stands for the capacity of a point heat source (sink) at the point
X = (J. Being dependent on x, the quantity q varies very widely.
Specifically, q -^ ±oo as X ^ 5 ± 0. Thus, the physical reason for the convergence
of scheme B) is that the heat balance (the conservation law of heat) is

150 Homogeneous Difference Schemes
violated, thus causing the appearance of the extra heat source (for q < 0)
or sink (for g > 0) at the point x = ^.
We call various schemes, in which conservation laws fail to be true,
nonconservative or disbalanced.
The above example shows that in designing difference schemes it is
very desirable to reproduce the appropriate conservative law on a grid.
The scheme,? with this property are said to be conservative. In .subsequent
sections the general method for constructing conservative .schemes, which
are convergent in the class of discontinuity coefficients, will be appreciated.
Before we undertake the complete description of this method, it is worth
noting two things.
Quite often, in practical implementations without concern of
theoretical estimations of the desired quality of a scheme, the convergence of
the scheme is verified by experiments in which the grids are successively
refined. Sometimes this approach may cause erroneous conclusions on
convergence of the scheme on account of some nearness between a solution of
the difference problem and some limiting function u(x) during the course
of successive grid refinement. In the above example the function u{x) may
deviate, generally speaking, from the solution u(x) of the original problem
as large as we like. That is why the method of successive grid refinement is
employed with caution. Anyhow it cannot exclude theoretical investigations
at least in model problems.
The method of test functions is quite applicable in verifying
convergence and determining the order of accuracy and is stipulated by a proper
choice of the function U{x). .Such a function is free to be chosen in any
convenient way so as to provide the validity of the continuity conditions
at every discontinuity point of coefficients. By inserting it in equation A)
of Section 1 we are led to the right-hand side / = {kU')' — qU and the
boundary values /ij = t/@) and i_i^ = t/(l). The solution of such a problem
relies on scheme D) of .Section 1 and then the difference solution will be
compared with a known function U{x) on various grids.
One more thing is worth noting here. It would be erroneous to think
that every scheme, which is convergent in the case of smooth coefficients,
is obliged to converge in the case of discontinuity coefficients. Further
explorations are connected with a family of schemes converging in the class
of discontinuity coefficients, thus expanding possibilities. Let us stress that
in the sequel we will deal with such schemes only.
2. The integro-interpolational method for constructing homogeneous
difference schemes. Various physical processes (heat conduction or diffusion,
vibrations, gas dynamics, etc.) are well-characterized by some integral

Conservative schemes 151
conservation laws (of heat, mass, momentum, energy, etc.). Usually the
derivation of a differential equation of mathematical physics is on an
integral relation (a balance equation) which expresses such a conservation
law for a small volume. Letting the volume to zero and assuming that the
derivatives involved in the balance equation are continuous, one can write
down the appropriate differential equation.
From a physical point of view, the finite difference method is mostly
based based on the further replacement of a continuous medium by its
discrete model. Adopting those ideas, it is natural to require that the
principal characteristics of a physical process should be in full force. Such
characteristics are certainly conservation laws. Difference schemes, which
express variou,s conservation laws on grids, are said to be conservative or
divergent. For conservative scheme.? the relevant conservative laws in the
entire grid domain (integral conservative laws) do follow as an algebraic
corollary to difference equations.
In this view, it seems reasonable to construct conservative difference
schemes with the aid of balance equations for an elementary volume (cell) of
a grid domain. Integrals and derivatives involved in these equations should
be replaced by approximate difference expressions, thereby completing the
design of a homogeneous difference scheme. The way of obtaining
homogeneous difference schemes is called the integro-interpolational method
or balance method.
In what follows the illustration of this method is concerned with
equation A) from (Section 1) capable of describing the .stationary distribution
of temperature over a homogeneous bar 0 < x < 1. The equation of the
heat balance can be written on the .segment x^^^,^ ^ x < X;_|_j .^ as
"•'i-l-i/a ^'i+-i/2
^ ^ dx.
A1) W;^i/2 - w^i-H/2 + / I{x)dx= / q{x)u{x)
w = —k u ,
where w{x) is the heat flow, q{x) u{x) is the capacity of heat sinks (sources
for g < 0) proportional to the temperature and f{x) is the distribution
density of external heat sources (sinks).
The existence of a heat sink owes a debt to the heat exchange with the
external environment on the lateral surface of the bar. The quantity w^_^,^
is equal to the amount of heat being supplied to the segment x^_^,^ < 2; <
^i+1/2 through the cross-section x = x^^j^j) while w^_^_J/^ refers in a similar
fashion to the heat transfer from this segment through the cut x = x^_^_^j^.
The third member on the left-hand side of A1) reflects the amount of

152 Homogeneous Difference Schemes
heat being emitted on the segment [Xj^j,^ > ^'i+1/7] ^y heat sources with the
distribution density f{x). The integral on the right-hand side of A1) is the
amount of heat being transferred to the external environment by the heat
exchange on the lateral surface.
In order to develop a difference equation from A1), we substitute
linear combinations of the values of u at the grid nodes in place of w and
the integral with ii that can be obtained through the interpolations in some
neighborhood of the node x^. The simplest interpolation gives
u = const = Uj for x^_^j^ < 2; < k^^j,^ ,
A2) / q{x) u{x) dx K hd^ u- , d- — - / g(x) dx ,
where d, is the mean value of the function q{x) on the segment Xj_-^,^ <
X < 2;i_|_i/2 of length h. Upon integrating the equality u' = w/k over the
segment Xi_. < x < X; we arrive at
'■'-i-l
for which the substitution w[x) = w^_^i2 = const on the segment x^_^ <
X < Xj yields
■"j-i ~ '"i ~ ^i-1/2 /
"^i — l
k{x)
From what has been said above it is clear that the approximate value w^_^i2
of the flow is
Uj — ti^_j 1 f dx \
A3) uj,_^^^ = -a,~~~~^-^ = -a,u,_,, a,= ^- J —j .
^'i-l
Here J ' dx/k[x) is the heat resistance of the segment [x^_■^,x^\.
Substituting A2) and A3) into A1) and denoting by j/, the unknown
function, we obtain the conservative difference scheme
,1.x 1 / Vi+i -Vi Vi- Vi-i \ ,
A4) ^ (^ai_^i -^^ cii j - di Vi = -(fi ,

Conservative schemes 153
where
0
O; = a,-
dx
h J k{x) I \ J k{x^+sh)
-1
1/2
A5) d^ = d.= J ,i..+sh)ds,
-1/2
1/2
ip. = ^i= / fi^i + sh) ds .
-1/2
We have written the difference equation A4) at a fixed node x =z x-. With
an arbitrarily chosen node x,; it is plain to derive equation A4) at all inner
nodes of the grid. Since at all the nodes x^, i = 1, 2,. . . , A'' — 1, the
coefficients a^ and h^ are specified by the same formulae A5), scheme A4)-
A5) is treated as a homogeneous conservative scheme. Because of this,
we may omit the subscript i in formulae A4)-A5) and write down an
alternative form of scheme A4):
{ayg)^ -dy= -(fi.
In the general case the coefficient a^ built into the formula for the heat flow
is some functional of the values of k{x) on the segment [x,;_j, x^].
Observe that the conservation law in the entire grid domain W;^ known
as the "integral conservation law" is an algebraic corollary to equation A4)
for any conservative scheme of the form A4) with arbitrary ingredients a,
d and (p. Indeed, with the notation w^.wj = —a^ {y^ — yi_-^)/h for the
difference expression of the heat flow at the point x = x^_-^,^, we can
rearrange A4) as W;„W2 — w^.wj + hip^ — hd^y-, which becomes after
summation over i = 1,2,. .. , N — 1 the difference conservation heat law
within the entire grid domain:
7V-1 7V-1
i=l J=l
meaning the difference approximation of the integral conservation law for
equation A) from Section 1.

154 Homogeneous Difference Schemes
3. Homogeneous conservative schemes. In the preceding section we have
designed the conservative scheme A4) by means of the integro-interpo-
lational method. In the general ca,se the coefficients a, d and (f of scheme
A4) are some functionals of the coefficients k(x), q{x) and f\x) involved in
the differential equation
a{x) = A[k{x + sh)] ,
A6) d{x) = F[q{x + sh)] ,
^{x) = Flfix + sh]].
The domain of the pattern functional A[^(s)] is Q^°^[—l, 1] (function k{s) G
g(")[-l, 1]), while the domain of F[f{s)] is Q(°\-1/2, 1/2]. In other words,
the functional A[A;(s)] (or F[f{s)]) is defined for all piecewise continuous
functions k{s) (or /(s)) given on the segment —1 < s < 1 (or —1/2 < s <
1/2). In trying to recover the coefficient a{x) the intention is to use formula
A6) with k[s) = k{x -\- sh). This is consistent with the passage from the
pattern —1 < s < 1, on which the function k[s) is defined, to the pattern
•T — /z < x' < X -|- /z, so that the function k{x') should provide a possibility
of subsequent calculations of a(x).
At the next stage we consider the homogeneous conservative scheme
A7)
y@) - Uj , j/(l) = u^ , a > C( > 0 , d>0,
for which the coefficients are expressed by formulae A6). Comparison of
conservative schemes A7) or A4) with the three-point scheme of general
form D) from Section 1 shows that scheme A7) is governed and constructed
in accordance with the rule h^ = aj_,_i. We will not pursue analysis of this:
the ideas needed to do so have been covered.
The requirement thai .scheme D) of Section 1 should be conservative
("divergent") is equivalent to being self-adjoint of the appropriate difference
Operator. To make sure of it, we refer to a second-order operator
{Ay)i = Ui j/,_i - q j/j. + h- j/._^i
0 _ _
ill the Space Qh of all grid functions y = {y^} defined on the grid Q/^ and
vanishing on the boundary: yo = Vn ~ 0- The space Q,h is equipped with

Conservative schemes 155
the mner product {y,v) = X];_j ViVih- Being elements of the space Q/^,
0
any functions y,v ^Q,h are subject to the identity:
7V-1 7V-1
E (a; Vi-i - c, % + bi j/i+J v^ = ^ F,_i t;,-,.! - q t;,- + a,._^i v^^^) % .
This provides enough reason to conclude that the condition
(Aj/, v) = (y, Av)
0
holds for arbitrary y,v G Qh if and only if b^ = aj-_,_i, f = 1,. .. , A^ — 1
(see Section 2, Chapter 1). The condition h^ = a^_^_-^ for scheme D) of
Section 1 means that we should have 5 [A;(x-|-s/z)] — A[k(^x + {s + l)h)'\ or
B[k{s)] = A[k{s + 1)] for any k{s) e Q'-°^[-l, !]■ Evidently, it is possible
only in the case when the functional A[A:(s)] is independent of the values
of fe(s) on the segment 0 < s < 1. The same applies equally well to the
functional B[k[s)] and the function k{s) on the segment —1 < s < 0, so
that
a[x ) = A[k{x+ sh)] for - 1 < s < 0 .
Conditions E) of Section 1 relating to the second-order local approximation
for the conservative scheme A7) acquire the form
h
A8) <1±R±<-1 ^ k[x) + 0{h'),
d{x) = q{x) + 0{h^), <p{x) = fix) + 0{h') ,
implying that a{x) ~ k(x)-■^hk'{x) + 0{h'^) or a{x) = k{x - ^ h)+ 0{h'^).
In Section 3.2 the iutegro-mterpolational method was aimed at
constructing the homogeneous conservative scheme A6) with the coefficients
a, d and (f of the special form A5), namely with pattern functionals such
that
/ 0 . -1 1/2
A5') A[k{s)] = (/ ^1 . F[fi^)] = / /» ds.
^-1 ' -1/2
In this case the coefficients a, d and tp are calculated by integrating
the functions k{x), q{x) and f{x) (see A5)).

156 Homogeneous Difference Schemes
In practice it is convenient to employ more simpler formulae for
determination of a, d and (f with using the values of k, q and / at the isolated
points. Usually a pattern consisting of one or two points permits one to
make considerable simplifications:
A9)
«, = ^.-1/2 = H^^ - 0.5/1) {A[k{s)] = fc(-0.5)),
d^ = qi, ^.^fi [F[f{s)]=f{Q))
a, = i (fc, + fc,„J [A[k{s)] = i (^(-1) + ^@)) ,
2k,ki__^ f 1 1 / 1 , 1 \\
d; =
k. + ki^i \{A[Hs)] 2U@)"fc(-l)
Conditions A8) are certainly true for all the schemes we have mentioned
above.
If the coefficient A;(x) is discontinuous at the middle nodal points x =
X;_j,2 and the coefficients q{x) and /(x) are discontinuous at the points x =
Xj, the half-sums of the left and right limiting values have to be substituted
into formulae A9). As a final result we get
«; = I (^(a;i-i/2 -0) + k{xi_^i^ -f 0)) ,
d^ = ^(^(^^ - 0) + ^(^i + 0)) .
^, = i (/(x, - 0) + /(x, + 0)) .
We note in passing that formulae A9) and some others for determination
of the coefficients a, d and tp can be derived through the approximations to
integrals A5) and other members
1 f dx 1 1 f dx 1 / 1 1
h J k{x) k'_^,^ ' h J k{x) 2 \k^ k
4. A primary family of conservative schemes. We spoke above about the
family of the homogeneous conservative schemes A7), whose description is
connected with some class of pattern functionals A[^(s)] and i^[/(s)]. For

Conservative Schemes
157
convenience in analysis, it is supposed that F[f{s)] is a linear nonnegative
functional such that
A) F[cJ,{s) + c,f,{s)] =:c,F[f,{s)] +c,F[f,is)],-l/2<s<l/2,
where Cj and c, are arbitrary constants;
B) F[fis)] >Ofor/(s)>0.
In Spite of the fact that A[^(s)] is usually a nonlinear functional (see scheme
A5')), we may assume for the sake of simplicity that A[A;(s)] is a linear
nonnegative functional and consider, in addition to schemes A6)-A7), those
with the coefficient a(x) still subject to the relationship (cf. A5'))
A6')
a(x)
= A
k{x + sh)
Conditions for the second-order approximation A8) imply some restrictions
on the pattern functional A[I;(s)] and F[/(s)]. In preparation for this,
plain calculations give
^(x) = F [/(x + sh)] = F [fix) + shf'ix)+ Oih^)]
= fix)F[l] + hfXx)F[s]+Oih').
Since f{x) is taken arbitrarily, it follows from the foregoing and A8) that
B0) i^[l] = 1 , F[s] = 0 .
By the same token.
a{x) = A k{x) + s h k'{x) + ^ s^ h^ k"{x) + 0{h^)
= k{x) A[l] + h k'{x) A[s\ + i ]-? k"{x) A[s'] + 0(/z3),
a{x + h) = k{x) A[l] + h k'{x) A[l + s]
+ i/i2fc"(x)A[(l+sJ] +0(/i3),
a(x -\- h) — a(x)
= {A[l+s]-A[s])k'{x)
^l.h[A[{l+s)']-A[s'])k"{x)+0{h'),
a[x + h) -\- a{x)
A[l] k{x) +^h {A[l +s] + A[s]) k\x) + 0{h').

158 Homogeneous Difference Schemes
Comparision of the resulting expansions with A8) permits us to assign the
values
B1) A[l] = l, A[s] = ~l
by virtue of the relations
A[l + s]+A[s] = A[l] + 2A[s] = 0,
A[(l + sJ] _ A[i'] = A[(l + sJ - s2] = A[l + 2s]
= A[l] + 2A[s] = 0.
To decide for yourself whether conditions A8) related to schemes A7),
A5') are met or not, we should take into account that
a(x + h) ± a{x) = a{x) a{x + h) —— ±
a(x) a{x + h) I
2 \ a(x -|- h) a(x) / k(x)
' 0{h'),
^ ^ i ^ ^ +0{h').
h \ a[x + h) a{x) I \ k{x) I
a{x)a{x + h) = P{x) + 0{h^).
We contrived to do it and give an answer to this question. When this is
the case, the functional A[^(s)] also satisfies conditions B1). As far as the
functionaLs
F[f(s)] = y /(.s) ds, A[f(.s)] = J f{s) d.s
-J/2 -1
are concerned, we thus have
1/2 1/2
i^[l] = / ds = 1 , F[s]= / sds = 0,
-1/2 -1/2
A[l] = / ds = 1, A[s] = /
0
s ds = —7^
1

Conservative schemes 159
provided conditions B0)~B1) hold.
In what follows we deal everywhere with the primary family of
homogeneous conservative schemes A6), A7) and A6'), A7) as well as with
linear nonnegative pattern functionals A[A;(s)] and F[f{s)] still subject to
conditions B0) and B1) of second-order approximation.
In the sequel scheme A7), A5) will be called the best possible.
3.3 CONVERGENCE AND ACCURACY OF HOMOGENEOUS
CONSERVATIVE SCHEMES
1. The error of approximation in the class of smooth coefficients. The main
point of the theory is the accurate account of the accuracy of the uniform
scheme A6)-A7) in the class of continuous and discontinuous functions
k{x), q{x) and /(x). In preparation for this, let u{x) be an exact solution
of the original problem
{k{x) u'{x))' - q{x) u{x) = -f{x) , 0 < x < 1 ,
A)
■u(O) = Uj , u{i) — u^ , k{x) > Cj > 0 , q{x) > 0 ,
and let y = y{x) be a solution of the difference problem
{ays)^~d{x)y=-^{x), x = ih , i = 1,2,. .. , N ~ 1 ,
B)
Vo = "i . Vn -  . «(^) > Ci > 0 , d{x) > 0 ,
which belongs to the primary family of conservative schemes designed in
Section 2.4.
Common practice involves the error z(x') = j/(x) — u(x) being a grid
function for x G w^j. Inserting the representation y{x) — z[x) -\- u{x) in B)
and assuming u{x) to be a known function, we may set up the difference
problem for the error
A z = (az^) — d z — —ip{x), X — ih , i — i,2, .. . ,N — I ,
C)
2@) = 2A) = 0, a>Ci>0, c/>0,
where the residual
D) ^(x) = A tj -|- i-p{x) = {aug) , — du -\- f

160 Homogeneous Difference Schemes
is the error of approximation of equation A) by the difference scheme B)
on the solution u = ii{x) of problem A).
In Section 1.4 we have established conditions of the second-order
local approximation for the conservative scheme B) with linear nonnegative
pattern functionals A[^(s)] and F[f{s)] such as:
E) A[l] = l, A[s]^-^, F[l] = l, F[s] = 0.
As can readily be observed, these conditions remain valid for any scheme
from the primary family.
In order to evaluate the order of accuracy for scheme B), it is necessary
to make the accurate account of the error z — y — u being viewed as a
solution of problem C). Moreover, the desirable estimate should be expressed
in terms of the right-hand side Jp. In this direction the error of
approximation to tp{x) is considered first. If A;(x) £ C^-^' and g(x),/(x) £ C^^^
then
tp{x) = Au + if- {Lu + f)
= [{au,)^ - iku'Y] -(d-q)u + {^-f)^ 0{h').
This means that scheme B) provides a local approximation of order 2, so
that \\^\\(^ < Mh'\ where M = const > 0 is independent of h.
In the sequel we succeed in showing that
7V-1
^iL= j: h j: hi.,
?=i
< M h^
if k{x), q(x), /(x) e C'(^\ meaning that k{x) posses,ses two (but not three)
continuous derivatives.
2, The error of approximation in the cla,ss of discontinuoxxs coefficients.
Our aim here is to justify that the error of approximation D) can always
be represented in the form
F) i; = ^^ + r,
G) %■=(«%),■-(^«'),_i/,.
1/2
(8) r^ = (^» ~ / /(^» +''') '^')
-1/2
1/2
-1/2

Convergence and accuracy of homogeneous conservative schemes 161
This can be done with the aid of the balance equation upon integrating
equation A) over x from X;_W2 to x^,-^,^:
^'i + l/2 ^i + l/2
r [{ku')i+i/2-{ku')i^i/2]--r / q{x) u{x) dx + J- / f{x)dx = 0.
With the new variable s = {x — x^)/h, the balance equation becomes
1/2 1/2
((A;u')t--i/2)^^. ~ / q{x-+ sh)u{xi + sh) ds + / /(x■ + s/z) ds = 0 .
-1/2 -1/2
Upon subtracting the last equation from D) we establish formulae F)-(8)
and deduce that if k{x), q{x), f{x) G C^^^ then
(9) % = 0{h'), ^P* = 0{h').
For this, we proceed as usual. What is available are the useful expansions
1 h'^
«i = «i-i/2 + 2 '^ <-i/2 + -g- <-i/2 + Oih""),
which emerge from the chain of the relations
a, = A[k{x^ + sh)] = A[A;(x,_j/2 + (s + i) h)]
= A[k,_,i,+hkl^^Js+l) + Oih')]
= k,_,i, + h k[_^,^ A[s + i] + Oih') = k,_,„ + 0{h^).
A simple observation that for any v{x) £ C^^^
1/2
/ v{x^ + sh) ds = Vi + 0{h^)
-1/2

162
Homogeneous Difference Schemes
helps motivate what is done with Jp* = (g^ u- — d, u^) -\- (cp^ — /j) + 0{h'^) =
0{h^).
We now turn to the case of discontinuity coefficients k(^x), g(x) and
/(x) and assume without loss of generality that k, q and / have
discontinuities of the first kind only at a single point x = ,J £ @, 1), so that
^ = x„ + eh, o<e = e{h) <i, o<n<N.
The solution u = u(x) to equation A) for x = ^ satisfies the continuity
conditions for the function u(x) and the fiow k(x)u'(x):
[u] = u{^ + 0) - u(,e - 0) = 0 , [ku'] = 0 for x = ^.
Under the assumptions k{x), q{x), /(x) £ Q^^^ we thus have u(x) £ Q'--^'.
Before going further, it is straightforward to verify for i]^ that
,?, = (at(,)^.-(fcw')^.__^^^=0(/i2) for all i^n + 1.
In dealing with tj^.^ = a,j , j u^. ,j^j — {ku')n+i/2 we make use of the
expansions in a neighborhood of the point x = ^■.
'7-i + l
u{O + il~0)hu'{^ + O) +
i^-0)\2..»
h\i"{^ + 0) + O{h^),
= u{0 - e h u'{^ - 0) + y h\"{^ - 0) + 0(/i3) ,
{ku')
n + l/2
(fcw')x-=e+o + @.5 -e)h (fcw'),Ue+o + 0{h'') for 0 < 0.5,
(A;w'),^=^„o + @.5 -0)h (A;w')Ue~o + '^i^'') ^°'' ^ > 0-5.
by means of which an alternative form of writing 7?„,i can be re-ordered
for later use;
Vn + l = fln-H (A ^ 0) ^'nght + ^^ieft) " ^"o + 0{h) ,
where w^ = {ku')right - (^"M')ieft, t'left = ^{i " 0) and t;„ght = v{i^ -|- 0).
At the same time, the continuity condition \kv!\ — 0 assures us of the
validity of the relations
(l-'')<,ht+''<ft
0 \-0\
"■left '^nght /
7n + l
V^-left ''right ^
- 1
w^ + 0{h).

Convergence and accuracy of homogeneous conservative schemes 163
It seems worthwhile to focus the reader's attention on the best scheme
A5) from Section 2 for which the chain of the relations occurs:
0 1
1 f ds f els
° J k{x„^i +sh) J k{x^ +sh)
n+l _1 0
1
ds f ds
J k{x^ + sh) J k{x^ + sh)
0 e
n ^
h^) ds
' \ ''■right
d)h'(\) +0{h'')\ds
'""+0^,
^left ^i-ight
SO that 7/„_|_i = '?„_!_! = 0(h). For all other schemes ri^_^^ = 0A).
In the estimation of Jp* it is necessary to distinguish two possibilities
of interest: d < 0.5 and d > 0.5.
A) Let 6 < 0.5. In every such case ^* = 0{h^) for i ^ n, while
0 0
?/)* = 0A) and only for the best scheme with (p^ = f^ and d- — di
0 1/2
incorporated i/)* = / q{x^ + sh) {u{x^ + sh) — u^) ds = 0{h)
-1/2
is attained for i = n, since w(;k„ + sh) = u{^) + 0(h) for any
s e [-0.5,0.5].
B) If 61 > 0.5, then ^* = 0{h^) for i^ n+1, while i>*^^ = 0A) and
C+i = 0{h).

164 Homogeneous Difference Schemes
It follows from the foregoing that
% = 0{h^) for i^n+1, v„+,=0{l), V„+, = 0{h) ,
r, = 0(/i') for I ^ n, r„ = 0A) ,
A0) r„ = 0{h) if (?<0.5,
^*=0(/i2) for i^n + 1, rn+,=0{l),
k+i=0{h) if e>Q.d.
Using these estimates behind we draw the conclusion that at the nodes
X = x„ and x = x^,^ the function Jp{x) can be expressed by
A1) ^„ = ^ + o(i), ^^^^ = ^!h±i + o{i), ^„+i = o(i),
thereby clarifying that at the nodes adjacent to the discontinuity point
X = ^ scheme B) does not approximate equation A) in light of the limit
relations
i'n=0{ - ) ^ oo, i>„^^ =0{ J ) ^ oo
1 \ , „/ 1
h
as h ^ 0. From asymptotic formulae A1) it is readily seen that the main
summands in the expressions for ^„ and V'n+i have equal modules and
Opposite signs, so that
^„ + ^„+i-0(l), /z(^„+^„+J = 0(/i),
meaning that the error of approximation is of dipole character in
neighborhoods of discontinuity points of the coefficient A;(x). This provides enough
reason to conclude that the conservative scheme B) is of order 1 in the
norm
II^IL = (i,hl] + (i,l/^|] = o(/i)
with fi^ = '}2'k = i ^"^l for * = 2, 3, .. . , A'' and /^i = 0.
3. A priori estimates of the error. We now estimate the error z — y — u,
which is a solution of problem C):
i\z={azg.) ■~dz = -4>{x), 0<x~ih<l, z@) = z(l) = 0,

Convergence and accuracy of homogeneous conservative schemes 165
where i>{x) is the error of approximation having the form F): ^ = r]^ + Jp*.
We will show that the solution of problem C) with the right-hand side F)
admits the estimate
A2) \\z\\c = \\y-^^\\c<^ii^A^\] + i^'\^^\])'
where Pj = Yl\ = i^^ i'l for i = 2, 3, . .. , A'' and fi^ = 0. To that end, it
suffices to carry out the proper evaluation of a solution of the auxiliary
problem
A3) Av={av,)^~dv = ~i^^, t;@) = Kl) = 0.
Other ideas are connected with solving the difference problem
(aw^s).^ = ~'?x-. t«@) = w(l) = 0,
from which follows immediately that aWg-\-i] = const = c^. As its corollary,
we establish the recurrence relation for w^:
W/t=W/t~i ^ . k = 1,2,... ,N .
Summation over k from 1 to i yields
fc = l *■■ /5: = 1 ^
Putting i = N and keeping in mind that iuq = Wj^ = 0, we find that
_ J_ v^ l^^lk
^°^ An ^ a
fc=i
TV
E" 'Ik
n. '
yielding
Ev^+^f:
or
h
N
a,. Am ^-^ ill.
k = \ ^ ' k = l '-
■". = -(i-^)E'^ + ^ E^
Am I ^-^ Oj, An
" k=\ >" k=i+l

166 Homogeneous Difference Schemes
As far as A, < A/v, the preceding decomposition implies that
' ' ' ~ V Aw/ ^^ oj. An ^-^
The difference ^^ = v^ — w^, solving the problem
{a^,)^.-^d^ = dw, e@) = e(l)=0,
with a > Ci > 0 and d > 0, is aimed at the further estimation of v^ in
complete agreement with the lemma from Chapter 1, Section 2.8. The
results of this section provide support for the view that ||<J||(-. < ||t«|U
and, therefore,
IIHIc<ll'»^llc + ll^llc<2lkllc<r(i.l'?l]-
.Summarizing, we arrive at
A4) lhllc<-(l.kl]-
The function tp* is representable by
A5) ^*=/^,.,
where ji^ = '}2'k^i^^^l 1°^' * = 2, 3,. .. , A'' and fi^ = 0. Because of this
form, it is possible to write down jp = (^rj + ^)x and involve estimate A4)
in subsequent reasonings;
||z||^<-(l,h? + p|]<-^((l,h?|] + (l,|p|]).
4. On convergence and accuracy. The results obtained in the preceding two
sections may be of help in establishing the rate of convergence for scheme
B).

Convergence and accuracy of homogeneous conservative schemes 167
Theorem 1 The accuracy of any scheme B) with coefficients A6) is of
order 2 in the class of smooth coefficients A;(x), g(x), /(x) £ C(^^[0,1];
\\y-u\\^<Mh'
with constant M > 0 independent of h, whereas it is of order 1 in the class
of discontinuity coefficients A;(x), g(x), /(x) £ C^^-*[0,1];
\\y-u\\^ <Mh.
The best scheme also has the second-order accuracy in the class of
discontinuity coefficients.
These assertions follow from the representation of the approximation
error in the form F)~(8) and a priori estimate A2), On the basis of the
estimates for rji and ^1 obtained in Section 3,2 we find that
A6) (l,|'7l] = M'7n+il+0(/i'),
^i = 0{h^) for i < n,
//, = /i(^*+^;^J+0(/i2) for i>n + l,
yielding
A7) {h\f^\] = h\r„ + r„^,\ A - xj + o{h').
Combination of relations A6)~A7) just established and estimate A2)
provides the sufficient background for the validity of the assertions of the
theorem in light of the asymptotic representations
C + C+i = o(i),
k + k+i = 0{h) for any 9 G [0,1] ,
'7n+l=0(l), ^„ + ,=0(/l).
In the case of smooth coefficients we achieve for any admissible scheme
7y. = 0(/i2) and ^; = 0{h'^) for all i = 1, 2,, ,. , jV - 1 and, therefore,
Reraiark It can be shown that the approximation in the class of smooth
coefficients is necessary and sufficient for the convergence of the homogeneous
scheme B),

168 Homogeneous Difference Schemes
3.4 HOMOGENEOUS DIFFERENCE SCHEMES ON
NON-EQUIDISTANT GRIDS
1. Schemes on non-equidistant grids. Quite often, in practical
implementations difference schemes on non-equidistant grids are in common usage for
solving differential equations. In Chapter 2, Section 1 we have produced for
the simplest equation u" = — f a difference scheme on the non-equidistant
grid
^h
{.Tj , i - 0,1, ... , N, Xg = 0, Xj^r = I, /^^■ = x'i - ^i~i}
and have determined for this scheme the error of approximation.
In order to develop a homogeneous conservative scheme on the non-
equidistant grid w;j, we write down the balance equation
^i + l/2 ^'j-l-l/2
A) »._i/. - -.,1/. - / ,i^Mx)dx = - I fix)dx, r.= -ku'.
on the interval (a^i_i/2 > ^2+1/2) with the ends x^__^i^ = x^ — 0.5/ij and
X,
J,2 = x^ + 0.5 /ij_|_i. By analogy with Section 2.2 the forthcoming
substitutions
■^'j-l-1/2 ■^■!:-l-i/2
f 0 0 1 /■
/ qu dx -^ h^ di u^ , di — T~ I l{^) dx .
h, = ^{h, + hi.,).
_ _ 0 Uj - Ui_, _ ^ 0
^2-1/2 ~ ^2-1/2 ~ '^i I, " '^i ^S,i '

0 1 f dx \
hi J k{x)
■^2-1

Homogeneous difference schemes on non-equidistant grids
169
lead to the difference scheme
B)
0 Vi+i - Vi _ 0 Vi - Vi-i
i+l
hi+1
1 f dx
hi J k{x) ^
0 1
di - T-
••^2 + 1/2
g(x) dx ,
=^2-1/2
•'^i + 1/2
!{x)dx ,
^2-1/2
which IS the best possible in the same sense as we spoke above about this
on an equidistant grid. On the same grounds as before, the coefficients a^^
0 0
di and fj are representable by
/ 0 , -1
of ds \
k{xi + shf)
C)
hi
q{Xi + s/ij ds + -^
1/2
q{xi + s/i,;+i) ds ,
-1/2
0
fi =
hi
hi+i
1/2
f{xi + shi)ds+^-- / f{xi + shi_^_^) ds .
-1/2 0
Retaining the notations of Chapter 2, Section 1
Vx.
Vi - Vi-
Vx.
Vi+i - Vi
hi+i
y$.
Vi+i - Vi
h,
which allow a simpler writing of the ensuing formulae, we refer to the three-
point scheme
D)
(aj/,f).« - dy = -<p,
X ^ Xi euj,,,
j/@) - u^ , y{l) = u^, a > Cj > 0 , d > 0

170 Homogeneous Difference Schemes
When the available functions fe(x), q{x) and f{x) happen to be of the class
Q^'^'[Q, 1] and their discontinuity points are known, a non-equidistant grid
can be made so that all discontinuity points of the coefficients k, q and
/ would be nodal points of such a grid. We denote by <jjf^{K) any such
grid depending on concrete functions k, q and / and arising in subsequent
discussions. It is easily seen from C) that the simplest expressions for a^,
d; and i^c, on Cji^{K) are given by
,,, , , hj q~ + /i,^i q+ hif-+hi^J+
E) a,=fc,_,/,, ./, = , ^. = —-^^r -,
where ff^ = f{x^ ± 0), etc,
= HkU + K)'
2k+ ,k-
l — l I
a,- =
kU+K'
hi qi-i/2 + "i + i 91-1-1/2 "i Ji-l/2 + "i-l-l Ji + ll2
di = ~ , fi ■=
2/i, ' ^^ 2h,
In the case of continuous coefficients expressions E) imply that a^ = ki_^i^,
d^ = 5; and tpi = /j. If discontinuity points coincide with nodal points of
the grid Cjj^, that is, x = x^_^,^, then the members Oj, d^ and (p^ can be
found in simplified form:
E') flz = , ' ,'"' ' d^ = (li^ fi = Ii
9/-+ h-
^'-1/2 '"i-l/2
a,. =
'^i-l/2 + '^i-1/2
hi+i
K+i
'?i + l/2 + h^ it
2/i,
Ji + ll7 + '^J ^j-
1/2
1/2
2/..
2. The error of approximation. We now investigate the error of
approximation for scheme D) on the non-equidistant grid w/j by considering the
equation for the error z = y — u:
{azj,)i - dz = -t/j{x) , x e Wft ,
F)
Zg ~ Zj^r — 0 , a > Cj > 0 , rf > 0 ,

Homogeneous difference schemes on non-equidistant grids 171
where i>{x) = {aUj,)i — du -\- (p{x) is the error of approximation,
On account of the balance equation A) the error of approximation ^^
reduces to
where
^2 + 1/2
0 1 [
G) r, = {fi - fi) -d,Ui + - J q{x) u{x) dx .
^!-l/2
In what follows the functions k, q and / will be smooth for Xj_j <
X < Xj and .t, < x < .Tj-_|_j and possess discontinuities of the first kind at a
single node x-. Combination of the expansions
for s < 0 and /(x^ + s /zj = /^.^g + s hi^^ /^_^g + 0{hl^^) for s > 0 with
0
formula C) for fi gives
Likewise,
^i + l/2
- J q{x)u{x)dx= +( ^ j,,.+^(^^-)-
•'^i-l/2
With the relations O; = k^_j,^ and Uj, ; = t't-_i/2+C'(/ii^) in view, we deduce
from the foregoing that for the scheme with coefficients E)
(8) ' ^, = ('? + ^),,, + C.
whose ingredients behave as follows:
h'^ (qu — f)'i , „
% = fli ugj - {ku')i^^i^ = 0{hf).

172 Homogeneous Difference Schemes
3. The order of accuracy on non-equidistant grids. As before, the inner
products are defined by
TV-l TV
(y. v)^ = Y. Vi ^i K. (j/. ^]* = E Vi ^i hi ■
t=i t=i
For problem F) with the right-hand side (8) the estimate
A0) \\z\\c< ^{(i,hl + h?l] + (i,|p|]}
-1
k = l
is valid with //,; = Yl ^\ i'l ^or i = 2, 3,... , A'' and i^i^ = 0.
The derivation of tliis estimate coincides with tliat of inequality A2)
carried out in Section 3 by inserting a^/h^ in place oia^/h.
In light of the asymptotic relations (9) estimate A0) implies that if
k, q, f & E^^-*[0, 1], then scheme D)~E) is of second-order accuracy on the
sequence of non-equidistant grids Cjf^{K):
\\z\\c = \\y-u\\^<Mh\
where h = 1^A, K^] is the mean square step.
When only one coefficient k{x) G Q^-^' is discontinuous, while other
coefficients q, f (z C^"^' are continuous, any conservative scheme D)
generating a second-order approximation is of second-order accuracy on the
sequence of non-equidistant grids u)f,(K). This fact is an immediate
implication of the expansions r]^ = a^u^. , — (A; ii')j_i/2 = 0{h^), valid for the
aforementioned schemes, and ip* = O(fi^).
We are now jnterested in the question concerning the accuracy of
scheme D) with second-order approximation on an arbitrary
non-equidistant grid. A discontinuity point a; = <^ is free to be chosen for the relevant
coefficients:
■■Cn < ^ < '-^n + i , ^ = X^ + Sh, 0 < (? < 1 .
Being concerned with ^* specified by G) and involved in the representations
■>Pi = Vi.i + i'i: Vi = (a%), - (A;w')i-i/2 ,

Homogeneous difference schemes on non-equidistant grids 173
which are always valid, we follow the same procedures in the accuracy
analysis as we did on equidistant grids in Section 3. This amounts to
regarding q, f & C'--^' to be continuous and establishing the relations:
m = 0{h^) for i^n+1, !!„+,= 0{l).
Only for the best scheme C) we might achieve Ty^^j = 0(/i„^j) and
h'^(qu-f)'. ,„
r. = %i + rr, v. = '' ^''~"', ^r = o{n])
for alli = 1,2,... ,7V- 1,
Using estimate A0) behind we deduce that the best scheme C)-D)
retains the second-order accuracy in the class of discontinuous coefficients
on an arbitrary sequence of non-equidistant grids, while any scheme D)
appears to be of order 1.
In the class of continuous coefficients k, q, f £ C^-^^[0,1] any
scheme D) retains the second-order accuracy on an arbitrary
sequence of non-equidistant grids.
Before we undertake the proof of the last assertion, it is worth
mentioning the estimate obtained in Chapter 2, Section 4 for the operator
equation A z = ip\
0 0 0
where A > c^ A, A = A* > 0 and A = A* > 0. In the case of interest
, 0 0
Ay = -{ay^)^ + dy, Ay = -y^^ , J/e fi ,
where Q = H is the space of grid functions defined on uj/^ and vanishing
for i = 0,A'', The negative norm ||'(/'||°_i was estimated in Chapter 2,
A
Section 4 as follows:
A
Here h'"^ denotes, as usual, the mean square of h'^ and so the desired assertion
follows immediately from the preceding,

174 Homogeneous Difference Schemes
4. A greater gain in accuracy on a sequence of grids. The Runge method.
In a common setting we are dealing with a linear mathematical-physics
equation
A1) Lw = /(x), xeG,
regardless of additional conditions on the boundary F, Let w;j be a grid in
the domain G with step h, by means of which a difference scheme
A2) LhU^ = f{x) , X e w
h I
can be put in correspondence with problem A1).
When this scheme happens to be stable, so that ||j/;j|L ■, < Af||iiC;j|L ■.
with a positive constant M independent of h (see Chapter 2, Section 2),
the approximation implies the convergence
A3) ||j/;.-'"JI(i,)< M ||'^J|B,.)'
where ^tp^ = cpf^ - L^u^ - {cp^ - fh) - {Lh u^ - {Lu)^) is the error of
approximation (residual) on the problem A1) solution u = u(x), It is
readily seen from A3) that the order of accuracy is not lower than the order
of approximation and the behaviour of the residual || tpj-^ \\, . = 0{h"') is a
corollary to the estimate for the relevant deviation || y^^ — Uj^ \\, ^ — 0(h").
In order to improve the accuracy of the approximate solution, it is
necessary to diminish the grid step h or make the approximation order
n of the scheme higher. An elementary example illustrating how to raise
the order of approximation on a solution is available in Chapter 1,
Section 2.2. However, in trying to develop higher-accuracy schemes for many
problems, especially for equations with variable coefficients, considerable
obstacles of technical character do arise. Moreover, the transition to such
schemes except the schemes for the heat conduction equation with constant
coefficients and the Laplace equation which will be given special
investigation in Chapters 4-5, should cause the essential growth of the volume of
computations.
The gain in accuracy provided by refining the step h is limited by
requirement,? of economy. Such an approach is equivalent to minimizing the
execution time nece.ssary in this connection in obtaining the solution, But
if the .solution of the original problem u and / both are smooth functions
of X, the accuracy of numerical solution can be increased by performing
calculations for the same problem A2) on a sequence of grids W;j , . . . , W;j .
In the sequel we assume that u = u(x) possesses all necessary
derivatives which do arise in the further development. In order to understand

Homogeneous difference schemes on non-equidistant grids 175
this approach a little better, we consider the simplest case, provided that
the asymptotic expansion
A4) y,ix) = u,{x) + a,{x) h"^^ + 0{h>'-), k, > k, > 0 ,
holds, where a^{x) does not depend on h. In such a setting it is required
to find a grid function yi^{x) satisfying the relation
A5) y,{x)=u,ix) + Oih''^).
The traditional way of covering this is to introduce two grids w^j and w^j
with steps /jj and h^ and common nodes, the set of which is denoted by
W;j, and to form the grid function
A6) y^{x) = c,yf^^[x) + c.,yf^^{x), x G W;, ,
while Cj and c^ remain as yet unknown, Substitution of expansions A4) for
J/fti and y,^^ into A6) gives
hix) = (ci + cJ «„(x-) + (cj /if 1 + c, /if 1) a,(x) + 0{h^-) ,
whence it follows that i/^ — Uf^ = 0(h '^) if
Cj + C2 = 1 , Cj h\^ + c^h'l^ = 0 .
This is certainly so with
A7) c, = 1 - c,
hi
K' - K
In particular, keeping h^ = h and h^ — ^h, we find w^ = w^j and Cj =
-1/B^1 - 1),
Thus, the improved accuracy of a grid solution on some set of nodes
ujf^ is connected with solving problem A2) twice (first on the grid w^j and
then on the grid w^j^) and drawing up the linear combination A6) with
coefficients A7). The grids w^^^ and ujf^^ are chosen so that their intersection
coincides with w^. For instance, by applying a scheme of second-order
accuracy satisfying the relation
yh{^) = u^^{x) + cx^h'^+0{h'^)

176 Homogeneous Difference Schemes
and having k^ = 2 and A;^ = 4 to the difference problem with steps h^ = h
and /i2 = i /i we calculate the solutions j/^j and yi^/2 and the coefficients
Cj = — i and c^ = |, leading to the function in question:
y{^) = t yh/2 - kVh'
which is defined on the grid ujf^ and approximates the exact solution u(x)
with accuracy Oiji^)'.
yh = Uh + Oih'^) for X euj^.
One way of proceeding is to take for granted the expansion
j/^ = Ui^ + a^{x)h''' + a,(a;)/z*^^ + 0{h'''), k^ > k^ > k, > 0 ,
where the functions aj(x) and a^i^) are independent of h. Because of
this, we must solve the difference problem concerned three times with steps
/ij, h^ and h^, respectively, in an attempt to obtain a solution with the
prescribed accuracy 0{h^^). In what follows j/;, (.t), j/^^ [x) and j/^^ (x) will
stand for the appropriate solutions. Also, it will be sensible to introduce
their linear combination
yh{x) = c^yh^{x) + c^y^^(x) + c., j/;,^(x), X e w,,.
As before, the grid Cof^ is the intersection of the three grids tOf^ , w^j and
W;j^. In particular, Cjj^ = W;j under the agreements h^ = h, h^ = ^h and
/ig = i/i. When the approximation t/f^ = u^j + 0{h^^) is accepted for later
use, the following equations
CJ+C2+C3 = 1, Cj/zf'+c^/i^'+Ca/ij' = 0, Ci/ZjHc^/i^'+Cs/ij' = 0
constitute the system for finding the coefficients Cj, c^, Cj. Evidently, the
determinant of this system is nonzero.
In the general case the expansion of the error j/^j — Uf^ in powers of h
is of the form
n-l
A8) J/;, = M;,+ ^ a,(x)/z'' + a„(x,/z)/z",
s = l
where (^^(x), s = 1,2, . .. , n—1, are independent of/z and the absolute value
of a„(x, /z) is bounded by a constant M = const > 0, which is independent
of h. Our next step is to establish an a priori expansion of the form A8).

Homogeneous difference schemes on non-equidistant grids 177
When m(x), f{x) and the coefficients of equation A1) are smooth enough,
the expression for the error of approximation ■(/';j(k) is simple to follow:
n-l
A9) M^) = Y,p,{x)h^ + p„{x,h)h\
s = l
where Ps{x), 1 < s < n — 1, are independent of h and |/?„(x,/i)| < M,
M = const > 0 is also independent of h. From such reasoning it seems
clear that any sufficiently smooth function a^i^) admits the representation
n-l
B0) L„a^{x)=^ La^{x) + u^ + Yl 7s{^) h' + 7n{^,h) h".
s = l
In this view, it is not unreasonable to attempt the difference j/^j — u^
in the form A8). Applying the operator L^ to identity A8) yields
n-l
s = l
Using the relation L^^yf^ — iif^) = ?/)^ and formula B0) for LkCt^, we arrive
at
5 = 1 m = l
giving
n—i s—\
B1) ^. = E (^ "^ + E 7m h^) h^ + 0{h") .
Comparison of formulae A9) and B0) justifies the validity of expansion
A8) if functions a^{x) are solutions to the equations
s-l
La, =/?,(x) - E 7m(a;) foi' s = 1,2,... ,72-1.
m = l
Observe that expansion A9) is not obliged to contain all of the powers of
h, some of them may be omitted. In this case the appropriate coefficient
equals /?, = 0.
Adaptive grids may be of assistance in raising the order of accuracy
without increasing the total number of nodes, If the subsidiary information

178 Homogeneous Difference Schemes
on the behaviour of the solution of an original problem is available, a grid
can be made so as to achieve a prescribed accuracy at minimum nodal
points and, therefore, in a minimal number of the necessary operations,
For instance, successive grid refinement will be appreciated in the region of
widely varying coefficients and the right-hand side. Specifically, near the
point (the line) of discontinuity of coefficients, representing the boundary
between two media, the grid is refined in such a way to attain the minimum
of the step near the boundary. After that, the step is being enlarged for
instance, as a geometric progression in moving from the boundary. When no
information about the behaviour of the solution is available, a preliminary
computation proves to be useful on a sparse gi'id, after which we are working
on a new grid with a more smaller step in the regions with large deviations
of the solution. In practice non-equidistant grids are widely used. As
shown m Section 3, it is possible to construct special grids ujf^(K) so that
all discontinuity points of the coefficient k{x) involved in the equation
(A;(x) u') — g(x) u = — f{x)
fall into nodal points of the grid Cjf^{K). Under such a choice, any
homogeneous difference scheme (aj/^) . — dy = —ip generating approximation
of order 2 (in the class of smooth coefficients) is of second-order accuracy
in the class of discontinuity coefficients k{x). The accuracy of difference
schemes, which depends on the existing grid.s suitable to computer
calculations for equations with variable coefficients, needs investigation for each
concrete problem.
3.5 OTHER PROBLEMS
1. The third boundary-value problem. The main goal of our studies is
a homogeneous difference scheme for the boundary-value problem of the
third kind:
Lu— {k{x)u'^ — q{x)u = —/(x),
0 < X < 1, A;(x) > Cj > 0, g>0,
A)
A;@) u'@) = /?! u@) - 1^1, , -A;(l) u'(l) = p^ u(l) - p^ ,
A > 0, /?2 > 0, /?^ + /?, > 0 .
We approximate equation A) in the u.sual way:
B) Ay = -(p{x), Ay = (ayg) - dy, a>c,>0, d>Q,

other problems 179
where a, d and cp satisfy the approximation conditions A8) of Section 2,
First, we consider the simplest approximation of the boundary
condition at the point x = 0: a^ y^ ^ = P^ y^ — jj,^ and calculate the error of
approximation by inserting y =^ z + u:
«1 %,1 = A ^0 - ^1 . ^1 = fll ^^S,l - ^1^0+ l^i-
We obtain
through the approximations
a,=^k, + ^hk'^ + 0{h''), u^ 1 = < + i h < + 0(/i2) .
Substituting here {ku')'^ — q^Ug — fg emerged from equation A) we find
that
thereby justifying that the boundary condition
C) Oj j/g 1 =r /?j j/o - /7j, C^= C^ + \hq„, Ji^ = ^i^ + \ hf„
provides an approximation of order 2 on the solution u(x) of problem A).
In a similar manner we derive the difference boundary condition of second-
order approximation at the point x = 1:
D) -Uf^y^j^ = P^yj^-^2: /?2 = A + 5/j 9iv . 1^2 = 1^2 + k^ In ■
In this way, the third kind difference boundary-value problem B)-D) of
second-order approximation on the solution of the original problem is put
in correspondence with the original problem A).
2. A problem with periodicity conditions. First, we study the elementary
problem in which it is required to find on the segment 0 < x < 1 a solution
to the equation
E) u"{x) — q^u — —f{x), qg = const > 0, 0 < x < 1 ,
satisfying the condition of periodicity
F) u{x +1) = u{x) for all x G @, 1) ,
Here /(x) is a periodic function of period 1: /(x -|- 1) = /(x).

180 Homogeneous Difference Schemes
At any point x £ @, 1) condition F) is equivalent to the pair of
continuity conditions at the single point x = 0:
G) u@ + 0) = u(l-0), u'@ + 0) = w'(l-O),
Problem E)-F) has a unique solution having the estimate
Ihllc < ll/llc/?o
in complete agreement with the maximum principle.
For (/o — 0 *'h'^ statement of the problem is
u" = -f{x), w@ + 0)=: w(l-O), w'@ + 0) = w'(l-O),
which is solvable under the condition L f(x)dx = 0 and possesses a unique
solution u = u(x) only if
(8) Ju{x)dx = 0.
0
Indeed, the general solution to the equation u" = —f{x) admits the form
u{x) = C,x + C^~ f i I /(a) da\ dt = C,x + C^ - I {x - t)f(t) dt
0 ^0 ^ 0
with arbitrary constants Cj and Cj.
With this, conditions G) provide
1 1
Jf{t)dt = 0, C, = -Jtf{t)dt = 0,
0 - 0
it being understood that the function u(x) can be recovered from
conditions G) within a constant C\. Under condition (8) we find that Cj = 0,
extracting a unique solution of the problem,
As a first step towards the solution of the original problem, we initiate
the design of a difference scheme on an equidistant grid w;j = [x^ = ih, i =
0,1,... , A''} on the segment 0 < x < 1 with step h = 1/A'' and approximate
equation E) and the continuity conditions G). The first of these conditions
is satisfied if y^ = Un ■

other problems 181
At the nodes x^ = ih, i = 1,2,,,, , A'' — 1, we write down the three-
point equation
Vxx -loV = -Vi.^)^ x = ih, f = 0,1,. .. ,/v'- 1 .
In this context, the difference derivatives
u^,^ = u'{l - 0) - i /z m"A - 0) + 0{h''),
u^a = «'@ + 0) + i /i w"@ + 0) + 0(/i2)
will be given special investigation. Substituting here u" — q^u^ — f
recovered from E) we obtain
%,iv +kK% «A) - /(I - 0)) = u'{l - 0) + 0{h^),
«.,o - 5 '^('/o «@) - /(O + 0)) = «'@ + 0) + 0(/i2),
With these relations established, the second continuity condition u'@ + 0) —
u'{\ — 0) is approximated to 0{h'^) by the difference equation
(9) u^,o -^hq,y, + ^hf{0 + 0) = u^f, + Lhq,yf,-^h /(I - 0) .
By merely setting j/jv+i = Vi condition (9) becomes
%x-,iv - 90 J/iv = -Vn> fN = 5 (/(I - 0) + /(O + 0)) .
This is a way of establishing the correspondence between problem E), G)
and the difference scheme
A0) y^^-qay=-f{x), x-ih, i=l,2,... ,N,
with the conditions of periodicity
A1) ~ Vo = J/iv. Vi =J/iv+i •
We will elaborate on this for rather complicated cases and turn to the
equation with variable coefficients
A2) {ku'y -qu=-f{x), 0<.T<1,
where k{x), q[x) and /(x) refer to periodic functions of period 1:
A3) fc(x+i) = fc(x), g(x+l) = g(x), /(x + 1) =/(x),

182 Homogeneous Difference Schemes
which are continuous at the point x = 0 (x = 1), so that k(l — 0) =
A;@ + 0) = A;@), etc. Let
A4) A;(x)>Cj>0, g(x)>Cj>0.
Assuming this to be the case, it is required to find a solution to equation
A2) subject to the periodicity condition u(x + l) = u(x), which is equivalent
to the requirements
A5) m@ + 0) = 41-0), fc«'|,^o+o = ^«'lx- = i-o.
The maximum principle implies that problem A2)-A5) is uniquely solvable.
With this in mind, we start from the scheme for Q < x — ih < 1:
{(lyg)^- dy= -(p{x), x = ih, i = l,2,... ,N -1 ,
keeping y^ — yj^. The coefficients a, d, ip can be recovered from the
conditions of second-order approximation in Section 2,4,
With the aid of the equalities
(aj/g). = (k w')i-o + 5 /i (/ - <ly)^-l + OiU') ,
fli+i «x-,i = {^ ^'Oi+o - ^h{f - qu)i+o -F 0(/i2)
the condition A;«'|^._g_j_g = A;u'|^,_j_g is approximated to second order by
the difference relation
«i ?/.,o - 5 Kl{^) Vo - /(O + 0)) = a^ y.^f, + \ h{q{l - 0) y^ ~ /(I - 0)) .
Under the agreements j/^y+i = Ih ^ii"^ '^n+\ — "i '■^^'^ preceding reduces to
{ay^) ^ ~~ dy =-p{x) , x- = x^ = l,
with
d = d^ = i (g@ + 0) + g(l - 0)), p = Pn = \ (/(O + 0) + /(I - 0)) ,
leaving us with the periodic difference scheme
{"-Ih) ^ - dy =-"Pi^) ^ x-=ih, i - 1,2,... ,N ,
A6)
a>Cj>0, d>Cj>0,

other problems 183
subject to the conditions
A7) y^=y^ , j/j = y^_^^ , a, = a^^j ,
For determination of y^, i = 1,2,.. .N, we obtain the system of equations
a,
%__! - (a,- + a^^i +dih ) y^ + a^^j %^i = -ip^ h , {=1,2,,.. ,N
supplied by the conditions of periodicity, y^ = j/^ and j/^y+i — Vi- This
system can be solved by the cyclic elimination method (for more detail see
Chapter 1, Section 1,2),
Since a > Cj > 0 and d > c^ > 0, the maximum principle is still valid
for problem A6)-(i7), due to which
\\y\\c<^\\^\\c'
The inequality obtained permits us to derive for the error z = y — u the
estimate
\\-A\c = 0{h'),
since ip^^ = 0{h'^) for i = 1, 2,. . . , A''. Thus, scheme A6)-A7) is of second-
order accuracy in the space C when k[x) £ C^'^' and q{x), /(x) £ C^'-^\
3. Monotone schemes for an equation of general form. The object of
investigation is the boundary-value problem
Lu = {k u'y + r{x) u - q{x) u = -f{x) , 0 < x < 1 ,
A8)
■u(O) = Uj , ■u(l) = u^ , k{x) > Cj > 0 , I r(x-) | < c, , <? > 0 ,
The main idea here is connected with the design of a new difference scheme
of second-order approximation for which the maximum principle would be
in full force for any step h. The meaning of this property is that we should
have (see Chapter 1, Section 1)
A9) A,y^_,-Ciy, + Biy^^, = -Fi, i = 1,2,... , N - 1 ,
where Ai > 0, Bi > 0 and d - Ai - Bi = Di >0.
Any such scheme is said to be monotone, As before, the operator
Lu = [ku'y — qu is approximated to second order by the homogeneous
three-point scheme Ay = (aj/j;) — dy-

184 Homogeneous Difference Schemes
The natural replacement of the central difference derivative u'{x) by
the first derivative Uo leads to a scheme of second-order approximation,
Such a scheme is monotone only for sufficiently small grid steps.
Moreover, the elimination method can be applied only for sufficiently small h
under the restriction h \r{x) \ < 2k{x). If u' is approximated by one-sided
diff'erence derivatives (the right one u^, for r > 0 and the left one Ug for
r < 0), we obtain a monotone scheme for which the maximum principle is
certainly true for any step h, but it is of first-order approximation. This is
unacceptable for us.
It is worth mentioning here that the sign of r(x) has had a significant
impact on construction of monotone schemes. One way of providing a
second-order approximation and taking care of this sign is connected with a
monotone scheme with one-sided first diff'erence derivatives for the equation
with perturbed coefficients
B0) Lu = -f, Lu = x{ku'y + ru'- qu ,
where x = 1/[1 + R) and R = 0,5 h \ r \/k is the Reynolds diff'erence number.
By obvious rearranging r(a;) as a sum
r=r'^Jrr~, r+= i (r-^ t r t) > 0, r~ = )^{r - \r\) <Q ,
the expres.sion ru' is approximated by the formula
where b^ = F [f ^(Xj-|-s/z)], r^ = r^/k, i^ is a pattern functional being used
for calculations of the coefficients d and tp. We may accept, for instance,
6+ = r+/A; and h~ = r~ jk. As a result we get the homogeneous scheme
Aj/ = x(aj/j;)^-F 6+0*^+^^1/^ +b''ayj; - dy = -ip ,
B1) j/o = Uj ,. yj^=u^, a'^'^^^ = a{x + h), a > q > 0,
X- ^— R- ^-^
"" l+R' 2k ■
We are going to show that scheme B1) is monotone by observing that
B2) Ai yi_^^ -Ciyi + Bi j/,.^j = (p^ , y„ = u^ , yj^ = u^ ,
where
Ai = S (^. - h b~), Bi = -j^ (x, + h b+), a = Ai + B,+ d, .

other problems 185
As can readily be observed from the foregoing, Ai > 0, Sj > 0 and Di > 0,
since 6~ < 0, 6+ > 0 and d,- > 0.
Equations B2) can be solved by the elimination method for any h and
r. The error of approximation for scheme B1)
ij; = X (auj) . -|- h'^a^'^^'u^ + b~aug — du-\- (p — (^Lu -\- /)
is representable by 'tp = ip^-^' + 'tp^~' with the members
^A) = [(a%.).^ - du + f] - [{ku'y -qu + f] ,
'ijp) = [{x - 1) (aug)^, + b+a^+''^u^ + b'-au^] - r u' .
The first summand satisfies the estimate
By virtue of the relations
6+ = f++ 0(/i2), b~=r~ + 0{h^),
k f = r , r -\- r~ — r, r'^ — r~ = \r\ ,
au^. = ku' - lh{ku'y + 0{h'^),
a'- + ^'>u^ =^ku' + |/i(A;tt')' + 0(/i^),
{au,)^, = iku'y + 0{h')
we obtain
b+a'^+'\^ + b~aug = ru' + hi{ku'y -^ + 0{h'') ,
'^^'^ = -J^l (^"')' + ^(^"')' + ^('^') = -^l (^"')' + ^('^') = 0('^')
exploiting the fact that /? = 0.5 /i | r |/A; = 0(/z).
Because of this, the monotone scheme B1) generates an approximation
of order 2:
B3) ^ := 0(/z2) .

r
186 Homogeneous Difference Schemes
If g > Cj > 0, then for a solution of the difference problem B1) the
maximum principle provides the estimate ||j/||^. < ;r-||')^|lc with j/^ = j/^ = 0,
which implies on the basis of B3) the uniform convergence of scheme B1)
with the rate 0(/z^).
The monotone function B1) is quite applicable when r(x) is a fastly
varying function of the variable x and the condition R < 1 fails to hold at
some points with no influence on the accuracy to a considerable extent.
The same procedure with a monotone scheme works on a
non-equidistant grid.
One more method available for designing a monotone scheme is related
to equation A8). In preparation for this, let us multiply equation A8) by
the function //(x) and require that
li{ku'y -\- r jiu' = {k i_i u'y .
This is true only if r^i = kjj,', so that
fi{x) = fig explj r{t) dt\,
The equation
[^ k u'y — fiqu = —/.« /
can be approximated by the conservative monotone scheme
B4) (fiayg)^^- ^dy- -^ip, fi = ^(x-~ ^h),
where, for instance, the members are taken to be a^ = ^i-1/21 '^i — 1i)
Vi = .fi ■
The number [i[x^ may be very large when the ratio r = rjk grows.
Dividing both sides of the difference equation
B'^') /^ K+i/^i+i'"/2(j/i+i - Vi) - a./^i-i/2(j/i - J/i-i)] - A'j li Vi = -fii fi
by //j = i^g exp < Jq ' f (<) dt>, we obtain the noncouservative, but monotone
scheme
B5) /^
% = ^'i . Vn =^2,

other problems 187
with the coefficients
6, = a,;+i exp <^ / f-{t) dt \ ,
B5') 1..
a, = a. exp <^ - / r{t) dt \ .
It is straightforward to verify that the scheme concerned provides an
approximation of order 2 due to the expansions
*LZA = k[+ r, + Oih^), ^ = k, + 0{h^) .
Replacing the integrals in B5') to 0{h'^) by the expressions | Cfj- -|- r^^j)
and I Cfj -|- fj_^), respectively, we arrive at the monotone scheme with the
coefficients
K = a^+i exp 11 Cf, + fi+i)| and a,; = a,; exp |- | C?',: + ?,__,) j ,
generating an approximation of order 2,
4. Difference schemes for a stationary equation in cylindrical coordinates.
The stationary diffusion or heat conduction equation
div {k grad u) — qu — —f{r, (p, z)
taiies in the cylindrical coordinate system (r, (p, z) the form
- — (rk{r) -—) - q(r) u = -f{r), 0 <r < R,
B6) ^ ''^ ^- ''^^
q{r) > 0, 0 < Cj < k{r) < c^,
in the case when the solution u ~ u(^r) depends neither on z nor on (p, that
is, in the case of the axial symmetry.
When 7' = 0 we impose the boundedness condition | u@) | < oo being
equivalent to the requirements
, , s du
B7) hm rk{r) ^ = 0.
r^o dr

188 Homogeneous Difference Schemes
VVhen r = R one of the usual boundary conditions applies here, for instance,
B8) u{R) = ^2 .
Let WjG') and u^i^r) be linearly independent solutions to equation B6),
Uj(r) being bounded for r £ [0,i?]. A brief survey of their properties is
presented below,
1) If g@) and /(O) are finite, then Uj@) 7^ 0 and ^^(O) = 0.
2) liq{r), f{r) G C<'-^\0,R] and k{r) G C'^^\0,R], then the derivatives
u' u'' u\ and u\ are bounded for Q < r < R.
3) The second solution u^ij) of equation B6), which is linearly
independent of u^{r), has a logarithmic singularity at the point r = 0.
Conditions B7) and B8) together provide the existence of a unique
solution to equation B6). By virtue of property 1) condition B7) can be
replaced by
B9) w'@) = 0.
We proceed as usual and introduce on the segment 0 < r < R the
equidistant grid u^ - {r,- = ih, i = 0,1,... , A^, hN = R).
Still using the framework of Section 2, a difference scheme for equation
B6) can be obtained by the balance method:
''i+1/2 ''i+1/2
''i-1/2 ''i-1/2
vfhere
ri = ih, i=l,2,... ,N - 1 ,
Approximating the flow w by the expression
Wi__i/, ~ r,.__j/2 o-i (w, - w,_i)//i
and substituting d^ u^ r^ h and ^p^ r^ h, respectively, for the integrals in the
balance equation C0), we arrive at the difference equation
1
C1) A j/i = ~ ('■i-1/2 o-i yr,i),. , - rf, Vi = -'^i, f = 1, 2,,,, , Af - 1,

other problems 189
where
_ Vi - %-i _ Vi+i - Vi
yr-,i ^ > yr,i ^
the coefficients a^, d^ and (p^ are chosen so that
C2) a, = fc,„,/, + 0(/i2), di = q, + 0{h'), ^^ :=/. + 0(/i2) .
In the simplest case we accept
C3) fli = ^"i--i/2. di = qi, fi = h-
Let us approximate the boundary condition at ?' = 0 that can be
declared to be the condition of the zero flow at r = 0: w@) — 0. We are
going to show that the difference boundary condition
C4) a,j/,@)=^(g@)j/@)-/@))
has the approximation error 0{h^) on a solution to equation B6) satisfying
the boundary condition B7).
Indeed, the residual for C4) is equal to
C5) f> = a,.v@)-^(g@)j/@)-/@)).
The forthcoming substitutions
a, ^k^ + lhk'^ + 0{h''), «,@) = w'@) + i/i"«@) + O(/i2)
yield
C6) ,) = iku')o + I h {ku% - ^ {q, u, - /J + 0{h^) .
From equation B6) we deduce that
{ku'y = qu- f .
Since u' —*■ 0 as r —*■ 0, we have
*■ (k u'Yq as r —*■ 0

190 Homogeneous Difference Schemes
and
C7) {k u% = {qu~f)o- {k uja = ^ (g« - /)o .
Substituting C7) into formula C6) and taking into account B9), we get
i> = 0(/z2).
The difference boundary condition C4) can be expressed by
--j^ 9oJ/o = -/o. h, = !i.
With these, the difference scheme
A J/ = ~ ((?■ - i /i) a j/,,)^, -dy = -(fi, 0 < r = ih < I ,
C8)
—j^ 9o% = -/o. "* = 4' ?/iv=/^2.
is put in correspondence with problem B6)-B8).
The statement of the difference boundary-value problem for
determination of j/j is
A Vi^, - Ci Vi -F Ai+1 j/i+i =-Fi, f = 1, 2,.. ./V - 1 ,
V / r- a-
which are supplemented with the boundary conditions
D0) ~ % = ^1 ?/i + H > ?/iv = /^2 ,
where
^1 = ai/(ai + ^9o) and ^i = y-^o/("^ "^ T ^V '
This problem can be solved by the elimination method (see Chapter 1,
Section 1), for which the stability conditions are satisfied, because Ai > 0,
Ci > Ai + A'+i and 0 < Xj < 1, Xj = 0,

other problems 191
We now estimate the accuracy of scheme C8). Substituting into C8)
y = z + u, where u is a solution of problem B6)~B7) and j/ is a solution of
problem C8), we may set up the problem for the error z = y — u\
Az = - {{r - ^ h) a z,r)^. - dz = -ip , 0 < r = ih < 1 ,
D1)
r
a, z,
1 ^r,0
9o^O = -^> ^iV=0:
where ip and v are the errors of approximation to the equation
D2) '/'i = 7 (^i-i/2 "i ^f,i)r,i ~diUi + ipi
and the boundary condition
D3) 1^ = ^ ' - qoUo + fo,
respectively.
The balance equation C0) with regard to 'ip gives
■>Pi = T ''Ir.i + i'* > % = ^i-l/2 (Cii t'f, J - (^ M')i~l/2) ,
''i + 1/2 , ''i + 1/2 ,
Next, we set r = r^^ + s/z and develop the expansion of integrals involved in
the formula for ip* in powers of h:
'■j + l/2 1/2
1
/ /(r) r dr ■= — / /(r,- + sh) (r,- + s/i) d.s
h r
'■i-l/2 -1/2
1/2 1/2
/ /(r, + s/i) ds + ^ j (/^ + s/»/; + 0(/i2)) s ds
-1/2 --1/2

192 Homogeneous Difference Schemes
By the same token,
1 f K^
y / qur dr = q^ u^ + ^^ {q u)'- + 0{h'^) ,
SO that
D4) r, " 1^ ^'^''" ^^'' + ^^^'^'
giving II rip* \\^. = 0{h'^).
0
It is clear that the function J]^ = a^u^ ^ — (A;m')j_i/2 = O(h^), meaning
D5) Vi = ri^i/2 \ , \ = 0{h^).
Comparison of formulae C5) and D3) yields v = h^v. This provides
support for the view that v = 0{h), because i> = 0{h'^) as stated above.
In order to estimate the error z = y — u, v/e shall need an auxiliary
lemma.
Lemma Let z be a solution of problem D1) and let v be a solution of the
same problem for d- = 0, i = 1,2,... , N — 1, q^ = 0. Then the inequality
is valid:
D6) ||z|L, = max | zJ < 2 ||v|lr. •
To prove this assertion, it suffices to use only the lemma of Chapter 1,
Section 1, Subsection 8 and to write down the equations for z and z — t; in
the form D1).
The function v^ can be recovered in explicit form from the conditions
''i+i
w.
/2
=-'/'r. J= 1,2,.., ,/v'- 1, w,. = 6^t;-. , 6i = air,._i
Vpf = U, = -V, that IS, w =--~!y
where ip^ is specified by formula D2) for dj = 0. Summing the equations
w^/t+i = "Wk- hr,.ip,. over k -1,2,... ,j yields
D7) t« ■ + ! := Wj - V /l I'k Ipk, ^1 = -^ "■
k = l ^

other problems
193
Substitution of Wj_^^ = fe^^j (^j+i ~ ^j)/'^ ^^^° D7) leads to the relation
D8)
Vi = V,:, -
""'"' ' E^r,^^
°3+l "j+l ^^1
Recalling that t;^ = 0 and summing equalities D8) over j = i, i+1, . . . , A'' —
1, we obtain
D9)
"]+i
3=t
"i+1 fc =
Substituting i/)^ = — r].^ j. + 'tpl into D9) yields
1 ■ ^ 1 1 ■^
"^■+1 J,-:.! "j+l "i+l k:.l
^i+1
X] '^'■fc'^'t
/t = l
1 ., "
< —(hi+il + hil) +
\rr\i
>■ 'J+1 I ' I '1 1/ ' J
< 7A Vi 1 + 1^.1) +711^'^* lie
in light of the relations
0
Furthermore, we might have
J2'' = ^j ^ 'V
+1/2
'i+i
^j+1/2 "j+i
fc=i
/z" I !^ I h
< -TT
^1/2 ^1 4 Cj
As a final result we get the inequality
E0) \\v\l^<?l^ + ^J2h{\V,^,\+\h,\) + ~\\rr\\c-

194 Homogeneous Difference Schemes
By inserting in E0) the estimates
u^O(h), \?,^\ = 0{h'), \\rr\\c = 0{h^)
we conclude that || v \\^ — 0{h?) and, hence,
||^|lc<2Mlc<M^'-
This means that scheme C8) is of second-order accuracy in the space C.
Let us turn to a scheme of the second type known as a "scheme on
the current grid". We split up the segment [0, i?] into iV parts by the nodes
(current points)
r^ = 0, f^ = \h, r.^ = pi, ,,,, 7\-=(i-i)/}, ,,,,
rr,-i = {N-l)h, f^=GV-i)/j = i?,
Denote by y^ = y{ri) the values of a grid function at those nodes.
The balance equation for B6), which is an analog of C0), is aimed at
designing the difference scheme, making it possible to write on the interval
r^_j = fj_w, < r < rj_|_j/2 = r^ the difference scheme
E1) — {ri_,a^^,y^-i)^.- diyi + (p{ri) = 0, i = 2, 3,... , TV - 1,
where r^ ~ ih, r^ = [i — ^)/h and a, d, cp are chosen by analogy with
C2)-C3), so that in the simplest case
E2) ai = k{ri), ipi = f{fi), di = g(f;) ,
The balance equation written on the interval 0 < r < r^ = /i
■ft
(/(r) — q{r) u{r)) r dr = 0, w{r) = r k{r) u'{r).
0
and the condition w^ — Q imply the difference equation
E3) ^-^Yir -diVi+Vi^^
for r = Tj = I /i, where a^, dj, ip^ are specified by formulae E2).

other problems 195
The standard condition
E4) y^ = IJ.2
is imposed for r = R. As a final result we get the difference equation E1)
subject to the boundary conditions E3) and E4).
Let y = y(r) be a solution of this problem. We begin by placing the
following problem for the error z — y — it:
E5) ^^ = (i^i)/^, i = 2,3,...,7V-l, zw = 0,
—:-—^ dj Zj + V-i = 0 ,
Tj h
where V" is the approximation error equal to
i>i = V'(?=,) = — (''i-i «,;-!%,»),.,,■ - '^i "i + ^i ' I = 2, 3,... , TV - 1,
i
V
i'l = T—- «! Mr,l - d^ Wi + ^1
with
Combination of the resulting expressions and the balance equation on the
interval rj__j < ?- < r,; gives
' i
Vi = ai__^Ufi- ki__^u.__^, i = 2,3,... ,7V, V^ = 0 ,
''i-i
^i = —J, / '' /('') '^''' "'^'■^ j:h I '' '^'^''^ '^'' ■

196 Homogeneous Difference Schemes
It follows from the foregoing that
V-=^, f^. = 0(h^).
An estimate similar to that established for problem D1) is valid for a
solution of problem E5) with the right-hand side
^. = r(''-i'^i)r + ^»- = ~
yielding
\\z\\^^\\y-u\l. = 0(h').
This means that scheme E1), E3), E4) is of second-order accuracy if we
agree to consider k{x), g(x), f{x) G C'^^-'[0,1].
5. Diiference schemes for an equation in spherical coordinates. If a solution
to the equation
div (/cgrad u) — qu — —/('', f?, <^)
in the spherical coordinate system is centrally symmetric, that is, is
independent of d and <^, then the function u — u[r) satisfies the equation
E6) 1 dr \ drJ
0 < q < k{r) < c.^ , g(r) > 0 .
In the general setting the function u{r) is supposed to be bounded at the
point r = 0. This property is equivalent to the condition
E7) r'Kr)-r
= 0.
r = 0
At the point r = R we may impose, for instance, the standard condition
E8) u(R) = 1^1^ .
A bounded solution of problem E6)-E8) possesses the same
properties as in the case of the axial symmetry (for more detail see problem
B6)-B8)).

other problems 197
We introduce on the segment 0 < r < i? the equidistant grid lo^ =
{r^ = ih, i = Q,l,. .. , N, hN = R} and by analogy with C1) may attempt
the scheme in the form
E9) Ayi = ^{rf__^^^aiy,-j)^^.-diyi = -<Pi, i = 1,2,... , N - 1 .
i
At the point r = 0 we impose the difference boundary condition
«! y,. 0 h
F0) —j^ ?o J'o = -/o . '** ~  '
and for i = TV set
F1) Vn = 1^2-
These equations admit the form C9) and can be solved by the elimination
method. The main difference from the cylindrical case lies in the selection
rules for >p and d. To obtain the formulae for (p and d, we consider the
residual
F2) Vi = A w; + >pi = — {rf__,f2 Hi "f,i),.,i - di u^ + ipi
i
and subtract from equality F2) the balance equation written on the segment
''i + l/2 ''i + l/2
0= '—-—;: —^ / gMr^dr+—^ / fr-dr,
hrf hrf J hrf J
'■j^l/2 ''i-l/a
lu = r ku'.
The outcome of this is
V-i = ;^ Vi;i + i>*, Vi = 'f-i/2 '^i. *?; = «i «f,i - (^ "')^-i/2 >
i
where
''2+1/2 , ''i + i/a .
F3) ^;-^,-^ y /r^dr- |f/,w,-^ y gwr^drj.
' ^i~l/2 ' '•i-1/2

198 Homogeneous Difference Schemes
After replacing the variable by r = r^ + sh we find that
^i + l/2 1/2
fr' dr=^ f{r, + sh) (r^ + shf ds
hr"^ J r?
1/2 1/2 1/2
/(r; +sh) ds + — s f{ri + sh) ds + — / /(r^ + sh) s^ ds
-1/2 -1/2 ' -1/2
Also, a similar expression can be derived for the second integral by merely
setting qu in place of /. From here and formula F3) it is plain to show
that
r, = ^ (?«- /)■ + o{h'), II rr \\c = o{h')
if (p^ and d^ are specified by tlie formulae
F4) ^.= (i + tC^)/>:. ^■-(i + iIt^)'?- i=l,2,,,.,7V-l,
i i
or by other formulae differing from F4) only by the quantity 0{h'^).
Arguing as in the preceding section, we verify that the residual in the
boundary condition reduces to
F5) V = —^ g^ w^ + /o = 0{h).
To develop those ideas, we contrived to do it with further reference to the
problem for the error z = y — u
F6) Az = -V', reoj^, ^^l' - q^Ug = -;y, z^ = 0.
A solution of this problem can be estimated in a similar way as was done
in the preceding section, but with b^ = r'^__^,.ya^. In concluding this
section we establish through such an analysis that scheme E9)-F1) converges
uniformly at the rate 0(/i^):
\\z\\c^\\y-u\\^ = 0(h').

Difference Green's function
199
3.6 DIFFERENCE GREEN'S FUNCTION
1. Diiference Green's function. Further estimation of a solution of the
boundary-value problem for a second-order difference equation will involve
its representation in terms of Green's function. The boundary-value
problem for the differential equation
du\
A)
dx
[k(x) —-) — q(x) u = —f(x), 0 < X < 1
V dx /
u@) = 0, w(l) = 0, k{x) > Cj > 0, q{x) > 0 ,
can add interest and aid in understanding. As known, the solution of this
problem arranges itself as an integral
B)
i{x) = jG{x,OmdL
0
where G{x,£^) is the source function or Green's function. Function B) is a
solution to equation A) subject to the boundary conditions m@) = 0 and
w(l) = 0 if Green's function G(x,£^) as a function of x for fixed ^ satisfies
the conditions
C)
L, G{x,0 = ^ {k{x) ^^^^^) - q{x) G(x,0 = 0 ■.
x^^, 0<j;<l, G@,O = G(l,O = 0,
dG
[G] = G(e + 0,0-G(e-0,0 = 0,
dx
1 for X ~ ^ .
The very definition implies that Green's function is nonnegative and
symmetric:
G(x,O>0, G(x,0 = G(Ox).
The function G(a;,0 so defined can be written in the explicit form
( a(x)/3@
D) G(x,0=<
a(l)
«A)
for s < O
for X > ^ ,

200 Homogeneous Difference Schemes
where a{x) and P(x) are solutions of the relevant Cauchy problems:
La=0, 0<x<l, a@) = 0, A;@) a'@) = 1,
E)
LP^O, 0<x<l, /?(!) = 0, A;A)/?'(!) = -1.
The functions a(x) and P{x) are linearly independent. This is due to
the fact that the Wronskian is nonzero A(x) 7^ 0. Moreover, a{x) > 0 for
X > 0 and /?(x) > 0 for 0 < x < 1.
We now turn to a difference equation of second order. We learn from
Chapter 1, Section 1 that any difference equation of second order Ai j/;„j —
Hi + Bi yj_,_j — —Fi can be treated as an equation of divergent type,
meaning
Hi+i (%+i - Vi) - ai ivi - y;„i) - di Vi = -ipi .
By replacing here cp^ by h'^(Pi and d^ by h'^d^ we obtain
^yi= -f^ [a,+i {y^+l - Vi) - o-i {iJi - yi_,)] - d^y^ = -^i ,
i = 1,2,... ,7V- 1,
which is more convenient for the further comparision with the differential
equation. For the moment, we write down this equation without subscripts
^y = (ay^) -dy = -(p{x), x = ih ,
F)
a(x)>Cj>0, d{x)>0, i = 1,2,... ,7V-1.
For i = 0 (x = 0) and i = N {x = I) the boundary conditions of the
first kind are imposed as usual:
G) yo = 0, yw = 0.
Common practice in numerical analy.sis involves the inner products
N~l N
{y,v)~ E yi^i^' {y,v] = YlviVih.
8=1 »=1
The traditional way of covering this is to seek a solution of problem F) in
the form
N~l
(8) %= E Gik(Pkh = {G^k,Vk)
k = l

Difference Green's function 201
under the agreement that this expression solves the equation Ay^ = —<^;.
From the equality Aj/,; = Ylk=i ^@ ^ik Vk ^ ^^ i^ easily seen that equation
F) holds true only if A(j) Gik = —6^j./h, where 6^-f. is Kronecker's delta:
1, i = k .
ik
0, i^k.
The equalities y^ = y^ = 0 are certainly true for the choice Gok = GNk = 0,
thus formula (8) gives the solution of problem F)~G) if Gik = G{x^ ,Xf.)
as a function of i for fixed A;=1,2,...,7V— 1 satisfies the conditions
A(i) Gik - [o-i {Gik)x,i) ■ - di Gik - —7- ,
(9) ■ ^
i,k^l,2,... ,N-l, Gofc = Gwi = 0.
We must show that Green's function specified in such a way exists and
find its explicit representation similar to expression D). With this aim, the
functions a^ and /?, will be declared to be solutions of the corresponding
Cauchy problems
Aai = 0, i= 1,2,... ,7V-1,
A0)
tto = 0 , a, a^o = a, = 1,
A/?. = 0, i= 1,2,... ,7V- 1,
Pn = 0, aj^P^j^ = aj^ = -1.
It is straightforward to verify that the functions a^ and /?, possess the
following properties:
1) ttj and Pj are positive functions, a^ being monotonically increasing
and Pi being monotonically decreasing:
tti < tti+i < aw . A < A-i < Po .
Oi > 0 for i = 1, 2,... , 7V and /?,. > 0 for i = 0,1,... , 7V - 1. Indeed,
conditions A0) imply that
^~i h
«j ttg.i = 1 + E '^ 4 "i . «! = — > 0 .
k = l "l

202 Homogeneous Difference Schemes
li Ok > 0 for A; = 1, 2 .. . , i — 1, then a, a^, > 0 and a^ > aj„i > 0.
In a similar way we can be pretty sure that Pg i < 0 and 0 < /?, < /3j-„i.
2) Oj^ — Po or a(l) = /3@). From the second Green formula
(a, A/?) = {P,Aa) + a^ {a P^ -Pog)^ -a, (a 4. -/Ja^)^
it follows immediately that a^ = /?o on account of conditions A0).
3) The determinant A,' = a^ (a^,/?; — a^Pg^) = const = a^ is
positive for 0 < i < TV. Applying the second Green's formula in the domain
{0 < Xf := ih < Xig = x} we obtain
0 = ^ (aAP- pAa)ih = a-^ {a P^ ~ Pa^).^ - a^ {a P^ - Pax)^
1 = 1
Since X; = x is an arbitrary node of the grid w^j, we might have
A(x) = const =/?@) = a(l).
Other ideas are connected with a possibility of arranging the Green
function such as
■ <^^Pk
A1) Gik = <;
for i. < k
a
N
^^ for i>k.
On
Whence it follows that G'ofc = GNk = 0.
The preceding representation will be proved if we succeed in showing
that the function specified by formulae A1) solves the equation Aa^Gik =
— S^i./h. Indeed, A(;) Gifc = 0 for i 7^ A; due to the facts that Aa^ = 0 and
A/3; = 0. The expression A(^i^Gik for i= k needs investigation:
A2) (A(i) Gik)i=k = ri K+i (ttfc Pk+i ~ Ok Pk)
~ H («fc Pk - afc-i Pk)] - cffc Gkk ■
From the condition Ak+i = a^._^_^ {Q|._^_^ P). — a^ Pi.j^^)/h — a^ we
derive the expression aj._|_j 0^/3^+1 = a.k+i '^k+i Pk ~ haj^ and substitute it
into the right-hand side of formula A2). As a final result we get
Pk I , 1 d,p, p, 11
, - aj. = Atti, - - = —-
i-^ ■''" h Oj^ Ojy h h
{A^i)G,k)i=k = T- (««*),,. -T- —— a^ = ~~- Aaf,

Difference Green's function
203
as required.
Formula A1) means that Git > 0 for i,k ^ 0,7V and Git = Gti-
Moreover, Gik as a function of k for fixed i satisfies the conditions
A(jk) Gik
h
Gv
G,
N
0,
In the case of the boundary condition.s y^ = Xj y^ and yj^ = x^y^-i
the Green function can be constructed in a similar way. Of interest is the
0 °
special case where d{x) = 0. Here the functions a = a(x) and /? = /3(x)
can be determined from equations A0) in the explicit form:
S = l * S=J+1 *
and the Green function related to the problem
0, yw = 0 ,
i < k ,
>k ,
a, I ^-^ a.
\ s=l ' s=i+l " / s=l ^
For the best scheme A5), A7) arising in Section 2 Gik coincides on
the grid W/j with Green's function for a differential equation.
2. A priori estimates. The explicit representation (8) of a solution of
problem F)-G) in terms of Green's function lies in the background of
various a priori estimates of a solution in terms of the right-hand side. It
is easily seen from (8) that
A4)
reduces to
A5)
[ay 5-), =
-^. a; G Wft ,
f i . N .
Gik = <
^ a ^ a 1
s = l '^ s=k + l " 1
k , N ,
Y^ n Y^ n
Vo =
1 N
/e
s = l
/ N
/e
A6)
N~l
\y^\<{G^k,\Vk\)^ E Gik\Vk\h
k = l
and, therefore, a uniform estimate of y^ can be obtained by estimating
maXi^fc Gik- In this line, ipi arranges itself as
A7)
f^V,,-,
'?l = 0:
Vi
J2 hep.
2,3,
,W
Upon substituting these expressions into formula (8) we deduce by the
summation by parts formula that
A8) y{x)={G{x,0,Vi) = -{Gi{^,0^v{0], x,U^h-
The next step is connected with | GV"(x,,^)|.

204
Homogeneous Difference Schemes
Lemma For the Green function G{x,£^) related to problem F)-G) the
uniform estimates
A9)
B0)
G(x,0<
1
G,{x,0\<-, |G^-(^,0|<
are valid for all x, ^ (^ cOj-^.
Proof 1) In Chapter 1, Section 1 we have obtained for a solution of problem
F)-G) the estimate
N~l
w-i i
B1)
» = 1 ' + ^ s:zl 1 i = l s = l
fs
Replacing in B1) y,- by Gik and cp^ by S^i^/h, we establish the relations
w-i
?iA: < —■ } h Sik
max G.
i,k C
1 — X,. 1
1 < _
» = 1
where
Sik = Yl ^sk> Sik = 0 for i < k, Sik = 1 for i> k.
s = l
2) Let d{x) = 0 and the Green function G = Go{x,£^) be specified by
formulae A5). Assuming this to be the case, we find that
Gos;(x,£,) = <
a(l)
a(l)
for X- < ^ ,
for X > ^ .
By virtue of the relations
0
1 1
— < —
1 1
— < —
«@<«(i). /?@<«(i)

Difference Green's function
205
we arrive at
B2)
By the same token,
G(ix{x,i)
G,^{x,0
<
Let v{x,i^) = Gq{x,Q — G(x,£^). The equations for Go and G imply
that
Aav{x,0 = -d(x)Go{x,0, v{0,0 = t^(l,0 = 0.
We can calculate the left difference derivative of both sides of this equation
with respect to £^, whose use permits us to establish for w{x,£^) = vf
A:,w(x,^) = {a{x)iug{x,£,))^. - d{x)w{x,^) = -d{x) Go^{x,^),
w@,O = 0, w(l,O = 0.
By the lemma from Chapter 1, Section 1,
B3)
max I iu(x,£^) \ < max | GQf(x, ,^) | <
Using estimates B2) and B3) behind, we derive from the obvious inequality
\G^{x,0\<\Go^{x,0\+\w{x,^)\
the desired estimate B0).
Theorem For a solution of problem F)~G) the estimates are valid:
B4) ||2;|L <- 1,
B5)
\y\\n<- 1'
J2 ^^s
J2^^s
N~l
E^
N-l
N-l
izzl
i
s = l
When (p{x) happens to be of the form ip = rj^ + cp*, a solution of problem
F)-G) satisfies the estimate
B6) \\y\\c<^{i^'\v\] + {h\^^]},
where f.i^ = Yl'kzzi ^ V*k ^O-^'« = 2, 3 .. . , TV and f.i^ = 0.

206 Homogeneous Difference Schemes
Proof Putting fi = rj^ and applying then formula A8), we deduce by the
preceding lemma that
B7) |yi(x)|<(|G^-(x,OI,h(Ol]<-(l>l'?(OI]-
The function ri{x) can be recovered from the condition 1]^ = ip within an
arbitrary constant. A solution to the difference equation
can be expressed either by
N~l N~l
S~i S~i
or by
i i
r]i+, = r],+Y hip^ = Yh(p^ if r], = 0 .
s = l s = l
Substitution of these formulae for r|^ into the right-hand side of inequality
B7) leads to estimates B4) and B5).
To prove inequality B6), it suffices to set ip* = /i^, so that (p = (ri+ij,)^
and to repeat the preceding arguments.
Remark Formula A6) entails the e.stimate
\\y\\c<^(^A^\)<h\^\\<h\^\\c
in the norm || <^ || = \/{tp, tp). ft is plain to show the obvious inequalities
fc=i
With the aid of the estimate
\\G,{x,0\\c<^
an a priori estimate in the space C for the difference derivative of a solution
of the boundary-value problem F)-G) can be derived without difficulty.
Indeed, the relations
iy,i = l(G,(x,o,^(e))|<-(i,ki)
lead to
^1

Higher-accuracy schemes 207
3.7 HIGHER-ACCURACY SCHEMES
1. An exact scheme. For equation A) in Section 1 it is possible to make
the design of a homogeneous conservative exact three-point scheme so that
a solution y^ of the difference problem is identical with the exact solution
u — u{x) of problem A) from Section 1 at all the nodes of any grid tOf^:
y,=u{x^) for A;,g,/Gg(°)[0,l],
Before giving further motivations, it will be convenient to set up problem
A) arising in Section 1 m the form
m@) = Mj , mA) = Mj 1 p[^) = k~^{x), 0 < p{x) < — , q{x) > 0 .
It is worth noting here that the best scheme A4)-A5) of Section 2 is exact
for q = f = 0. Indeed, the function
X , 1 . .
B) u[x) = u^ + c J p[t) dt, c = {u^ — Uj) if p[t) dtj
1 N -1
/
0 '0
is just the solution of problem A) for q = f = 0. From such reasoning it
seems clear that
C / . , C 0
%,» = T / P(i) dt = — , «; u^^i = c ,
where a^ = [h~^ J,' p[t) dt) and, therefore, function B) solves the
equation (au^.)^. = 0.
We now turn to equation A) on an equidistant grid uj^. A key idea in
the further development of an exact scheme is to express at any inner point
(and, particular, at x = x,) of the interval (Xj„j , X;_|_j) a solution u = u(x)
to the second-order equation A) in terms of the values W;„i, Wj_|_i and the
right-hand side f(x). This can be done using u(x) in the form
C) u{x) = Ai vl{x) + Bi vl{x) + vl{x), Xi__^ < x <
'i + l

208 Homogeneous Difference Schemes
where Ai and 5,; are some numbers which will be specified in the sequel,
vUx) and v'Jx) are linearly independent solutions to the homogeneous
equation IAP''''u = 0 (pattern functions) and v^^[x) is a particular solution to
the nonhomogeneous equation A) subject to the homogeneous conditions:
D) L^P-^^vi = f{x), Xi__, <x< x,.+, , vlixi^,) =. vi{xi__,) = 0 .
The accepted view is that the pattern functions v'{x) and v'Jx)
will be declared to be solutions of the appropriate Cauchy problems
E) L^P'"\\=Q, x,„, <x<x,+, ,
F) ^•^^1 = 0, x;„, <x<x;+,,
By merely setting in C) x = .'c^„j and x = a;;_,_j we find that
G) A, =
"(a-'j + i) o _ M(^'i-i)
Pattern functions so defined possess a number of nice properties
(compare with Section 6), so there is some reason to be concerned about this:
1) v'^(x) is positive and monotonically increasing for a;;„j < x < Xj_,_j;
vUx) is positive and monotonically decreasing for X;_j < x < Xj_|_i;
2) the equality is certainly true:
(8) ^U^-i+i) = ^2(^Vi);
3) the relation occurs:
(9) vli^i+i) = v\{Xi) + vl(xi) + vl(xi) J v\ q(x) dx
^i + 1
+ v\{xi) I vl q{x) dx ;
4) the recurrence relation is valid:
A0) vi{x,) ^ v\+\x,^,) .

Higher-accuracy schemes
209
We will prove the indicated properties,
1) Property 1) follows directly from problem statement E)~F).
2) Taking into account E) and F), we have for L = L^P''')
0= j {vi Lvi-vl Lv\) dx=[v\ ~yj~vl ~K)')[^'^]
'■i -1
3) Applying Green's formula on the segment [Xj_j , xj yields
0= / {v\Lvl-vlLv^dx=[v\^~{vl)'-v\^^{^)'
1
1
Pi Pi
Upon substituting here the relations
1
Pi
{v\)'{x,) = l+ / q{x)v\{x)dx
i+i
P,
{viy(x,) = -l- / q{x)vl(x)dx
we establish property 3).
4) Having stipulated the conditions
L(p.*)^;+i = 0, x,<x<Xi+,, vl+\x,) = 0
p dx
the function v''^'^(x) does follow
"^j+i
0= / {vi+'Lvi-viLvi+') dx
-U^'li-iy-<li-'^'y
'i + l
vl+Hx,+,) + vl{x,).

210
Homogeneous Difference Schemes
The function vUx) admits an alternative form
"^i+i
A1) vl{x)= / G(x,0f{0d^, xgK„, ,a.v+J
where G{x,£^) is the Green function related to problem D) (see A1) in
Section 6):
A2)
G{x,Q= <
v\{x)vl{0
X "^ C, <. X^,j
By placing successively expression A2) and x = x, in A1) we obtain
A3)
v'M)
v'M) / viiomd^
H+1
+ v\{x,) J vKOfiOdn-
By virtue of relations G), (9) and A3) we deduce from C) that
A4)
where
\_ / Mj + i - Mj _ Mj - "i-1
h \ v\{xi) v\{xi)
diUi
-fi
'»+i
A5)
fi
hv\{xi)
hvlix,)
y[{Omd^-
v\{0.f(Od^-
hvi(x,)
1
hvi{x,}
viiOqiOd^-^
^i+i
yliOfiOd^

Higher-accuracy schemes
211
We now introduce a local coordinate system at a point x = x, by merely
setting X = x^+sh, s = (x — x^)/h, in which the segment [x;_j , x^_|_j] carries
into the segment (pattern) — 1 < s < 1 with the node x = x^ corresponding
to the point s = 0. Then we concentrate primarily on v'^{x) = v^^{x^+sh) =
ha^{s,h) and v^^{x) = t;*(x,-+ s/i) = hp^{s,h), -1 < s < 1, v'^{xf) = ha^
and, because of A0), v^ix^) = ha^_^_J. Being the pattern functions, a^{s,h)
and P'{s,h) are subject to the conditions
c^" ^» = ^s^s -"'*'"-»•
■1 < s< 1
a(-l,/j) = 0, a'{-l,h) = p{-l),
Lp = 0, -l<s<l, p(l,h)^0, p'(l,h)=^p{l),
where p(.s) = p(x^ + sh) and q{s) = q{x^ + sh) depend only on the values
q{s) and p{s) \p{x) and q{x)) on the segment — i < s < 1 (on the segment
^i~i "£ ^ "£ ^i+i)- Omitting subscript i in formula A4) we obtain the
homogeneous conservative scheme for y{x) = u{x), x G W/^:
A7) {a~'y,)^-dy
where
G Wft, y{0) = Ml, y{l)
A8)
a{x) = a@, h) = A [p{x + sh), q{x + sh)]
0
d(x)
1
a(x)
a{x + h)
a(s, h) q(x + sh) ds
/?(s, h) q{x + sh) ds ,
>p(x)
a(x)
a(s, h) f(x + sh) ds
1
a(x + /i)
/?(s, /i) /(x + s/i) ds .

212 Homogeneous Difference Schemes
The coefficients d(x), >p{x) can be calculated by one and the same formula
(p{x) = F[p{x + .sh), q{x + sh); f{x + sh)] ,
d(x) = F [p{£ + sh), q{x + sh); q{x + sh)] .
Pattern functionals are defined in the class of piecewise continuous
functions:
A[p{s),q{s)] for p(s),g(s)GgW[^l,0] and
F[p(s),q(s); f(s)] for p(s),q{s), f{s) G ^("^[-1, 1] .
It is easily seen from A7)-A8) that the exact scheme does not belong
to the family of schemes A6)-A7) in Section 2, whose pattern functionals
^[j3(s)] and F[f{s)] depend solely on a single function. In the case of
equation A) with constant coefficients p(x) = p^ = const and q(x) =: g^ =
const the pattern functions a(s, h) and /?(s, h) can be determined explicitly:
X h H h
At the same time the coefficients a{x) and d{x) become constant values
sh (x/}) 2^0 , x/i
a{x) = p,^^^, d{x) = ~-j^th — .
2. Schemes of arbitrary order accuracy. There is no difficulty to construct
a scheme of arbitrary order accuracy by means of appropriate expressions
for coefficients of an exact scheme.
We now know from A6) that a{s, h) and /3(s, h) are analytic functions
of the parameter h"^ and, therefore, can be expanded in the series
oo oo
A9) a{s, h) = E «fc(s) h'', P(s, h)^J2 p,{s) h^\
where ai.{s) and /3j,(s) are calculated by the recurrence formulae
S , t \ S
a,{s) = J p{t)U a,_,{X}q{X)d\] dt, k>0, a„{s) = J p{t) dt,
1.1 ^ 1
P,(s)^ f p(t)l f p,^,{\)q{\)d\] dt, k>0, p^{s) = j p{t) dt .

Higher—accuracy schemes 213
One trick we have encountered is to take only a finite number of the
members in series A9):
m m
k=0 k=0
making it possible to determine the coefficients a'-'"^ d^^' and <^('"^ by
formulae A8), where the polynomials a^™^ and /J^™) stand for a and /?. As
a final result we get a scheme of accuracy 0(/i^™"'"^) in the class of piecewise
continuous functions k{x), q{x), f{x) G Q^ ''[Q, 1]. Any such scheme is called
a truncated scheme of rank m.
For m =: 0 it refers to schemes of zero rank and accuracy 0{h'^) with
regard to k, q, f (^ Q^^^- When providing current manipulations, the
expressions for d and ip involved in this scheme
-j(o) — _ — / p(x _|_ g/j") dg ^ P = 7
dW = d + {hd,).,, ^(°) = ^+(/j^,),.^
0
of ds f
q[x + th) dt,
f{x + th) dt
are different from those being used in the best scheme A4)-A5) of Section 2.
Truncated schemes-provide a possibility to attain any order of accuracy for
arbitrary piecewise continuous functions k{x), q{x) and f{x) and appear to
be useful in many aspects.
An exact scheme and truncated schemes can be designed on an
arbitrary non-equidistant grid w,j by the same methods as we employed before.
In practice the use of truncated schemes in the case of equation A)
with variable coefficients necessitates carrying out calculations of multiple
integrals on each interval of the grid. Replacing those integrals by finite
sums we are able to create more simpler schemes of accuracy 0{h'^) and
0{h^), whose coefficients can be expressed through the values of k, q and /
'^^
^:¥
— u.
0
= a
J
-1
0
/
-1
k (x + ■
ds
k{x +
sh)
sh)
J
-1/2
s
[
J
-1/2

214 Homogeneous Difference Schemes
at some isolated points on each segment [Xj_j , Xi_|_i]. Such schemes permit
one to retain the order of accuracy in the case of discontinuous coefficients
k, g, / on the grids loj^^K) when discontinuity points fall into the nodes of
the grid ^^(A').
Exact and truncated schemes are quite applicable in the estimation
of the accuracy of schemes A6)-A7) in Section 2. This approach allows to
weaken or get rid of the smoothness of the functions k, q, f involved in the
estimation of the accuracy order of schemes A6)-A7) in Section 2.
3.8 METHODS FOR DESIGNING DIFFERENCE SCHEMES
1. General remarks. From such reasoning it seams clear that in the space
of grid functions difference schemes should retain the basic properties of
differential equations such as self-adjointness, the validity of certain a priori
estimates (for instance, the maximum principle), etc. Moreover, a scheme
which interests us must satisfy, first at all, the requirements of solvability,
stability, approximation and, hence, accuracy of a certain order; at last,
an algorithm must be expedient in computer resources. The expediency
depends not only on a scheme, but also on making a substantiated choice
of the method for solving difference equations and a grid which,
generally speaking, may be non-equidistant and depends on the behaviour of a
solution.
The construction of schemes with the indicated properties and a
desired quality is one of the outlines of the possible theory.
As a result, a considerable amount of effort has been expended in
designing various methods for providing difference approximations of
differential equations. The simplest and, in a certain sense, natural method
is connected with selecting a suitable pattern and imposing on this pattern
a difference equation with undetermined coefficients which may depend on
nodal points and step. Requirements of solvability and approximation of
a certain order cause some limitations on a proper choice of coefficients.
However, those constraints are rather mild and we get an infinite set (for
instance, a multi-parameter family) of schemes. There is some consensus of
opinion that this is acceptable if we wish to get more and more properties of
schemes such as homogeneity, conservatism, etc., leaving us with narrower
classes of admissible schemes.
Actually this way has been demonstrated in Sections 1-3. Several
methods find a wide range of applications in designing difference schemes
of a desirable quality, among them
1) the integro-interpolational method (see Section 2);

Methods for designing difference schemes 215
2) the variational difference methods (the Ritz method and Bubnov-
Galerkin methods) and the finite element method;
3) the method of the summator identity;
4) the method of approximating a variational functional.
2. The integro-interpolational method (IIM). In Section 2 we have already
studied the IIM, but its possibilities and potential have not been illustrated
in full measure. Here we consider other ways of its applications by appeal
to the problem
A) {ku'y -q(x)u= ~f{x), 0<a;<l,
B) ku'— a^u = —jj.^, x = 0, —ku' — (T^u=—ij,2, x=l,
0 < Cj < A;(x) < C2, ""i > 0, o'2 > 0, g(x) > 0 .
We introduce an equidistant grid on the segment 0 < x < 1
cD^ = {x^^ih, i = 0,l,... ,7V, hN = 1}.
Integration of equation A) is accomplished over the segment x^ < x < x^_^^,
leading to
C) Wj-_|_i — Wj = / (qu — f) dx = $^+1, w = ku'.
Here, in contrast to Section 2, the flow —w is taken at the same node x^
as the unknown function u, which is sought. Therefore, the intention is to
use instead of t«j_|_j,2 the approximation (Wj_|_i + Wj)/2 by accepting
D) ^(Wi+i +w,) ~ Hi+i
'S.J + l '
where Ojij is some functional of k[x) on the segment x^ < x < x^.^
satisfying the relation a^ = k[x^_^,2) + O(h^). In the course of the elimination
of Wj from C) and D) we find that
Subtracting from here the relevant expression

216 Homogeneous Difference Schemes
and taking into account C), we establish
•■^'i+i
■^■j-i
The approximation of the integral on the right-hand side of E) can
be done using various quadrature formulae, for instance, by the formula of
trapezoids
^ / (?" - /) dx Ki {qu - f)i
If 1
^ / (qu-f) dxKi ~{{qu-f)i_i +2 {qu~ f), + (qu-f)i+i)
= (?'»' - /).: + -J- {([U - f)s;x,t ■
With these, we arrive at the schemes of accuracy Oih?):
F) {°-yx):c - qy = -/,
G) {ay, - ^ (g yh)^ -qy=-{f+~ 4,) .
To approximate the boundary condition, for instance, at the point
X = 0, we apply C) for i = 0
h
Wj — Wq = / {qu — f) dx
0
and then adopt here
h
Wi ?a tti u^Q + ^ f{qu- f) dx , lUo = {ku')Q = a^u^- fj.^,
0
so that
h
«i "^x 0 - (c^i "o ^ /^i) ~ i / (<?" - /) dx .
0

Methods for designing difference schemes 217
Making use of the approximation
h
f{qu - f) dxK. {qu - f)o h ,
0
we impose the boundary condition for y^ with the appi-oximatioii error
0{h'^) at the point x — 0:
(8) «i Vxfi = ^iVo - 1^1, ^i= ^i + \hq^, fJ-i = IJ-i - ^ h fa .
The boundary condition B) of the third kind can be established at the
point X = 1 in a similar way.
So far we have studied some versions of the IIM on the basis of the
balance equation (the balance method). We now consider the second method
for the design of homogeneous difference schemes by means of the IIM, in
the framework of which equation A) has to be integrated twice: first, we
integrate equation A) from x^ to x:
(9) w{x) - iu{xi) = / {qu - f) dx .
Second, we integrate the preceding over x from Xj to x^_^_^ and from x^_^
to Xf:
A0) / w{x)dx-w,h= I dxi l{qu-f)dt\,
A1) / w{x) dx ~ w^h = / dx i / (qu — f) dt
The interchange of the integration order leads to
dx I [{qu- f) dt] = f {x,^, - t) {qu - f) dt ,
, X \ ^l
dx I / {qu - /) dn = - / (i - x;_j) {qu - f) dt

218 Homogeneous Difference Schemes
The function m(x) is linearly interpolated on each of the segments [x^_j , xj
and [Xj ,Xi_|_i], so that
{Mj + (x — Xj) u^^^ for X,- < X < x,-_,_j
ttj + (x — X J Mg J for Xj_j < X < x^
which simplifies the huge job done with the integrals
"-i+i ■'•i+i ■'-i+i
/ w dx = / A;m' dx ?a u^ , / A;(x) dx ,
w dx ~ Wj i / A;(x) dx
/ gM(i — Xj_j) di ?a —Mj / qit) (t — x,-_i) dt
x;
+ i / 9(^)(»i-^)C^---'=i-i) c?^;
■'•i-i
(x,_^i - t) qu dt ^ M, / q(t) (x,-_^i - i) dt
Xi
+ "x-,i / (l{t){xi+i-t){t-Xi) dt.
Substituting the resulting expressions into identities (lO)-(ll) and
subtracting the second identity from the first one, we arrive at the difli'erence

Methods for designing difference schemes 219
scheme
A2) {ay,)^,-dy=-^,
^i = \ I k{t)dt-^ / q{t){x,-t){t-x,_,)dt.
The right-hand side Lp^ can be determined by the same formula as we used
for dj with f{t) standing in place of q{t). The formulae for a^ and dj can
be rewritten as
0 0
U' — k{x^ + sh) ds + h"^ / s(l -|- s) q[X' + sh) ds ,
-1 -1
A3)
0 1
d,- = A + s) q{Xi + sh) ds + A — s) q{x^ -\- sh) ds .
-1 0
If k and q are constants, then
A3') aj = A;-yg, di - q ■
In the sequel we will show that scheme A2) is identical with the scheme
emerged in variational difli'erence methods (the finite element method).
In what follows we share our practical experience of the design of
difli'erence schemes for problems with lumped parameters by means of the
IIM. Suppose, for instance, that a single heat source of capacity Q is located
at a point a; = ,^ so that a solution of problem (l)-B) satisfies the conditions
A4) [m] = 0, [k ^] =-Q for x=i.

220 Homogeneous Difference Schemes
Let ^ = a;„ + 6'/i, 0 < 6* < 1, that is, x^ < £_ < x„_,_j, n > 0. The
balance equation C) on the segment x,-j < x < x^,^ is of the form
or
A5) to„+i - w„ = $n+i + Q .
On the remaining segments [x^ ,K(_|_i], i ^ n, identity C) holds true. As a
final result, instead of A2), we get the scheme
where
'Pi = fi, i ^n, i^n + 1 ,
^''^ f.^ f .^
fn — In ' n 1 ' fn + l — Jn + 1 ' O U '
In the physical language, this is a way of saying that the source is spread
over two intervals.
Rewriting the same identity on the segment [xi_i,2 ,x^_^_^j2] reveals
another scheme
A7) (ayj;)^ -dy = -(p, ^„^i = /„+! , ^„ = /„ + -
for 0 < 0 < |,
f ^^ f
Pn + i = Jn + 1 + — . Vn = In
for I < 0 < 1.
Let us stress that the integro-interpolational method is a rather
flexible and general tool in designing difli'erence schemes relating to stationary
and nonstationary problems with one or several spatial variables.

Methods for designing difference schemes 221
3. Variational difference methods (the Ritz method and the Bubnov Ga-
lerkin method). The Ritz and the Bubnov-Galerkin variational methods
have had considerable impact on complex numerical modeling problems and
designs of difference schemes.
Let A be a self-adjoint positive definite linear operator in Hilbert space
H equipped with an inner product (,) and let / be a given element of the
space H. The problem of minimizing the functional
A8) I[ti] = {Au,u)-2{uJ)
is equivalent to the problem of solving the equation
A9) Au = f.
The element Ug (E H satisfying the equation Aug = / and realizing
min I[u] = I[ug]
is unique.
The main idea behind the Ritz method is to take into consideration
a sequence of finite-dimensional spaces Vn with basis functions (p\ , i =
1,2,... ,n, and look for an element m„ G 14, minimizing the functional I[u]
in the space V„.
Still using the framework of this method, we may attempt an
approximate solution tt„ in the form
n
B0) w„ = E Vj Vj
7 = 1
with unknown coefficients y^, y^, ■ . ., y„. By inserting this expression in
the formula for I[u\ we find that
n n
B1) I[Un] = J2 ^ij yi 2^7 ~ 2 X] Pj Vj .
2,7=1 7 = 1
where
B2) a,^={A^^,^j), /?, = (/,^,).
Since A = A* is a self-adjoint operator, we have a^j = a.^. The functional
I[u„] is a function of n coefficients y^, y^, .. ., y^. By equating the
derivatives dl[uj^]/dyi to zero and using the symmetry of coefficients a^j = a,-j,
we obtain n equations for determination of y^:
n
B3) ^ a,^. y^. -/?, = 0, i = l,2,...,n.
7 = 1

222
Homogeneous Difference Schemes
The Ritz method proves to be useful in studying the problem
B4) {ku'y -qu = -f(x), 0<x<l, w@) = 0, w(l) = 0.
Other ideas are connected with the function
0, s < -1, s > 1 ,
B5)
r](s) =<(l + s, -l<s<0,
1 - s, 0 < s < 1 ,
which guides the choice of the functions
B6)
fii^) = V
Vi{x):
where x^ = ih, i = 1,2,... ,7V — l,isa node of the grid W/j = l^x^ = ih, i
0, 1,. .. , TV, hN = 1}. It follows from the foregoing that
B5') 7?,(X) =: <^
and, hence.
X
X
'^i
h
+1 ~
-1
X
B7)
dVi
dx
0 for X < X;_j and x > x^,^
for Xj„j < a; < x^ ,
for Xj < X < Xi_^_^ ,
' 0 for X < a;j_j and x > x^ , j ,
1
S h
1
for x^_^ < X < x^
for Xj < a; < x^.j
Upon substituting
into B2) we get
d / du\
All = --- [k —-] + qu
dx \ dx /
B8)
1 1

Methods for designing difference schemes 223
In light of the properties of the function rii{x) and its derivatives the matrix
{a^j} is tridiagonal, because only the elements with j = i — 1, j = i and
j = i + I are nonzero.
In the new notations
B9) ai = -haii_^ , ^^c^i = ^ «;,; + ^ (a^.i-i + ai,i+i)
we obtain
If If
a^ — — / k{x) dx — — I q{x) [x — X;_i) (Xj — x) dx ,
C0)
/ ^i ^i + 1 \
d; = —^ ( / q{x) {x — x,-_j) dx + / q{x) (Xj_,_i — x) dx
Then the system of equations
can be rewritten as
a-i yi-i - ("i + fli+i + /»"<■) % + "i+i ^i+i + /jV, = 0
or
C1) (ay^)^ - dy = -ip ,
where
C2) "Pz = j^ i / /(x)(a;-Xi_i) da;+ / f{x){xi_^_,-x)dx\,
thereby clarifying that <^j are calculated by means of the same formula as
So, the three-point scheme C0)-C2) constructed by the Ritz method
is identical with scheme A2) obtained by means of the IIM. In contrast
to the Ritz method the Bubnov-Galerkin method applies equally well to

224 Homogeneous Difference Schemes
problems, whose ingredients have no fixed sign and turn out to be non-self-
adjoint. In this case the coefficients y^ of the approximate solution B0) are
to be determined from the orthogonality conditions for the residual Au„ — f
with respect to all of the basis functions rii{x):
C3) (Aw„-/,7?i)=0, z=l,2,..,,n.
A non-self-adjoint boundary-value problem acquires the form:
[ku'y + r(x) u' — g(x) u = —f{x) ,
C4) 0<a;<l, w@) = w(l) = 0,
k[x) > 0, q{x) > 0.
We introduce the grid d)^ = {x^ = ih, i = 0,1,. .. , TV, IiN = 1}. Then the
dimension n of the space Vn equals TV — 1. The functions
7?,(X) = 7?(^^), i=l,2,..,,TV-l,
where the function ??(s) was specified by B5), provide the background for
subsequent constructions.
In this context, condition C3) becomes
N-l
C5) ^ a,,.y,.-A = 0, f = l,2,...,TV-l,
7 = 1
where
1
0
C6)
1
P^ = J f[x)i^,{x)dx, z,i = l,2,...,TV-l.
0
By definitions B5), B5') of the function i]i{x), the coefficients a^ j are
nonzero only for j — i — 1, i, i + 1. Retaining notations C0) for a,; and d^
and C2) for cp^ and accepting
r[x) (x — Xj_i) dx = j r(Xj- -|- sh) A -|- s) ds ,
/l2
'-i-i
-1
C7)
'->+ = J^ '■(*') (*'« + l ~ ^') ^^ = / ''(•^i + ^^) (^ ~ *') '^^ >

Methods for designing difference schemes 225
we reduce the system of equations C5) with the members 6+ and h~ to
C8) -^ K+i (^i+i -Vi)- ai ivi - y^.j]
+ 1 + 1 d,y, = -^,,
i= 1,2,... ,iV- 1 .
Thus, we arrive at the difference scheme
(«fe)^. + ^"Vx + b'^Vx - dy= -(p{x), 0 < X = i/i < 1 ,
C9)
Vo = 0- Vn = 0.
whose coefficients can be recovered from C0), C1) and C7).
For r(x) = 0 this scheme is identical with scheme C1)-C2) obtained
by means of the Ritz method. In the case of constant coefficients k(x), ?'(;c)
and q[x)
K^ _ r _
a-i = k- — q, di = d= q, b^ = b+ = -, b. y^ + b+y^ =ryo .
When the coordinate functions (Pi{x) = i][{x — Xj-)//}) are chosen by
an approved rule as suggested before, the Ritz and the Bubnov-Galerkin
methods coincide with the finite element method.
4. The method of approximating a quadratic functional. The boundary-
value problem
Lu= {kv.'y-q(x)u = -f{x), 0<a;<l, m@) = 0, w(l) = 0,
is equivalent to the problem of searching a minimizing element for the
functional (see Section 3)
1 1
D0) I[u] = / [k{u'f + qiP] dx - 2 [ fu dx .
Recall that the equation Lu = ^f{x) is Euler's equation related to such a
functional I[u].

226 Homogeneous Difference Schemes
We introduce the grid tOf^ = {xj = ih, i = 0,1,... , TV, hN =1} and
approximate on it the functional I[u\ by appeal to one of the auxiliary
modifications
I[u\ = Y. ^("')^ dx + J2 / (?"^ - 2/«) dx .
with this in mind, we approximate the integrals
k{u'Y dx Ki ttj {u^iYh ,
I (qu'' - 2/m) dx « - ((gw^ - 2 fu), + (qu'' - 2/«),-_i)
where a^ is a functional depending on k[x) on the segment Xj_j < x < x^.
There are many ways of taking care of these restrictions. For Instance, we
might agree to consider
1 f
a, = — / k{x) dx , etc .
^'i-i
Thus, instead of/[m] we deal with the functional
N W-1
D1) h[u] = Y^a, {u,^,f h+Y. i'i^ y' - 2 /.: Ih) ^ ,
i=i 1=1
where y is an arbitrary grid function vanishing for i = Q and z = TV:
y^ = yj^ = 0. The functional //Jw] is a function of TV — 1 variables y^, y^,
. . ., y]\/_i ■ Equating the first derivatives
-Q^ = 2 a^'+i y^/-l) + 2 a,; y^^ + 2q,y,h~2fih
to zero leads to the difference equations
D2) (ay^)^- qy = -f{x), x = ih.

Methods for designing difference schemes 227
Adopting those ideas to problem A), B), A4) concerning a point heat
source, an excellent start in this direction is to replace the function f(x)
involved in formula D0) by f{x) + S{x — (^)Q, where 6{x — ^) is Dirac's
delta-function. Recall that 6(x — ^) = 0 if a; :^ ^, S[x — ,^) = oo if x = ^ and
If-e H-'^ — £,) dx = 1 for any £ > 0. As a final result we get
1
D3) I[u] = [ [k{u'f + g«2 - 2 /«] dx-2Q w@ .
If ■? = ^'n + ^h, 0 < 6 < 1, then u[£^) must be replaced by u„ for S < ^
and by w„_|_i for ^ > |, whose use permits us to establish
N N-1
D4) h[u] = J2 «i {y^.ifh + J2 i"^' yl -'^^iyi)^-^
where
i=l 1=1
D5) ^, =
ii + f ^^,n fol- e < i
/i + -^ '^i,n+i for e > ^
and S^. is, as usual, Kronecker's delta. Equating dlh/di/i to zero we arrive
at the scheme {ayg)x — dy = —ip with the right-hand side specified by
formulae D5).
In the case of a non-equidistant grid w^ == {x, , i = Q,l,... , N, Xg =
0, Xj^ = 1} we obtain instead of D4)
N W-l
h[y] = J2 «i (yij)^f'i + J2 ("^i yl - '^^i y^^^i ■
, 1=1 »=i
In particular, it is always possible to choose a grid so that ^ = x„ would be
one of the nodal point and
fn = fn + ^ . Vi^^- ft, ii^n.
We are led by equating the derivatives dlh\u\ldy^ to zero to the scheme
D6) {ay^)ij- d^y, = -(fij , i = ] ,2,. .. , N - 1 .

228 Homogeneous Difference Schemes
5. The method of the summator identities (the method of approximating
an integral identity). A solution of the problem
Lu = {ku'Y — qu = —f{x) , 0 < X < 1 ,
D7)
ku = (Tjtt — yttj , X = 0, —ku = a^u — H^ , X = 1 ,
satisfies the integral identity
1
D8) /[w, v]= {ku'v' + quv- fv) dx + ff, w@) v{Q)
0
+ a^u{l)v{l) - n,v{{)) - n^v{l) = Q ,
where v = v(x) is an arbitrary function being continuous on the segment
0 < X < 1 and having the summable derivative in the space L2[0, 1]. This
identity will be used for the determination of the generalized solution of
problem D7).
The design of a difli'erence scheme on an equidistant grid w^ = {Xj =
ih, 1 = 0,1,... , TV, hN = 1} is based on the approximation of the integral
identity D8) by the summator identity for grid functions, for instance,
N N-l
D9) Ih [u, v] = Y^ Ui y^^ i Vg^ ,.h.+ Yl (?'■ ^i " ^'^ ^i '*
!=1 i=l
+ ^i Vo ^0 + ^2 Vn ^n - fii ^0 - A'2 ^w = 0 .
where v^ is an arbitrary grid function. Here a^ is any of the coefficients
having the form a^ = A[k{x^ + sh)], —1 < s < 0, and providing an
approximation of order 2: a, = kj_^,2 +0{h'^). There is no difficulty to verify
that
o-j = cTj + i /i go, 0-2 = cr, + I /i g^ ,
/ii = /ii + I /i /o , ft2 = fi2 + ^hfj^
if the trapezoids formula is in hand in computing the integrals
[qu — /) dx

Methods for designing difference schemes
229
by means of the relation
{qu - f) dx= -h(qu- f)o + -h (qu - /)i
etc. Accepting, for instance, v^ = S^ j„, 0 < ig < N, and recalling that
%_i = 0 for i < ig and i > ig + 1, v^ ^^ = 1/h and Vg ,;^_,_j = —l/h, we find
for i = L that
'^i+l Vx^i '^i yx,i
h+{diiM-f,)h = 0
or (ayj)x. - dy= -/.
If Vf = Ej Q, then v^ , = (~i//i) E, j and identity D9) yields
{-l/h)a,y^j +a,y„ - ft., = 0
or «! J/.f, 1 = ^1 ^0 - /ii ■ Likewise, for v^ = S^j^
With these, we arrive at the difference boundary-value problem
(«J'x)^ -dy=-f, 0<x = ih <l ,
«i Vx-, 0 = ^1 J'o - a"'! , -«w yx,N = ^2yN - N ■
3.9 STABILITY WITH RESPECT TO COEFFICIENTS
1. Stability of diiference schemes with respect to coefficients. In
solving some or other problems for a differential equation it may happen that
coefficients of the equation are specified not exactly, but with some error
because they may be determined by means of some computational algorithms
or physical measurements, etc. Coefficients of a homogeneous difference
scheme are functionals of coefficients of the relevant differential equation.
An error in determining coefficients of a scheme may be caused by various

230 Homogeneous Difference Schemes
factors: an error in calculations of pattern functionals, an error in specifying
coefficients of a differential equation, rounding errors, etc.
We say that a scheme is stable with respect to coefficients (co-
stable) if a solution of the boundary-value problem has slight variations
under small perturbations of the scheme coefficients, Iii order to avoid
misunderstanding, we focus our attention on the scheme with coefficients
a, d, cp
A) Ay= {ay^.)^-dy = -ip, 0 < x = ih < 1, y@) = 0, y{l) = 0 ,
and the same scheme with perturbed coefficients a, d, if (for the sake of
simplicity the values y[0) and y{l) remain unperturbed)
B) Ay={~ay,.)^-dy=-^, 0 < x = ih < 1 , y@) = 0, y(l) = 0 ,
under the conditions
a(a;) > Cj > 0, d[x) > Cj > 0,
C)
d{x) > 0, d{x) > 0, Cj = const > 0 ,
with constant Cj independent of a grid,
We estimate the difference z = y — y in terms of perturbed coefficients,
Substituting y = z + y into B) and taking into account A), we get
D) Az = (azj)^ -dz = -'i, Zo = Zn = Q,
where
E) ^ = <f-cp + {A-A)y=<p-ip+{{a-a) y^)^ -{d-d)y.
From D) it is easily seen that ^ is representable by
F) , * = /i. + V. ,
where
G) /( = («- a) y^.
and 7] is determined from the conditions r]^ = if — ip — (d — d)y and 1]^ = 0,
so that
(8) m=E h[{^k-n)-{dk-dk)yk], i = 2,3,...,N, t?, = 0 ,

stability with respect to coefficients 231
To evaluate a solution of problem D)-G) on the basis of F)-(8), we
rely on the estimate obtained in Section 3 for D)-G), making it possible
to establish
\\^\\c<^{{^'\(i^-^)y^-\] + (^Av\]}-
"-1
We shall need the estimates for y and yg from Section 6 such as
Combination of the inequalities
[h\v\]<[h\v\] + {^Ad-d\)\\y\\^<{l,\V\]+-{l,\d^d\){l,\^\),
where Vi = Yl K^k - ^t), ^i = 0, i = 2, 3,... , TV, and
fc=i
{lMa-a)y,\] < 1 A, | ^ |) A, | S - a | ]
gives the estimate
(9) ||^-y|lc<f {(l-l'?l] + f (l,l^l)((l,|d-d|) + (l,|«-«|])},
here y^ is a solution of problem A) and y,; is a solution of problem B),
provided conditions C) hold.
Relation (9) can be replaced by a more rough estimate
A0) ||j/-y||,^, <-{(!,||^-^||+-(l,|^|)((l,|d-d|) + (l,|5-a|])|,
If
{l,\i^\) = p{h), {l,\h-a\] = p{h), {l,\d-d\) = p{h),
where p{h) —> 0 as /i —> 0, schemes A) and B) are said to be co-equivalent
and for p{h) = 0{h™) they are of the mth order of the co-equivalence.
If schemes A) and B) are co-equivalent and scheme A) is convergent, then
so is scheme B). This fact follows immediately from the inequality
||y-«|lc<ll^^2^llc+ll2^-"llc—^0 a« ^'^^-
The co-equivalence property of homogeneous schemes lies in the main idea
behind a new approach to the further estimation of the order of accuracy
of a scheme: on account of (9) or A0) its coefficients a, d, ip should be
compared with coefficients a, d, (^ of a simple specimen scheme, the accuracy
order of which is well-known (see Section 7).

232 Homogeneous Difference Schemes
2. Stability of the first kind operator equations with respect to coefficients.
Here we give the general formulation of the concept of stability with respect
to coefficients for the difference scheme in Section 1 by having recourse to
an operator equation of the first kind
A1) Au = f, feH,
where A is a linear operator acting from Hilbert space H into H, A: iJ —>
H, f (^ H is & given vector, m G iJ is the unknown vector.
Problem A1) is said to be well-posed if there exists a unique solution
of equation A1) for any f (z H and this solution continuously depends on
the right-hand side /, so that
A2) ||«-«||A)<A//0||/-/||B),
where m is a solution of equation A1) with perturbed right-hand side /
A3) Au = f,
here || ■ |L^ and || ■ \\f^. are some suitable norms on the set H.
A case in point that in the statement of problem A1) we must specify
not only the right-hand side, but also the operator A. If, for instance, A is
a differential or difference operator, the coefficients of the equation should
be known in advance.
It is natural to require a solution of problem A1) to depend
continuously not only on perturbations of the right-hand side, but also on
perturbations of the operator A (for instance, on the coefficients of a
difference operator). As in the case of difference schemes arising in Section 1,
this property of operator equations is to be understood as stability with
respect to coefficients or co-stability of an operator equation.
The stability of a solution to equation A1) with respect to
perturbations of the right-hand side / and perturbations of the operator A is called
strong stability. 'The problem statement here is as follows: with regard
to the equations
A4) Au = f, Au^f,
where A and A are linear operators, whose domains coincide with the entire
space H, f and / are arbitrary vectors of the space iJ, it is required to
estimate the perturbations of a solution
A5) z ~ u — u

stability with respect to coefficients 233
via perturbations of / and A.
Suppose that the inverse operators A~' and A"^ exist. Moreover, we
assume that A and A are self-adjoint positive operators. Substitutions of
u = A"^ f and u — A"^ f into A5) yield
A6) z = A~'f~- A~'f = I-i (/-/) + (I-i - A-i) / .
Applying A^'"^ to both sides of equality A6) we find that
A^/^z = A''/^ if ~ f) + A'/^A-' - A-') f .
We will estimate the vector z in the norm ||^||^ = y(Az,z) of the
space H^ together with the same things for / and / in the negative norm
II-^IIa"' = y(^~V,/) of the energy space iJ;j_i. Via transform
^1/2 B-1 „ 4^1) y. ^ (^ _ 21/2^-1^1/2) B-1/2/)
we obtain
A7) \\A''^{A~' -A~')f\\ < 11(^-11/2^-111/2 II . ||2-i/2/||.
As a measure of perturbations of the operator A we adopt a relative
variation of the energy (Ax, x) of the operator A. The meaning of this is that
we should have for all x G -ff
A8) \{(A- A)x,x)\ < a{Ax,x), a>0,
whence it follows that
A9) (l-a)A<A<{l + a)A,
B0) {l~a)A~^ <A-^ <{l + a)A-\
Observe that B0) is an immediate implication of A9). To make sure
of it, we compose the difli'erence J = A -|- a){Ax,x) — [Ax,x) and insert
Al/2;c = y.
J = {l + a)\\ y\\' - (A-i/^lA-i/V y) = {l + a) || y\\' - (Dy, y) ,

234 Homogeneous Difference Schemes
where D = A~'^I'^AA'^I'^, and, after this, pass to 0^1"^%) = z:
J = (l + «)(Z3-i2,2)-||2||2
=: {\ ^ a){A'l'^A~^ A^l"^ z,z) -\\zf
= (l^a){A~^v,v) -{A~^v,v) = A^l'^z.
As far as J > 0, we have A + a) A~^ > A"^, thereby justifying that
for any operators A = A* > 0 and A = A* > 0 the inequality A~^ <
A + a)A~^ follows from the inequality A < [1 + a)A. Moreover, we claim
that inequalities B0) are equivalent to
{l-a)E <C<{l + a)E, C = A^/'^A-^ A^/"^.
Indeed,
A + a){A-'x,x) - {A~'x,x) - A + a) \\y\\^ - {A'/^A-'A'/^y,y)
= (l + a)\\y\\'-{Cy,y)>0.
Thus, A8) implies that -aE < E - C < aE and C = A^I'^A-^A^I'^.
By the definition of the norm of self-adjoint operator,
||^-C|| = \\E^A^I''A-^A^I''\\<a.
Substituting this estimate into A7) we deduce from A6) that
II^IU-<II/~/IIa-+«II/IU--'
or, what amounts to the same,
ir«-«iiA<ii/-/iiA-+«ii/iii—
If we are in possession of an operator Aq = A*^ of rather simplified
structure than the operator A satisfying the condition A > CjAq, Cj > 0,
then
A-'<yA-,\ ||/||;r-<^ll/IU--'.
provided the inverse A'^ exists. Thus, we have proved the comparison
theorem,

stability with respect to coefficients 235
Theorem Let u be a solution to equation A1) and it be a solution to
equation A4), where A, A and Aq are self-adjoint positive operators for
which the inverse operators exist. If condition A8) and the inequality
A > CjAq, Cj > 0 hold, then the estimates are valid:
B1) II«-«IIa<II/-/IIa-' + «II/IIa-'.
B2) ll«-«IL<fll/"/IU- + f ll/IU--
The first summand on the right-hand side of B1) is the value of
perturbations of the right-hand side / and the second one involves the coefficient
a, which is the value of a relative perturbation of the operator.
Example Let H he the set of all grid functions defined on ui^ = {xj =
ih, 0 < i < TV} and vanishing for i = 0 and i = N. We refer to the
difference operators
Ay=-{ayg)^, + dy, a > Cj > 0, d>0,
Ay =-{ays)^. + dy, a > Cj > 0, d>0,
^oy = -ys^ ■
By introducing the inner product in the usual way and applying Green's
formulae we get the inequalities A •> CjAq and A > c-^Aq. In conformity
with Chapter 2, Section 5,
,N-1 ,N-l ^2^1/2
ll/IL- <II/||(„i)=(E /^(E hf,) ) , f€H.
with these relations established, estimate B2) reduces to
1 ,. ~. „,, a
U.ll<-Il/-/Il(„i)+-Il/Il(„i)
or, on account of the inequality || z ||^, < | || ^j ||,
In concluding this chapter we clarify the meaning of condition A8) by
observing that
{l-a){{a,yl] + {d,y^)) < [h,yl] +{d,y^) < {I + a) {{a,yl] +{d,y^)) .
From such reasoning it seems clear that the fulfilment of A8) is ensured by
the inequalities
\a — a\ < aa , \d—d\<ad.

Difference Schemes
for Elliptic Equations
This chapter is devoted to various difference approximations of second-order
elliptic equations. In Sections 1-3 we present results of more a detailed
exploration of the Dirichlet difference problem for Poisson's equation. The
approximation technique for the Laplace operator and formulations of
difference boundary conditions are described for regions of arbitrary shape.
The maximum principle (Section 2) a.nd all of its corollaries are established
for grid equations of common structure. These tools are aimed at
establishing the uniform convergence with the rate 0{\ h p) for the difference scheme
constructed in Section 1 for the case of an arbitrary domain. In Section 4
we study the properties of the difference Laplace operator and develop the
difference operators corresponding to elliptic operators of general form with
variable coefficients'. In Section 5 higher-accuracy schemes are designed for
Poisson's equation in a rectangle.
4.1 THE DIRICHLET DIFFERENCE PROBLEM
FOR POISSON'S EQUATION
We now turn to the design of difference schemes for solving the Dirichlet
problem in which it is required to find a continuous in G -{■ T function u{x)
237

238 Difference Schemes for Elliptic Equations
solving the Poisson equation
A) ^" = Z! ^5^ = -•/'(^) ' XeCr,
« = 1 a
subject to the boundary condition
where x = {x^ ,x^ ... ,£„), G is a p-dmiensional finite domain with the
boundary F.
1. The difFerence approximation of the Laplace operator. We begin by
defining a difi'erence analog of the Laplace operator in the plane x ~
\X ^ , ^2/ •
B) Au=Liu + L2U, ^°'^~~^Z2' a =1,2.
In this direction the operator Li u — ^^ or L^it = g-r is approximated at
a point X — (x'j , Xg) by the three-point operator Ai or A2, respectively:
C) Liv -^ Aiv - %j.^j = j^ {v{x^ + h^ , x^)
— 2v (x^ , X2) + v(x^ — h^ , X2)) ,
D) L2V ^ A2V = v^^^.^ = — {v{x^ , X2 + h,)
~ 2v (x-j , x^) + v{x^ , X2 " h^)] ,
where approximation is denoted by the symbol ~ and h^ > 0, h^ > 0 are
the steps along the axes Xj and x^, respectively.
The operators- Ai and A2 are specified on the regular three-point
patterns
and
V^l ! ^2 2/' VI '^2/ V^l 5^2 ~r 2) '
respectively. Taking into account C) and D), we replace the Laplace
operator B) by the difi'erence operator
E) Av = Ai i; -F A2 ^y = %^,^^ -F %.^^.^ ,

The Dirichlet difference problem for Poisson's equation
239
2 9
4 d.
Figure 6. The regular "cross"-pattern
which is defined on the five-point pattern consisting of the nodes (xj ±
h^ jXg), (x'j jXg), (xj jXgi/ig)- Any such regular pattern is called a "cross"
pattern and is depicted in Fig. 6.
Here the symbol O corresponds to the point (xj , x^), while the symbol
1 corresponds to the point (x^ -|- /ij jXg), etc.
From formulae C)-E) and Fig. 6 it follows that
F)
1 1
In particular, for h^ = h^ = h (on any square pattern) we thus he
G)
^ '"o = 72 ("l +  +V3+V,-~4 Vo)
The next step is to calculate the error of approximation of the Laplace
operator B) by the difference operator E). Since for a = 1,2
(see Chapter 2, Section 1), it is plain to derive the expression
Av-~Av='^L'^v+'^Llv + 0{h1 + hi) .
This provides support for the view that
A^-A^ = 0(|/ip), \h\' = hl + hl,

240 Difference Schemes for Elliptic Equations
if v(x) is an arbitrary function having no less than four derivatives with
respect to x^,, a = 1, 2, that are bounded at least in the rectangle {x^ — h^ <
x' < x„ -|- he,; a = 1,2} for /i„ < /i„. So, the Laplace operator B) is
approximated to second order by the difference operator E) on a regular
"cross" pattern. A difference approximation of the p-dimensional (p > 2)
Laplace operator
(9) Lu='^LaU, LaU=.—-^,
can be arranged in just the same way. This can be done by replacing La
by the three-point difference operator A„ and accepting the decomposition
p
A0) Av = ^ hav, AcV = Vj.^^^,
a = l
SO
that
1
A1) A„. = .,^,,^ = ^(.^+^"^~2.+
,(+!") „ 9„ a-„(-!»)
v^
where t;(^i<») — t;(x(^-'")). Here x(+'''') (or x^"-^")) is a point into which
the point x = (xj ,. .. , x ) moves after the shift by one interval h^ along
the direction x„ to the right (or to the left) (see Fig. 7).
Figure 7.
Evidently, the pattern for operator A0) consists of 2p-|- 1 points: x,
x(i'-<^), a = 1,... ,p G points in the case p = 3) and the approximation
here is of order 2.
2. Approximation of the Laplace operator on an irregular "cross" pattern.
We now consider a difference approximation of the Laplace operator on
an irregular "cross" pattern. In the two-dimensional case (p = 2) such a
pattern consists of the five points
(X'j—/ij- ,^2), (Xj-|-/}2+ , X2), (Xj,X2), (Xj,X2—/l2-j> (^^i i 2!2"l~+j i

The Dirichlet difference problem for Poisson's equation
241
2?
'2 +
/h- 0
4 4
'1 +
Figure 8. The irregular "cross" pattern
where /ijj_ > 0, /i2± > 0 and /i„+ ^ /i„- at least for one value of a (Fig. 8).
We approximate either of the operators L\ and L2 at three points
(xj —/ij- , Xg), (xj +/ii+, X2), (xj , Xj) (the points 3,1, 0),
(xj , X2 —/ij-), (j^i , 2:2 + ^2+)' (^i>^2) (the points 4, 2,0),
respectively. These approximations can be arranged via transformations
(Chapter 2, Section 1) such as
Li t; ~ Aj t; =;
' (Xj + /ii+ ,X2) - v{x^ ,x^)
V (Xj , Xj) — V (Xj — /ij- , X2)
A2) {
Lo V ~"Ao V
t; (xj , X2) — V (xj , X2 — h^- )
where /i„ = \{h^- + /i„+) for a = 1,2. On any irregular pattern the
difference Laplace operator takes the form
A3)
A*v = A^v + Alv = Vg^ ^^ + Vg,_ J,,

242
Difference Schemes for Elliptic Equations
If, for instance, hi- = hi+ — h^, then Aj t; = Ai t; = 1;^ ^ , etc. The
procedure of writing out the arguments is somewhat cumbersome. This is
especially true for the case p > 2. To make our exposition more transparent,
it will be sensible to introduce the notations
a;*-'*'"''■' = (Xj +/zi+, x,^), x'- "^'^ = (x^ —/ij- , Xg),
t;(+i'') = t;fx(+^'')V v = v{x), t;(~i-) = t; fx^^^-^V a =1,2.
The disposition of the points x and x'^^"'") is shown in Fig. 9.
,-lo
„ + lo
+
Figure 9.
The expression for A*^ can be rewritten as
A4)
A*v - v^
„( + lc) _
V V — V
_„(-!<.)
ha+
^a = ^{ha- +ha+), a = 1,2.
In Chapter 2, Section 1 (see formula B7)) we have approved the expression
for Vg^. — v", whose use permits us to establish
A5)
a3„
Alv- Lav^-^ (/i„+ -ha-)-7^ + 0{hl).
Thus, on any irregular pattern the Laplace operator is approximated to
first order by the difference operator A* specified by formula A3).
The approximation of the type A2) applies equally well to a non-
equidistant grid and at the near-boundary nodes in the case of an arbitrary

The Dirichlet difference problem for Poisson's equation
243
domain. We follow the second way of approximating the Laplace operator
on an irregular pattern (see Fig. 8). The expression
A6) A> =
1
m( + U) _
V V ~ V
i-U)
ha +
h^ = max(/i„-. , ha+) ,
reveals A*^ instead of formula A4) as
'^ y. ^ ci ^ ci
"■a
We claim that in this case the operator h*^ generates the local
approximation of zero order
V-a = A^M- AaM = 0A).
Indeed, taking into account A5) we arrive at the chain of the relations
V-a = A^ M - Aa W = - 1 - 7^ la M
+ ^{K.-K~)^^ + o{nl)
±
3/ia
'''0'+ ^~" '''Q~
9x^
ia« + 0(?ia) = 0(l).
We clarify the situation, in which approximation A6) is quite applicable,
on the basis of one possible example.
Example For the boundary-value problem
w" = -/(x), 0<x<l, w@)=:0, w(l) = 0,
we form the grid
'^h — \^i > ^1 = fti , Ki_|_i =2;j-|-/l, 2= 1,2,..., TV— 1, Xj^_^i = Xj^ -|- /ijj ,
which is everywhere equidistant, but near the boundary h^ < h, h^ < h,
/ij + h^ + (TV — l)h = 1. At the regular nodes Xj, 1 < i < TV, we find that
M. ~ Ujf
Uj_|_j / Uj -|- Uj__ J
_ "i+

244
Difference Schemes for Elliptic Equations
By the same token,
A*u, = -
"■N
A*M/,, =
h
^N
'W — "W-l
/ij h
With these relations established, we arrive at the difference scheme
Vxx = -fix), Xi - h^ + (i - 1) h , 1 <i< N ,
A*yi=-/(xi), A*y^ = -/(x^), y^=y^_^j=0,
which will be convenient to be expressed with respect to the error z = y — u:
A7)
A8)
Az = —tpix), 0 < a; < 1, z^
^N+i
0,
where Az = z^.^ for Xj < e,- < x^, Az^ = A*z^ and Az^ = A*2^, ^p^ =
0(/i2) for i = 2,3,... ,7V- 1, Vi = 0A) for i = 1,7V.
In spite of the fact that this scheme generates no approximation at
the near-boundary nodes i = I and i = N, scheme A7) is of second-order
accuracy in the space C: \\ z ||^ = 0{h?). In order to obtain this estimate
at the points x = Xj, x^, we rewrite equation A8) as
1
Z, Z-. Zf.
0.
1 / 5
N+1 — ^N ^N " ^N-i
h^
= 0
where Zg = hh^tp-^ and Zw_|_j = /i/igV'w- Thus, problem A8) is equivalent
to the following one:
^xx = ~V'(^) I Xj < X < x^ , A*2j = 0 , A*Zj^ = 0 ,
On account of the a priori estimate
N i
II ^ lie < ™ax (I 2o I. I ^N+1 I) + E ^ E ^ I V-fc I .
i=l k=l
arising in Section 2 of Chapter 1, it is easily verified that
lU lie = II2^ - " lie < '^ '^i I ^1 I + '^ '»2 I ^w I + ^max^ lAl^Mh''.
This means that scheme A7) provides an approximation of order 2.

The Dirichlet difference problem for Poisson's equation
245
X2
K
0
K
(i^h
1 ' 'h'^2}
I
Xj
Figure 10.
3. The Dirichlet difference problem in a rectangle. Let now
Go = {0 < Xi<ti, 0 < x\_ < l^)
be a rectangle of sides /j, /g (see Fig.10) with boundary F.
Of our initial concern is the Dirichlet problem in the rectangle Go =
Go + F for the Poisson equation
(!')
Am = -f(x), X = (.i.-j , x^) G Go
//(a;).
In order to form in Go the grid w^ with steps h^ = /j/TVi and h^ = I2/N2,
where TVi and 7V2 are positive integers, we draw up two families of straight
lines such as
n(^0
xy = i^ h^, ij = 0, 1,,
.Wi
n(i2) _
X^ ' — *2 > *2 — U, i.
.No
We call the points of intersection x = (i^h^ ,«'2'*2) of those straight lines
with the coordinates i^h^ and 22/12 nodes. If a nodal point x = (i-^h^ ,i2h2)
is inside the rectangle, that is, 0 < i^ < Ni , Q < i^ < N2, it falls within
the collection of inner nodes. Let W/j be the set of all inner nodes. The
total number of inner nodes is equal to (TVi — 1){N2 ~ 1).

246 Difference Schemes for Elliptic Equations
The nodes on its boundary (for i^ = 0,Ni or i^ = 0,7V2) except for
four points @,0), @,/2), (/^ ,0), (/j ,12) ^^^'^ called boundary nodes and
are labeled crosses in Fig.10. They constitute the set jf^ = {(ij/ij ,^2^2)}-
The set of all inner and boundary nodes is known as the grid w^ = w^ + 7^
in a rectangle Gq.
Following established practice, at each of the inner nodes x G w^ we
compose a five-point regular "cross" pattern, whose nodes x^^", a = 1,2,
belong to w^, that is, either to w^ or to 7^. For this reason at all inner
nodes the Laplace operator Am can be replaced by the difference operator
In this view, it seems reasonable to approximate the right-hand side —f{x)
of equation A') by the grid function —<^(x) so as to achieve the error <^(x) —
/(x) = 0A K^ I), f{x) G C^'^y Assuming the function f{x) to be continuous,
in what follows we accept f{x) = f{x).
As a final result problem A') is associated with the Diriclilet difference
problem relating to tlie determination of a grid function y{x) defined on
the grid w^, satisfying at the inner nodes, that is, on w^ the equation
A9) Ay = -/(x), ^y = ys^x^ + yx2^2' ^^'^n-,
and talking the assigned values on the boundary 7^
B0) j/(x) = /i(a;), x G 7;, .
It is worth noting here that the grid w^(Go) becomes rectangular for
/ij ^ h^. In the case where h^ =^ h^ = h it refers to a square grid. More
a detailed expression for Ay on any square grid is of the form
Ay= T^ U^^^^ + 2^^"^^ + 2^^+ '^ + 2^^" '^-4y
Let ip — Q. The equation Ay = 0 can be solved with respect to y:
The value of y at the center of the pattern is the arithmetic mean of the
values of y at the remaining four nodes of the pattern. This formula gives a
difference analog of the formula for the mean value of a harmonic function.

The Dirichlet difference problem for Poisson's equation
247
It is easily seen from A9) and B0) that no values of/.t(x) at the vertices
of the rectangle appear in this matter. This feature has some influence on a
proper choice of 7;j. For the third boundary-value problem and the scheme
of accuracy Od/il"*) (see Section 5) the boundary 7/j consists of all the
nodes on the boundary of the rectangle including its vertices.
To evaluate the accuracy of the difference scheme A9)-B0), we pass
to the difference z = y -- u, where y is a solution of problem A9)-B0) and
M is a solution of problem A'). Substituting y = z + u into A') we set up
the problem for the error z
B1)
Az
-^,
X G w
h 1
z = Q, X ejh
where V" = Am-|-/ is the error of approximation of equation A') by scheme
A9). Since Lm + / = 0, we have
tp = Au + f — Lu + Lu = Au ~ Lu ,
yielding V" = Am — Lu. From (8) it follows that
^
h d^
hi d^
12 dxf 12 9x4
for
G C(^),
where the symbol over-bar designates that the values of the arguments are
taken at some intermediate points of the intervals (a;^ —h^ , X2), (xj +/ij , Xg)
and (xj , X2 — h^), (xj , X2 + h^), respectively.
Within the notation M4 = max
G ,a
IV" I < M4
[hi
12
, we get
The proof of convergence of scheme A9) reduces to the estimation
of a solution of problem B1) in terms of the approximation error. In the
sequel we obtain such estimates using the maximum principle for domains
of arbitrary shape and dimension. In an attempt to fill that gap, a non-
equidistant grid
^h = {^. = (^f'^4''^). 'a = 0, ],.. . ,7V„, xW =: 0, a;(;^°) = /„, a = 1,2}
with steps h\ ^' = Xj
(ii) _ ^(»i)_,.(»i"l)
id /ll'^) =
(i2) Ji2~-1)
can be introduced
in the rectangle in the usual way for later use of the difference operator A3).
Thus emerged instead of A9)-B0) the problem
B2) Ay = -/(x-). Ay = y^^^^ + y^^^^
X (z LOf,
y\j, = t^i^) ■

248 Difference Schemes for Elliptic Equations
This scheme provides the local approximation of order 1 since
V-, = (A« + f{x))^ := (%^j^ + Ug^^^ + f(x)).
^0{\h\), \h\' = hl + hl.
However, by analogy with Chapter 3, Section 4, ?/;, is representable by
a-l a
From such reasoning it seems clear that scheme B2) provides an
approximation of second total order in the negative norm.
4. The Dirichlet difference problem in a domain of rather complicated
configuration. If a solution of the Dirichlet problem needs to be determined
in a domain G with a nonlinear boundary, the grid i^h{G) is, generally
speaking, non-equidistant near the boundary. We describe below such a
grid and give the possible classification of its nodes.
Consider an arbitrary finite domain G with the boundary F in a p-
dimensional space, where a point with coordinates Xj , Xj , ...,»„ is denoted
by X = (xj , Xg , ... ,Xp). We proceed to construct a grid in the domain
G = G + T and confine ourselves, for the sake of simplicity, to the case of
a two-dimensional domain (p = 2). Here a constructive supposition about
the shape of the domain G is taken into account that the intersection of
the domain G with any straight line passing through an inner point x € G
in parallel to the coordinate axis Ox^ (a = 1,2) consists only of a finite
number of intervals.
If the origin of coordinates is inside the domain G, we draw up two
families of uniform-straight lines
x$^') = ij/ij, ij = 0,±1,±2,... , x'-^^'^ =. i^h^, ^2 = 0,±1,±2... ,
where /ij > 0 and h^ > Q are fixed numbers. Such straight lines split up
the plane (xj , Xg) into rectangles of sides /ij and /ig. The vertices of these
rectangles with coordinates Xj = i^h^ and x^ = i^h^ are called nodes and
the set of all nodes is known as a grid in the plane (xj yX^)- The nodes
x^ = {i^h^ , 22/12) lying inside the domain G refer to inner nodes with the
notation w^ = {x^ G G} for the set of all such nodes. The points of the

The Dirichlet difference problem for Poisson's equation
249
intersection of the straight lines x^*") = i^h^, a = 1,2 with the boundary
r of the domain G are called boundary nodes along the direction x^.
We denote by 7,^ ^ the set of all boundary nodes along the direction x^.
Let 7/j = 7/j 1 + 7/j 2 be the set of all boundary nodes. It represents such
nodes that are boundary at least with respect to one direction x^. The set
of all inner and boundary nodes is known as a grid w^j = ui^ + j^ in the
domain G (see Fig. 11).
Figure 11.
We offer below more a detailed classification of inner nodes. With this
aim, let us draw up a straight line parallel to the axis Ox^ through an inner
node X (z LOf^. Its intersection with the domain G is an interval (or several
intervals), whose ends are boundary nodes along the direction Ox^. The
nodes from this interval can be distinguished by the approved rules. We call
the nearest to the end of the interval node a near-boundary node along
the direction Ox^ (with respect to x^). If the distance to the boundary 7,^
is h*^ j^ h^, any such node is said to be irregular with respect to x^. Let

250 Difference Schemes for Elliptic Equations
Lol be the set of all near-bouiidary nodes with respect to a" and col* be
the set of those near-boundary nodes, which are irregular in the direction
Xq,. Obviously, wj^*^ C w^ ^. The notation wj^ stands for the set of all
near-boundary nodes (that is, near-boundary at least in one direction)
and io*^* stands for the set of all irregular nodes (that is, irregular at least
in one of the directions Xj or Xg). Let w^ be the complement of lo*^ to w^:
W/j = w* -|-w,j. All the nodes belonging to w,j are called strictly inner
0 . .
nodes. The symbol lOj^ ^ is in common usage for all strictly inner nodes
with respect to x^ (that is, the nodes adjacent to the node x G loj^ ^ in the
direction Ox^ are inner ones).
0
In Fig. 11 the aforementioned nodes w^ are depicted by the signs o,
the nodes being irregular only with respect to x^, (a = 1, 2) — by the signs
Aq,, the nodes being irregular both with respect to Xj and with respect to
Xg — by the signs Aj 2 and, finally, near-boundary nodes regular both with
respect to Xj and with respect to x^ — by the signs •.
The grid tOf^ in view is supposed to be connected, it being understood
that any two inner nodes can be joined by a polygonal line, the parts of
which are parallel to the coordinate axes and vertices coincide with inner
nodes of the grid. Then at least one of the four nodes x*^^"), a = 1,2, of
the five-point pattern (x'^^"''), x, x'^^"'^)) (regular or irregular) falls within
the collection of inner nodes. The assumption on connectedness of the grid
entails some limitations both on the choice of spacings h^ and /ig and on
the shape of the domain and its position with respect to the grid w^ for
fixed /ij and h^.
Examples of disconnected and connected grid domains are shown in
Fig. 12.a and 12.b, respectively. The assumption on connectedness for a
domain with a narrow bridge will be satisfied if we make the step h^ small
enough or refine the grid in this part of the domain. Fig. 12.b illustrates
the case where the connectedness of the grid is stipulated by the proper
choice of its step h^ rather than by successive grid refinements.
The procedure of constructing a grid in the plane domain we have
described above can easily be generalized to the case of an arbitrary p-
dimensional domain. A grid so constructed is a result of the intersection of
hyperplanes (planes for p = 3 or straight lines for p = 2)
^'a~'^aha, i„ = 0,±l,..., tt=l,2,...,p,
where h^ > 0. The preceding classification of nodes remains unchanged
here.

The Dirichlet difference problem for Poisson's equation
251
a) b)
Figure 12. a) A disconnected grid, b) A connected grid.
Our purpose here is to construct a difference scheme for solving the
Dirichlet problem in the domain G = G + T, the complete posing of which
is to find an unknown solution to the equation
I i6 2 , i6 2 ) tl v_T ,
which is continuous in the closed domain G = G + T and satisfies the
boundary condition u\ = ij,{x).
At each of the inner nodes x G w^ we approximate the difference
operator
L.
dx'^
by the three-point difference operator Aq,.
If a node x 6 w^ is regular with respect to x^,, then the difference
operator Aq, on the regular pattern (x^~^°\ x, x^'^^^'j is similar to A1):
y(+ic) _2y+y(-ic)
Aa y = Vs^ ^„
a
But if a node x G ^"q,, that is, a node is irregular with respect to x^ on
the irregular pattern, then the difference operator Aq, can be rewritten as
B3a) Aly
.(+i») _
w(-i»)
h*
for a;( i") G 7ft.«

252 Difference Schemes for Elliptic Equations
where h* is the distance between the nodes x and x^^^^"), or
B3b)
where h* is the distance between the nodes x and x^'^^°\
In this context, it is worth noting that the reader can encounter the
case when x^'^"^ G 7^ ^^ and x(+^<') G 'lh,a- If this happens, it is
recommended to refer to the difference operator
B3c) A„y = -(^^^^- ^-—j for x(±Me7.,„,
where h* . 9^ /i„ is the distance between x and x^^''-°\
Typical situations corresponding to the forms B3a)-B3c) for p = 2 are
shown in Fig. 13. Using one of the formulae A1) and B3) for approximating
La'u — f^ by the difference operator, we get instead of A) the difference
equation Ky + ip{x) — 0 for all x G w^, where A = X!«=i^a- Here the
exact value y\ = /i(x) is taken on the grid boundary 7^.
Finally, we arrive at the Dirichlet difference problem of
determining a grid function y[x) defined for a; G lj^ = u)j^-\- 7,,, satisfying sit the
inner nodes the equation
B4) Ay-|- (^(.-c) = 0 at the regular nodes,
B5) A* y -\- fix) = 0 at the irregular nodes,
and taking the assigned values at the boundary nodes x G 7;j;
B6) y-f.i{x), xejh-
By analogy with Section 3 we formulate conditions for the accuracy
of a scheme under the agreement that y{x) is a solution of the difference
problem B4)-B6) and u = u(x) is a solution of the original problem A).
Substitution of y — z + u into B4)-B6) yields
Az — —'ip at the regular nodes,
B7) A"z = —'tp* at the irregular nodes,
2 = 0 on 7,, ,

The Dirichlet difference problem for Poisson's equation
253
a)
b)
a) Aj y
^ (Vi-Vo Vo-Vs
hj \ /ij
^2y = ys2.,, A* = At + A2.
b) Aly^
1 (Vi-Vo Vo - y^
h, V /i*
A2 y = ?/£2-^2 ' A* = A^ + A2
c) Aiy- '
/ij V /i*^
A*2?/
1 (Ih-Vo Vo -Vi
K ^ K
A*=A*+A5

254 Difference Schemes for Elliptic Equations
where ^p is the approximation error equal for (p{x) = /(x) to
tp = Au + ip = Au — Lu at the regular nodes,
B8)
^p* = A*u — Lu at the irregular nodes .
Let u G C^'^\G), where C^'^^ is the class of functions m(x) with four
continuous in G derivatives with respect to Xj , . .. , Xp. As stated in
Section 3, we have at the regular nodes
B9) |^|<M4^, |/j|2 = /j2 + /j2+ ...+/j2.
Furthermore, in giving the approximation error at the irregular nodes as a
sum
C0) ^* = f] ^; , ^; = A„ w - L„ « ,
a=l
we apply the results obtained in Section 2 to the current situation:
C1) K = Hif^ 1^ + ^('^"^ = ^^'^' "^* = ^^'^'
meaning that at the irregular nodes the scheme does not approximate the
equation Am + /(x) = 0.
Thus, in the |>dimensional case a difference scheme such as
p
Ay = 2, Aq, y = —/(x) at the regular nodes,
a = l
p
A*y = y^ A^ y = -/(x) at the irregular nodes.
where A^y = y^ ^ and A^ is specified by the formula
° h^ y h„+ h^- J
is associated with problem A).

The Dirichlet difference problem for Poisson's equation 255
Remark Quite often, the Dirichlet problem is approximated by the method
based on the difference approximation at the near-boundary nodes of the
Laplace operator on an irregular pattern, with the use of formulae A4)
instead of A6) at the nodes x G w^. However, in some cases the difference
operator so constructed does not possess several important properties
intrinsic to the initial differential equation, namely, the self-adjointness and
the property of having fixed sign. For this reason iterative methods are
of little use in studying grid equations and will be excluded from further
consideration.
5. The canonical form of a difference equation. We now consider the
Bp -|- l)-point scheme Ay = —/ at a regular node
a=l "
which admits an alternative form of writing
C2) t^y(-) = t^{y''-'"' + y'-''') + fi-)-
a=l a a=l "
To avoid cumbersome calculations, we concentrate primarily on the
two-dimensional case. Fig. 6 demonstrates that at a regular node
2 Ij^+J^j 2/o = ;^ B/1 + 2/3) + ^ B/2 + 2/4) + /o ■
Let X G w* be an irregular node. In the case corresponding to Fig. 13.a
we obtain
■ * 1 /2/1-2/0 2/o-2/3\ 1 fVi 2/3 2/tj
' ^"' h,\ h, h* ) ' \\h, hi h, hi ^°
where h^ = ^ (h^ + h^), and
A22/0 = TI B/2-2 2/0+2/4)-

From the equation A*j/ = A* y -|- A2 J/ = —/ we find that
f 2n, 2\ 1 1 1 , , ^
' T^T7 + T^] yo = T2y^ + TT^ y^ + T^ (^2 + 2/4) + h
hi hi hi "" hi"' h,hl"' hi

256 Difference Schemes for Elliptic Equations
In the case corresponding to Fig. 13.c we deduce that
[ 2h, 2/I2 \ _ 1 1 1 1
where h^ = ^{h*_ + h*+) and h.^ = ^(/i2 + ''■2)-
Let W;j(G) be a grid in a p-dimensional domain and x G w^ be a
near-boundary irregular node. Then
^ ^ "'"^^^% ^*—
Substituting this expression into the equation A*y = —/ and regarding
formally the node x to be irregular in all directions x^, we finally get
K = UK* + I'l-) ■
This is acceptable for h* = h*^ = ha = hp when x happens to be regular
along some direction Xn. But if x is regular in all directions a;^, then
/i*_^ = h*^_ = /l„,= /i„ for all a = 1,2,... ,p, leading to formula C2).
Comparison of C2) and C4) shows that these equations can be represented
in the canonical form
C5) Aix)yix)= Y. B(x,OyiO + Fix), xElo,,
where PaU'(x) is the set consisting of 2p nodes of the Bp+ l)-point "cross"
pattern with center at the point x except for the node x itself, that is,
,f ^ X. We call the set Patt(x)'(x) the neighborhood of the node x. The

The Dirichlet difference problem for Poisson's equation 257
quantities A(x) and 5(x,^) are the available coefficients of the equation.
From C2) and C4) it is easily seen that
C6) A(x)>0, 5(x,O>0, J2 B(x,0 = Mx) iorallxEui^.
Equation C5) is put together with the boundary condition
C7) y\^^ = ii{x).
The Dirichlet difference problem is a special case of a more general
problem in which it is required to find a grid function y{x) defined on the
grid w^j = w^j + 7;j and satisfying on W/^ the equation
A{x)y{x)= J2 B{x,Oy@ + F{x), xElo,,
C8) eePaii'Gx-)
y(x)=fi(x), xEjf^,
where
C9) A{x)>Q, 5(x,O>0, D{x)=A{x)- J^ B{x,^)>0
^ePati'{x)
for all X G w^j.
Remark The third difference boundary-value problem for Poisson's
equation can always be represented in the form C8), equation C8) being satisfied
for all X G w^j and conditions C9) being valid. Here, in addition, _D > i5 > 0
on 7a ■
To prove the existence and uniqueness of a solution of problem C8)-
C9), it suffices to check that the homogeneous equation
C[y] = A{x)yix)^ ^ 5(x,O2/(O = 0, xEco,,
D0) (,ePatt'{x)
y{x) =0, a; G 7,^ ,
has only the trivial solution y{x) = 0 for x E uif^. We will show in the sequel
that this fact follows immediately from the maximum principle, valid for
schemes C8)-C9).

258 Difference Schemes for Elliptic Equations
4.2 THE MAXIMUM PRINCIPLE
1. The canonical form of a grid equation of common structure. The
maximum principle is suitable for the solution of difference elliptic and parabolic
equations in the space C and is certainly true for grid equations of common
.structure which will be investigated in this section.
Let w be a finite set of nodes (a grid) in some bounded domain of
the n-dimensional Euclidean space and let _P G w be a point of the grid uj .
Consider the equation
A) A{P)y{P)^ Y. B{P,Q)y{Q) + F{P), P e lj ,
QePatt'{P)
related to a function y{P) defined on the grid u). Here the coefficients of
the equation A(P) and B(P,Q) and the right-hand side of the equation
F(P) are given grid functions; PaU'(P) C w, being the set of all the nodes
of the grid u) except for the node P, is the neighborhood of the node P.
The pattern of the grid equation A) at the node P consists, evidently, of
the node P itself and its neighborhood Patt'(P).
Similar equations do arise in grid approximations of integral equations.
In what follows we will suppose that coefficients A(P) and B(P,Q) are
subject to the conditions
( A(P)>0, B(P,Q)>0 for all PEui, QePaU'{P),
B) { D{P)=A{P)^ J2 B{P,Q)>0.
[ QePatt'{P)
A point P is called a boundary node of the grid u) if at this point
the value of the function y(P) is known in advance:
C) yiP)=fiiP) for PEj,
where j is the set of all boundary nodes.
Comparing C) and A) we see that on the boundary j we must set
formally A{P) = 1, B{P, Q) = 0 and F(P) = fi(P).
We call the nodes, at which equation A) is valid under conditions B),
inner nodes of the grid; w is the set of all inner nodes and w = w -|- 7
is the set of all grid nodes. The first boundary-value problem completely
posed by conditions (l)-C) plays a special role in the theory of equations
A). For instance, in the case of boundary conditions of the second or third
kinds there are no boundary nodes for elliptic equations, that is, w ~ u).

The maximum principle 259
_The grid uj is taken to be connected, that is, for fixed points P (^ ui
and P E ui there always exists a sequence of neighborhoods {PaU'(P)} such
that the passage from P to P can be done using only the nodes of those
neighborhoods or, in other words, one can select nodes Pi ,P2 , . .. , -P,„ of
the grid ui such that
Pi G Patt'(P), P2 G Patt'(PL), ... , P,n E PaU'(P„_i), P G PaU'(P„,)
with
B(P,,Pi+,)^0, f=l,2,... ,m-l,
BiP,P,)^0, B{P„,P)^0.
In the case of the difference scheme for the Dirichlet problem B4)-B6) of
Section 1 the definition D) of connectedness coincides with another
definition from Section 1. The very definition implies that the point P may
be boundary and, hence, the connectedness is to be understood that every
point of the boundary belongs to the neighborhood Patt'(P) of at least one
inner node.
Within the notation
E) CyiP) = A{P)y{P)^ ^ B{P,Q)y{Q),
QePatt'{P)
we may attempt equation A) in the form
F) i:y(P) = F(P).
An alternative form of £ y(P) may be useful in the further development:
G) CyiP) = D(P)y(P)+ J^ B{P,Q) {y{P) ^ y{Q)) .
QePaii'{P)
In the preceding section the Dirichlet difference problem was set up
in the form A), C). Consider as one possible example the so-called scheme
with weights for the heat conduction equation
du d"u ^, .
37= 3^ + /(^.^)' 0<x<l, t>Q,
u
(x,0) = Mo(a.-)> u{Q,t) = Hi{t), u{l,t)^ i^i^{t)

260 Difference Schemes for Elliptic Equations
On the grid (ift^ = {(x,- = ih,tj = jr), i = 0,1,..., N, hN = 1, j = 0, 1,...}
this scheme takes the form
yll^^ = A (cT J//+1 + A _ cr) J//) + ^/ ,
(8)
These difference equations are convenient to be presented in the
canonical form A), giving _P as a node of the grid O^t'. P = -P(Xj , L-^J,
where Patt'(P) consists of the nodes Qi = (xf,tj), Q2 = (Xj_i , L-^i),
Q3 = (a-'i+i i^j+i)i Q4 = (a;,-! ,tj), Q5 = (a;i+j ,tj) and the boundary 7
consists of the nodes (x, , 0) and @,tj),(l,tA,{ = 0,l,...,N,j = Q,l,....
Next, we fix some moment t = tj^i and rewrite (8) as
1-0-
/l2
(j//_l +J//+i)+^/
From here it is easily seen that B(P, Q) > 0 only if r < h^/2(l — a) and
0 < 0- < 1. By the same token, D{P) = 0.
2. The maximum principle.
Theorem 1 (the maximum principle) Let y{P) ^ const be a grid function
defined on a connected grid ui and let both conditions B) and D) liold.
Then the condition Cy(P) < 0 (C y{P) > 0) on the grid ui implies that
y{P) cannot attain the maximal positive (minimal negative) value at the
inner nodes P E u).
Proof Let Cy{P) < 0 at all of the inner nodes P E uj. On the contrary,
let y{P) attain its maximal positive value at an inner node P G w, so that
y{P) = max y{P) = Mq > 0.
The theorem will be proved if we succeed in showing that there exists an
inner point P at which Cy{P) > 0, violating the condition Cy{P) < 0.

The maximum principle 261
Since y{P) > y{Q) for all Q G Patt'(P), we find by virtue of the relations
D(P) > 0 and y(P) > 0 that
Cy{P) = D(P)y{P)+ J2 B{P,Q){y(P)-y{Q))>D{P)y{P)>0.
QePatt'{P)
From such reasoning it seems clear that only the case C y{P) = 0
needs investigation. As formula G) shows, it is possible only if D(P) = 0
and y{Q) = y{P) for all Q G PaU'{P).
We now take the node Pi G PaU'(P) at which y(Pi) = y(P) = Mq
and adopt those ideas. Since y(P) ^ const on the grid w and the grid is
connected, there exists a sequence of nodes Pi , P2 ,. . . , Pm , P, satisfying
conditions D), such that
but
y{P„) = y{P) = Mo , y{P)<Mo.
P G PaU'{Pm) .
Then
C y{Pm) > D{Pm) y{Pm) + 5(P„ , P) (y{Pm) - y{P))
>B{P„,P){y{P)-y{P))>0,
meaning P = P^ and justifying the first assertion of the theorem. The
second assertion will be reduced to the first one once we replace y{P) by
-yiP)-
Corollary 1 Let conditions B) and D) hold and let a grid function y{P)
defined on w + 7 he nonnegative on the boundsay, that is, y{P) > 0 for
P (^ J and Cy{P) > 0 on u). Then y{P) is nonnegative on w + 7, that is,
y{P) > 0 for all P e u) + 7, But if y{P) < 0 on 7 and Cy{P) < 0 on w,
then y{P) < 0 on w + 7.
Proof Let £ y{P) > 0 on w and y{P) > 0 on 7. Suppose that y{Po) < 0 at
least at one inner node Pq ^ uj. Then y{P) should take its minimal negative
value inside the grid uj, what is impossible on account of Theorem 1, since
y{P) 7^ const on the grid iu{y{Po) < 0, y\j > O). The second assertion of
the theorem can be proved in a similar manner.

262 Difference Schemes for Elliptic Equations
Corollary 2 The homogeneous equation A) subject to the boundary
condition
(9) Cy{P) = 0 on ui, y{P) = 0 on j,
has only the trivial solution y(P) = 0.
It is straightforward to verify that y(P) = 0 is a solution of problem
(9). Further, assume to the contrary that there exists a solution y{P) ^ 0
of problem (9). If y(P) 7^ 0 at least at one point, then by Corollary 1 the
inequalities y{P) < 0 and y{P) > 0 must hold simultaneously. But it is
possible only if j/(_P) = 0. Thus, we have proved the following assertion.
Corollary 3 Problem (l)-D) possesses a unique solution.
3. Comparison theorem. The majorant.
Theorem 2 Let y{P) be a solution of problem (l)-D) and let Y[P) be a
solution of the problem
A0) CY{P) = F{P), PGw, Y{P) = fi{P), PG7.
Then the conditions
A1) \F(P)\<F(P), PEio, \fiiP)\<fiiP), PEj,
provide the validity of the ineciuality
A2) \y(P)\<Y(P) for PGw+7.
Proof By Corollary 1 the inequality Y(P) > 0 is valid on w + 7, The
functions u(P) = Y(P) + y(P) and v(P) = Y[P) — y{P) solve the equations
£.u=. Fy.=- F + F >Q And Cv= Fy = F - F >Q subject to the boundary
conditions u\^ — {Y + y)\^ = fi + fi > Q and v\.^ = (y — j/)|^ = jl — fi >0.
Since the conditions of Corollary 1 are satisfied, we have w > 0 or
y > — Y, V > Q or y < Y. It follows from the foregoing that —Y < y < Y
or I j/(-P)| < Y{P) on w+7.
The function Y(P) is called the majorant for a .solution of problem
(l)-C). Its determination entails immediately the validity of the desired
estimate for || J/ ||p.

The maximum principle 263
Corollary for a solution of the problem
A3) jCy=0 on ui, y{P) = n{P) on 7,
the estimate
A4) max |2/(/^)| = ||2/||c<ll/^llc,
is valid with \\n\\(j = maxpg^ | /i(-P) |.
Indeed, let us specify the majorant Y(P) by the conditions CY = 0
on w and Y = ||/-i||^ on 7. The function Y(P) is nonnegative on w + 7
and attains its maximum at some node of the grid. This node is none the
inner nodes if y(_P) 7^ const and, hence,
||y|U= max Y(P) = max Y{P) = \\i^\\
If y(-P) = const, then Y(P) = \\h\\q ■ In both cases ||y||(j = ||/i||^ .
Combination of this relation and the inequality || J/||(j < ||^ \\c gives
estimate A4).
4. The estimate of a solution to the nonhomogeneous equation. In the
further development a solution of problem A)^C) is viewed as a sum
y = y + v,
where y = y{P) is a solution to the homogeneous equation
A5) £t/ = 0 on w, ij = ii{P) on 7,
and V = v(P) is a solution to the nonhomogeneous equation
A6) Cv(P) = F{P) on ui, v{P) = 0 on 7.
We have already obtained estimate A4) for y{P) and so it remains to
evaluate the function v(P).
Theorem 3 If D[P) > 0 everywhere on the grid uj, then a solution of
problem A6) admits the estimate
A7) ll^ll^^l^lc'

264
Difference Schemes for Elliptic Equations
Proof Let a majorant Y(P) be taken such that
CY=\F{P)\, Y\^ = Q, Y{P)>Q for P E uj + j
and Y(P) attain its maximum at a node Pq E ui. As far as Y(Po) = \\Y \\^.
is concerned, the equation
D(Po)Y(Po)+ ^ B(Po,Q){Y(Po)-^Y(Q)) = \F(Po)\
QePatt'iPo)
yields
D(Po)Y(Po)<\F{Po)\, Y{Po)<
npo)
D{Po)
thereby completing the proof.
Remark Estimate A7) is still valid for the solution of problem A6)
provided that instead of B) other conditions
\A{P)\^Q, \BiP,Q)\^0,
D{P) = \A{P)\^ ^ \B{P,Q)\>0
QePatt'{P)
hold.
Indeed, let | v(P) \ > 0 take the maximal value at a node Pq- Because
of this fact.
A{Po) I • I v(Po) I = Yl ^(^0 ' ^) ^W) + ^(P°
QePatt'iPo)
< Yl \B(Po,Q)\-\v(Po)\ + \F(Po)\,
QePatt'iPo)
whence it follows that
DiPo) \viPo)\<\FiPo)\, \\v\\c = \viPo)\<
nPo) I
<
It may happen that D(P) = 0 on a subset u) of the grid ui and D(P) > 0
0 0 . ...
on the complement ofw tow: uj+uj* = u). This type of situation is covered
by the following assertion.

The maximum principle
265
Theorem 4 Let the conditions
D{P) = 0 for P G w ,
D{P) > 0 for P G
hold, where w is a connected grid. Then for a solution of problem A6) with
the right-hand side
F{P) = 0 for P G w
the estimate
F(P)^0 for PEui*,
A8)
IIHU<
is vaHd in the norm || / ||^» = maxpgi.
c
f{p)l
Proof Let Y(P) be a majorant and £ Y = | F(P) \ on the grid lo,Y\^ = 0,
y > 0. The function Y(P) should attain its maximum on a finite set w + 7
at some node, not belonging to the boundary, because Y\ = 0. Also, it
0 0
does not enter the grid w due to the connectedness of w and the maximum
principle. Hence,
max Y(P) = max Y(P) = Y{Po) ,
where Pq is a node on the set to*.
By the initial hypothesis, D(Po) > 0. Arguing as in the proof of
Theorem 3 we arrive at A8). An analog of the remark to Theorem 3 is still
valid for that case.
4.3 STABIIITY AND CONVERGENCE OF THE DIRICHIET
DIFFERENCE PROBLEM
1. Estimation of a solution of the Dirichlet difference problem. We make
use of a priori estimates obtained in Section 2 for a grid equation of common
structure for constructing a uniform estimate of a solution of the Dirichlet
difference problem B4)-B6) arising in Section 1:
A)
Ay = —(f
A*y = -if
y = K^)
at the regular nodes,
at the irregular nodes,
on the boundary,

266 Difference Schemes for Elliptic Equations
where
p
A 2/ = ^ A„ J/, A„ 2/ = j/j,^ j,^ ,
a = l
A*j/=^A;y,
a=l
Other ideas are connected with the operator A, which coincides with A* at
the near-boundary nodes and with A at the remaining inner nodes, leading
to an alternative form of writing
B) Ky=-f, xGw/j, y\^^-=ii{x).
In conformity with Section 1, problem A) can be recast as
C) A(x)j/(x)= Y. B{x,Oy{0 + F{x), xGc, j/|^^=/.(x),
(,ePatt'{x)
where
A(x)>0, 5(,T;,<e)>0, D{x)=A{x)- ^ B{x,i)>{).
iePatt'{x)
We now represent a solution of problem A) as a sum
y = y + y,
where y and y are,'respectively, solutions of the appropriate problems
D) A f/ = 0 , X euih, y = /i for X G 7,, ,
E) Ay =-(fi, xeuJh, 2/= 0 for x G 7,, .
An estimate for a solution of problem D) such as
F) ||2/|lc<ll/^llc.

stability and convergence of the Dirichlet difference problem 267
has been derived in Section 2 and may be useful in such a setting.
Having decomposed the right-hand side ip as
^ = ^ + ^*,
0 . . 0
where if = ip and <y9* = 0 at the strictly inner nodes x G ui)^ (see Section 1)
0
and <;5 = 0 and ip* = ip dX the near-boundary nodes x G a;*^, we agree to
consider
y — V -\- w
with V and to being solutions of the problems
G) Kv^-f> for x^LO^, ^L =0'
(8) Aty = -f* for x G w,, , tu\ = 0.
in.
We are going to evaluate separately each of the functions v{x) and
w[x). In order to estimate v{x), it is necessary to construct a majorant
y(x). Assuming that the origin is inside the domain G, we try to determine
a majorant of the type
Y{x) = K{B?-r-'), P=E^l
V
r
where K > 0 is a constant and R is the radius of a p-dimensional ball (a
circle for p = 2) with center at the origin containing entirely the domain
G■ The constant K will be chosen a little later.
By virtue of the relations A„ xE = 0 for a ^ (i and
A* X^ = —
0. = '"'
we find that
h,,~
2^«,
2/i.
AY = ^ A„y =-2pA' for x G W;, ,
a = l
A*y = -2p6'A' for x G w^ ,

268 Difference Schemes for Elliptic Equations
V
where 0 = p~^ J2 ^a- Here 6'„ = 1 if a node x G u)*^ is regular along the
a=l
direction x„. Thus, Y is a solution of the problem
where F(x) = 2pK for x G w^ and F(x) = 2p0 K for x G w*
0
Comparison with problem G), where F = 'fi, that is, _F = 0 for
0
X G wj^ and v\ =0, shows that F{x) > \ F{x) \ = \ f{x) \ if we accept
K = 2^11 'P II „. Here the conditions of the comparison theorem in Section
2 are valid as long as F(x) > \ F(x) \ = 0 for x G w*, assuring the relation
IIHIc<ll^llc-
It is easily seen from the expression for Y that || Y ||„ < KFl?. So, for
a solution of problem G) the estimate
(9) Mlc<gll^llc = |lIHI =
is valid in the next norm || lyS || o = max o | (p{x) \.
Our next step is the estimation of the function w[x). First, we are
going to show that for problem (8)
A0) Fl(x) > -t-t; , where h = max /i„ , x G w! ,
A1) ^(x) = 0 for xE^^.
Assertion A1) is simple to follow.
After that, we look at equation (8) at a near-boundary node x G w*^
Aix)wix)= Yl B(x,Ow@ + F(x),
A2) iePatt'{x)
F{x) = ^*(x), ^\,,=0-
If one of the nodes ^ = ^^^ say ^^ = x^'^^"-', happens to be a boundary
node, then w(S,o) = 0 and the neighborhood Patt'(x) contains no point S,o-

stability and convergence of the Dirichlet difference problem 269
In this case the function D(x) takes on the value
D(x) = A{x)- ^ B{x,0
= Aix)
iePaii'{x)
^ B{x,0-B(x,Q
.^ePatt'{x)
= 5(x,eo):
since A(x) = ^ B(x,S,) for the Laplace equation. This provides
^ePaii'{x)
support for the view that
D{x) = 5(x,x(+i")) >0.
If a node x is near-boundary not only with respect to x^, but also in
other directions, then sumA2) contains no other terms for ^ = ^^,^2' ■ ■■ I'ffci
so that
Dix) = 5(x,eo) + Bix,C,)+ ■■■+ 5(.T,e,) > 0 .
Let X G w,* be a near-boundary and irregular node only in some
direction x„ and ^0 = x^'^^"^ G 7/j, x^'^"^ G w^. From the equation
Pita
whe
A/3y = 2/s«:<:».
K*w =
1 /w(+l")
w w — w
(-1»)
/l„+
l I w w — w*- ^"-'
/i„ \ /^a+

270
Difference Schemes for Elliptic Equations
we establish the relations
A{x)
1
1 ''
E 5(x,o=^+i:
(,ePatt'{x)
lip
D(x) =
1
1
h^ h^+ h"^
If X is a regular near-boundary node only with respect to x„, then
a
When, in addition, it turns out to be near-boundary in other directions,
the function D{x) can only increase. Just for this reason estimate A0)
continues to hold on the same footing.
To evaluate a solution of problem (8), we apply Theorem 4 of Section
2 due to which
A3)
hHlc<
D
<h'^\\v*
ic ■
Collecting estimates F), (9), A3) and then involving the well-known
inequality II V lU < II ^ lU + II V II „ -|- II to lU, we establish the followins; theo-
rem.
Theorem 1 for a solution of the Dirichlet difference problem the estimate
A4)
holds with
\y\\c < ll/^llr'.
o il^lb, +f^ 11^ lie*
2p c '^'
^^'^h+ih
m
I/lie* = ™^? l-/'(^)
x£ujt
\ f \\o = max f(x)
/|L = max |/(x)
This theorem expresses the stability of the Dirichlet difference problem
A) with respect to the boundary data and the right-hand side.

stability and convergence of the Dirichlet difference problem 271
2. The uniform convergence and the order of accuracy of a difference
scheme. In the study of convergence and accuracy of scheme B) we begin
by placing the problem for the error
z = y-u,
where j/ is a solution of problem A) and u = u[x) is a solution of problem
A) arising from Section 1. Substituting y = z + u into A) or B) yields
A5) kz = -ijj{x), x£ijj, ir|^ = 0,
where ^(x) = A w + 'f{x) is the residual.
We stated in Section 1 that
^(x) = 0A h p) = OQi^) at the regular nodes,
^(x) = 0A) at the irregular nodes,
or, more specifically.
1^1
1^1
where
Mk = max
xeG
l<a<p
M4
<
\h? M4 ,,
'2 ^^ 12 ''
<pM2
dx^
a
, fc = 2,3,4,..
at the regular nodes,
at the irregular nodes,
P
|/ip = y^/iil, h= max /i„
Also, Theorem 1 of Section 1 asserts that estimate A4) is of the form
R II I II 7 2 II I II
Using the estimates of | ^ | obtained above we arrive at the relation
A6) \\z\\^ = \\y-u\\^<I^M4+pM2Jh\
making it possible to formulate the following statement.
Theorem 2 If u{x) G C'^{G), that is, a solution possesses continuous
derivatives in G = G + T of the first four orders, then the difference scheme
converges uniformly with the rate 0{h?), that is, it is of second-order
accuracy, so that estimate A6) is valid.

272 Difference Schemes for Elliptic Equations
4.4 SOME PROPERTIES OF DIFFERENCE EIIIPTIC OPERATORS
In this section we reveal some properties of difference operators
approximating the Laplace operator in a rectangle and derive several estimates for
difference approximations to elliptic second-order operators with variable
coefficients and mixed derivatives.
1. Eigenvalue problems for the laplace difference operator in a
rectangle. Let Go = {0 < x„ < /„ , a = 1,2} be a rectangle, w^j = {x =
(ij /ij , i2 h^), i„ = 0,1, 2,. .. , A^a , Na h^ = /„} be a grid in Go and Ji^ be
the set of all boundary grid nodes. The grid lOj^ is taken to be equidistant
in each direction x„ with step /i„.
The eigenvalue problem for the Laplace operator in the rectangle Go
subject to the first kind boundary conditions
At;-FAt; = 0, x G Go , v\^ = 0, v{x) ^ 0 ,
has an infinite set of the eigenvalues such as
fP P\
h = h,k. = ^'{jt + fj, ^„=1,2,..., a=l,2,
associated with the orthonormal system of eigenfunctions
so that
whei
(^fcifc, . Vk[kO = ^k,k[ h^k'^
0 0
This problem can be solved by the method of separation of variables. The
eigenvalue problem for the difference Laplace operator Ay = y^ ^ ~^ Vs x
supplied by the first kind boundary conditions may be set up in a quite
similar manner as follows: it is required to find the values of the parameter
A (eigenvalues) associated with nontrivial solutions of the homogeneous
equation subject to the homogeneous boundary conditions
A) Ay-FAt; = 0, x^ui,,, v\ =0, v{x)jtO.

Some properties of difference elliptic operators 273
Observe that ji^ and, hence, w^j contain no nodes @,0), @,A^2), {Ni ,0),
(A^i ,A^2) being the vertices of the rectangle.
We seek a solution of problem A) by the method of separation of
variables as a product
v(x^ , X2) = /-'(a-'i) i]{x^), Xj = «j /7.J , x, = I, /j, .
Substitution of this expression into the equation
Av + \v = v^^ ^^ + Vg^ ^^ + \v =0
yields
B) l-Ki,x,{^i} ri{x,) + ^{x^) iJi;,x2i^2) + ^K^i)v{x2} = 0.
Because we are interested only in nontrivial solutions of problem A), the
division of both sides of equation B) by l^{x^) i'j{x2) 7^ 0 is meaningful. As
a final result we get /.ig^ .^■^/A' + Is-^x^!''! + ^ = 0 or
fn\ A'Si Xi _ _Vx2jl^ — A — —A*-^^
with A*-^^ = const being independent neither of x^ nor of a;,. In view of
this, we might set up for /-i(xj) the eigenvalue problem on the grid
'h - {^i = ij/Ji , I'l = 0,1,... , A^i; /ijA^i = /j},
the statement of which is
Ai/i + A(^V = /,,,^,^+A^^V = 0,
Xj = ij/ij , ij = 1,2,... , A^i - 1, /<o=0i ^J■Nl-'^■
The conditions yUg = ji^q^ = 0 follow immediately from the relations
IJ.@)i](x2) = 0, ij{l^)r](x2) = 0 and rj{x2) ^ 0. As we learn from
Chapter 2, Section 3, a solution of this problem acquires the form
fi,^{x,) = J- sm-^, ^ = l,2,...,iVi-l

274 Difference Schemes for Elliptic Equations
where {l^kii^i)} i^ an orthonormal system of eigenfunctions: (yUj.^ }l^k')i =
i5j.j J.' under the inner product structure (y, v)^ = '}2i 'Ji vih^i) ^ih^i) ^i-
From the governing relations C) a similar problem arises for rj[x2)
with AB) = A-A(i):
%3j:2+^^^^'?B-2) = 0, X2 = i2K, I'a = 1,2,... ,A^2 - 1,
whose solution is given by
fe, = 1,2,... ,Af2-l.
Here (ryj,^ , 1]^'^^ = h^ k'^, where {y, v)^ = E^^^/ 2/(*2 ^h) ^ih K) K- Thus,
problem A) is completely solved, meaning that to the eigenvalues
A - A - a':^^ + a":^)
,.\ , ,11 . ^ IT k, h. 1 . ^ IT k, h, \
^j = 1,2,.., ,A^i - 1, ^2= 1>2,... ,A^2 -1
there correspond the eigenfunctions
E) v,,(x) = «tjfc3(xj ,x^) = Mi.,(xj)/ij,^(,T;2)
4 . TT fcj X, . IT k^ Xo
sm —^ sm
y 'I '2 M '2
^j = 1,2,... ,A^i - 1, ^2= 1.2,... ,A^2- 1-
The total number of the eigenfunctions is equal to (Ni — 1)(A^2 — 1) = A'^.
These constitute an orthonormal system
F) {""kik^ <'"k[kO = h^k[ h^k'^

Some properties of difference elliptic operators
275
in the sense of the inner product
G) iy>v) = Yl Yl y{hhi,i2K)v{i^h^,i^h^)h^h^
rew^j
y{x) v[x) /ij h^
Therefore, any function /(xj, x^) defined on the grid w^j and vanishing on its
boundary can always be expanded on the system of functions {vj.^^)^^^{x^, X2)}:
N
fix) = E Cj,Vk{x), Cj; = {f,Vk).
k = l
In the case of the second boundary-value problem with dv/dn\ = 0,
the boundary condition of second-order approximation is imposed on 7^, as a
first preliminary step. It is not difficult to verify directly that the difference
eigenvalue problem of second-order approximation with the second kind
boundary conditions is completely posed by
A t; -|- At; = 0 , X G W/j , A t; = Aj t; -|- A2 t; ,
f 2
-r- Vt,., a;„ = 0,
(8)
A„t; = <
ha<Xa<la-^a^
X = I
1,2,
Indeed, assuming that u{x) is a solution to the equation A u-|- A u = 0
subject to the condition duldn\ = 0, the error of approximation for the
boundary condition is represented by
v~ = (uj,.j -I- \ h^ A2 u -I- I /ij A u) I
du /ij d'^u h^ d'^u /ij Ai
dx, 2 dx'l 2 dxl
0{h])
Xl=:0
du
+ ^{Au + Xu) +0{hl) = Oihl)
a:i=0

276 Difference Schemes for Elliptic Equations
In a similar way we find that
v+ = (-Ug + I h, 7V2 u+\h,\u)\ = 0{hl),
etc. By means of the method of separation of variables problem (8) reduces
to the two one-dimensional eigenvalue problems for I-l{x^) = /.i(fi h^) and
%■,.. +A('^^ = 0, (ry,,^+|/i,AB),,)| =0,
.Ti=;2
Their solutions have been already found in Chapter 2, Section 3.
Along these lines, we can immediately write down the eigenvalues and
the orthonormal eigenfunctions
(9) K = h.,, = >^[\^ + ^Z^,
A0) Vkix) = Vk,k,iXi ,^2) = l^kAxi)VkAx^),
k^ = 0,l,2,... ,N^, a=l,2,
where
aO) = a(^) = o^
A1) ^L = T2 s"^
/i! 2/„
l<^a<^«. a =1,2;

Some properties of difference elliptic operators
277
//o(Xi) =
fJ-N.i^i)
A2) <
f^kA^i) = \l l =°s ^—
1 TT A^l X,
— COS
k, = l,2,...,N,-^l
VoiX2)
,1 IT N2X^
^N.M2) = \It cos
'2 '2
' 2 nk X
Vk^{'^2)=\lY cos ^ - , k., = l,2,...,N2~l
and
A3)
"'1 "'2 ' "'1 "'2-1 1 1 2 2
/ith
Ihh^
A4) [j/, ^] = {y, v) + ^ [2/@, 0) KO, 0) + 2/@, /,) ^0, /,)
K
N2-1
^ [2/@,1, /i2)^y@,23/12)
!,=2
+ y{Ni /ij ,i^h^)v{Ni h^,i^h^)\ h^
iVi-l
X] bC^i'^i .0)t^(Ji/ii ,0)
i-.-2
the inner product B/, t") being understood in the sense of G).
0
2. Properties of difference operators. Introduce the space fi/j of all grid
functions defined on the grid w^ = w^ +7^ and vanishing on the boundary
7;j and the space fi/j comprising all the functions defined on ujj^. The inner
product in the space fi/j is defined to be
A5)

278 Difference Schemes for Elliptic Equations
We list below the properties of the difference Laplace operator acting from
0 0
fift into fift C fift. By definition,
Av = Ai t; + A2 t;, A„t; = Vj^j.^, a =1,2.
1. The operator A is self-adjoint:
0
A6) (A J/, t;) = (j/, A v) for all j/, v G fi/j .
To make sure of it, it is straightforward to verify the chain of the relations
N2-I Ni-l
^2 = 1 Ji=l
N2-I Ni-l
~ Yl ^^ Yl '*iB/^i^)iii2 = B/.^i ^)
12 = 1 J'l-l
with a simple observation that the operator Aj is self-adjoint on the grid
Cj['^ = {x, = i,h,, z\ = 0,1,... , iVi , N,h,=l,}.
This is due to the fact that the order of summation over Zj and i^ may be
interchanged. In a similar way we obtain
(A2j/,v) = (j/, A2V),
which leads to A6).
2. The operator —A is positive definite:
A7) _ {-Ay,y)>6\\y\\\
where
A . ., irh, A . ., ttK 8 8
This property follows immediately from the relation
Ar.inll2/|l'<(-A2/,2/),
which is certainly true with 6 = X-^^-^ (see Section 1).

Some properties of difference elliptic operators 279
3. Tile operator A admits tlie bilateral estimate
A8) 5 II2/IP < (-Ay, y)< Ally IP,
where 8 is the same as before and
From the relations just established and the lower estimate for 6 we deduce
that
A9) 8,\\yf<{-Ky,y)<W\y\\\
These properties provide support for the view that the operator A = —A
is more convenient for many things than the operator A. The operator A
0
can be treated as an operator either from Hh = fi/j into fi/, C Hh or from
Hh — fi/j into Hk by merely setting
Ay = -Ay , y G 0,. ,
— 0
where Ay = Ay for y E Qh- In accordance with what has been said above,
we have in the space fi/j
A = A*, 6E<A<AE
with E denoting the identity operator.
0
Any function f E Qh, defined on the grid o)^ and vanishing on the
boundary 7^ or defined only on w^, can be expanded into a series of the
eigenfunctions of the operator A:
N
' f{x) = H Ck^k, Ck = if,Vk)>
fc=l
so that ^ ^ = 1 c^ = II / Ip, where Vj. are the eigenvectors of the operator
A: Av, = X,v„N = iN,^l)iN2-l).
In the case of the second eigenvalue boundary-value problem (8) the
space Hh = ^h comprises all the functions defined on the grid lo/^; the inner
product (,) on Hh is to be understood in the sense A4) and the operator
A is defined as a sum
A = A1+A2,

280
Difference Schemes for Elliptic Equations
Aay= ~A„j/ = -vi-^^^
for 0 < x-„ = i„ h„ <l^,
for x-„ = 0 ,
for x^ = Nah^ = l^, a =1,2.
With this in mind, problem (8) is convenient to be taken in the operator
form Ay = Xy. The operator A so defined is self-adjoint and nonnegative:
0 < A < A£',
A = A*,
/.-).
where A = A„,, = A{h^''
Observe that the operators Ai and A2 are commuting and self-adjoint
both for the first and second boundary-value problems. The methodology of
the general theory (see Chapter 2, Section 1) guides the choice of a common
system of eigenfunctions coinciding with the system of eigenfunctions for
the operator A = A[ -\- A^] m so doing, each eigenvalue of the operator A is
equal to a sum of the appropriate eigenvalues of the operators A\ and A2:
X{A) = X{A,) + X{A2).
3. The Laplace operator in a domain consisting of rectangles. We now
consider a domain consisting of a finite number of rectangles, whose sides
are parallel to the coordinate axes as shown in Fig. 14.
X2
1
h
^2
0
h,
G
I
X
Figure 14.

Some properties of difference elliptic operators
281
The sides of rectangles, which constitute the domain G are Eissumed to
be commensurable. All this enables us to place in the plane a grid with steps
/ij and h^ so that the boundary of the grid domain lies on the boundary of
the domain G. One trick we have encountered is to complete the domain
G to the rectangle and then denote it by G (see Fig. 14). After that, we
construct in G a difference grid w^ and extend it to G. The notation o)^
will be used for the grid in the domain G.
Let t; be a grid function defined on the grid ujf^ and satisfying the
boundary condition v\ = 0, Also, it will be sensible to introduce the
function
By definition.
where
v{x) =
v{x),
0,
ll?IU = IIHU=IIHI
\\vf = {v,v), {v,y)= ^ v{x)y{x)h,h^
\v\(k =
xeuih
v'^(x) /ij h^
x-eijjk
2
Ell%
2
E
a = l
With these relations established, we deduce that estimate A9) holds for
any function v{x) defined on the grid w^ in the domain G, providing the
values
where /j and l^ are the sides of the rectangle G.
4. The embedding theorem. Various a priori estimates for the equation
Ay = If can be obtained in the energy norms in light of the properties
of elliptic operators. One might expect that the energy estimates imply a
uniform estimate, that is, an estimate in the norm
II2/lie = maxl| j/(x)|.
The following grid theorem gives a definite answer to this hint.
The embedding theorem. For any function y{x) G 0/, defined on
tlie gridui^ = {x; = (i^h^ ,22/12); ^a = 0,1,... ,Na, h^ = ^c/^c a = 1,2}
and vanisliing on the boundary (for x„ = 0, /„; a = 1, 2) tlie estimate
B0)
||j/|L <Mo||4j/|

282 Difference Schemes for Elliptic Equations
is valid with
^2/=-2/.firi-2/i,:.2 - ^°^2~/rT' 'o = max(/i ./J.
Proof Suppose that v-k^]^^{x) and ^^.^^.^(x) are the eigenfunctions and
eigenvalues of the operator A that we have determined in Section 1 for k^ =
1,2,... , A^„ — 1 and a = 1,2. The expansion of the function y{x) with
respect to the orthonormal functions {vk^-k^{x)]
yields
Ay= ^ Ck,k^,h,i:,'"k,k, -
i^iP=E<
k2 >
' ) "•'2
ki, k2
The function y{x) satisfies the inequality
I Vi^) I < ( E I Cfcifc, I) max I v^^,,^{x) \.
From formula E) we get \vy^^^^{x^\ < 2/^/l^lQ. Applying the Cauchy-
Bunyakovskii inequality we establish the chain of the relations
B1)
i»wp<^(e
'^kik2 I
- / / 2^ ^kik2'^kik2 Z^ ;i^2
1 2 fcj j-j

Some properties of difference elliptic operators 283
With this in mind, it remains only to evaluate the remaining sums
ki+k^>2
Because the function sin ip/ip is monotonically decreasing on the segment
0 < iy9 < 7r/2 and, therefore, sin lyS > 2 (p/it, formula D) reduces in this case
to
where if„ = 7r/i^^^/B/„), a = 1,2. Using this estimate behind, we obtain
Ni-l N2-I j4 Ni-l N2-I ,4
E E \;l<i| E E i^l+Kr' = ^,J-
ki=l k2=l ki=l k2 = l
ki+k2>2 ki+k2>2
Majorizing the sum J by the integral
00 t/2
/■/■?' drdif IT
- J J r4 
1 0
and taking into account that A, ,> i5(, = — + •—- > —-, we deduce that
q q q
B2) ^'^^#+4TT6 = 6il^' 47-16
Substitution of B2) into B1) yields inequality B0).
5. Equations with variable coefficients. The Dirichlet problem for the
elliptic equation in the domain G + F = G comes next:
B3) Lu=~f{x), xGG, u = ^{x), x&T,
with a rectangle G = {0 < x^ < /„, a = 1,2} and
Lu='^La'U, LaU- -^ {k^{x)j~-
u <r. ^ \ cj X ^

284 Difference Schemes for Elliptic Equations
where ^„(x) are sufficiently smooth functions, 0 < Cj < k^{x) < c^, a =
1,2, with constants Cj > 0 and Cj > 0. We compose in just the same way
the grid with steps /ij and h^ and denote it by cD^j = w^ + 7^ as we did in
Section 1.
An excellent sta.rt in this direction is to approximate the operator L^
by the difference operator
where a„(x) is so chosen as to satisfy the relations
B4) AaU- LaU= 0{hl), 0 < Cj < a^,(x-) < C2.
This means that A„ provides an approximation of order 2, for instance, for
a^(X^ ,X2) — "'li'^'l" 21' '''2/' '^2l'''l)'''2J — "'21 ■''11 ■''2" 2 2/'
The next step is to put the operator Au = Yla=ii ^a u in correspondence
with the operator Lu due to which the difference Dirichlet problem
Ay=-ip, X G w,j , 2/ = fi{x), x G 7,, ,
B5)
Aj/= (Ai + A2)j/, Ac,ij= (a^ygj^^,^^,
having the approximation error (residual) i/j = Au + ip = 0{\ h p) will be
associated with problem B3).
Such an approximation is the result of a natural generalization of
homogeneous conservative schemes from Chapter 3 for one-dimensional
equations to the multidimensional case. These schemes can be obtained by
means of the integro-interpolational method without any difficulties.
0
We now investigate the properties of the operator A in the space fi/j
of all grid functions with the inner product in the sense of A5). In what
0
follows we accept Ay = —Ay for any j/ G fi.
Lemma The operator Ay = —Ay = — Yla=i ^a y, y E H, is self-adjoint,
positive definite and admits the estimates
B6) c^{Av,v) <{Av,v) <C2{Av,v),
B7) c,S\\v\\^<{Av,v)<c,A\\vf

Some properties of difference elliptic operators 285
for any v E H, where
0 0 2
A 2/ = - A 2/ = - E 2/£„ r„
a = l
and the constants S > 0 and A > 0 arise from formulae A8) and A9).
As far as v\ = 0, the first Green formula with regard to the operator
AaV= [a^ Vg ) implies that
iAaV,v) = -(a„%^ .%„]„'
where
;i=i ;2=i
iVi-l iV2
(^i', ^]2 = X^ X] "'(*! '^l ' ^'2 ^^2) ^(^l 'Jl . «2 ^12) 'Jl ^12 ■
»'l=l '2 = 1
With these representations in view, we have
2
I
a = l
On the other hand,
0 2
a=l
By condition B4) we are led to B6) and so it remains only to substitute
into B6) the bilateral estimate 8 \\v\\^ < (^^,^) < ^ ll^lP) which has
been obtained in Section 2. These manipulations permit us to derive B7)
with the ingredients
2 2
a = l a " a=l " "
In what follows inequalities B6) and B7) are adopted in operator form
and we agree to consider
B8) c, A<A<C2A,
B9) c,6E KAKc^AE,

286 Difference Schemes for Elliptic Equations
where E stands for the identity operator.
The self-adjointness of the operator A = Ai + A2 can be proved by
exactly the same reasoning as in Section 2 with further reference to some
properties such as
Al = Aa >0, A,yy = -A„ J/ = -(a„ iJsJ^.^ for y E Clh ■
To make sure of it, it suffices to calculate the row sum with a = 1 for fixed
i^ = 1,2,... , N2 — 1. The outcome of this is
Ni-l Ni-l
J2 (^1 2/)ii Vi, h, = ~J2 (°i VsJ^uh ^i, h,
ii = i ii = i
Ni-l
il=l
Wi-l
ii=i
Multiplying this identity by h.^ and summing over i^ = 1,2,... , A^2 — 1, we
establish {Aiy,v) = (j/, Ai t;) and, in a similar way, {A2y,v) = {y,A2v),
yielding
{A y, v) = {{Ai + A2) y, v) = {y, Av) .
By definition, this means A* = A.
6. Equations with mixed derivatives. In this section we consider problem
B3) involving the elliptic operator L with mixed derivatives
assuming the ellipticity conditions
22 2
C1) C, ^ e' < J2 ''api^)'^^P < ^2 E ^«' ^ ^ ^''
a = l a,/3 = l a = l
to be valid, where Cj > 0, C2 > 0 are constants and ^ = (^^ ,^2) is an
arbitrary vector. Setting first ^j = 1, ,^2 = 0 and then ^j = 0, ^2 = 1 i^ i^
not difficult to check that
0<Ci<^„„<C2, a=l,2.

Some properties of difference elliptic operators
287
In this line, the operator L^p u is approximated by the difference
operator
C2)
A„/3 U~\ {{k^p ■U,s/3)x<. + ikaC U^f3)s^)
which is defined for a 7^ /? on the 7-point pattern (see Fig. 15) consisting
of the nodes
(Xj , X2J, (Xj ± /ij , X2), (Xj , X2 ± n,), (Xj — ftj , X2 + ft2J ■
a 1
X
!
a 1
1 <
K
Figure 15.
The operator A„^ arranges itself as a sum
A«/3= K^a/J+^a/j)- A;^W= (fe„^WjJ^^, A+^ti= (^„^W,J_^.
We are going to show that Kf2 and A12 have the first and second order of
approximation, respectively:
C3) Kf2U = L^2U + 0{h^ + h2), Kyzu = Ly^u + 0{\h\^).
Indeed, we refer to the operator A~2 u = {k^^ '2)i- ■ Substituting here the
expressions
du /i2 d'^u hi d^u, _ .
^.•C2 ~ li W l!i 9 ~'~ ~C~ ir~3 V'^'l ' '''2/ ' ■''2 ^ [.■''2 ~ '^2 ) -^2] )
^ 0X2 / ox^ 0 0X2
(9t; /ij d'^v ^/,2
'' <9x, 2 5x2
OiK)

288 Difference Schemes for Elliptic Equations
for V = ^j2 Ug^ yields
A,2 " = ^12 t« + Y ^ i^i2 « - y i^i2 ^ + 0A h p).
^1
Likewise,
. + r h. d ^ hy ^ du ^,, , ,,,
7V+ M = Li2 u - y 3;;r^i2 " + y ^i2 1^ + 0A /i p) ,
from which estimates C3) follow immediately.
When a = /? = 1 and a = /i = 2 we might have
All u = i ((^11 w^Jx-i + (^11 w^-Jsi) = (flu %J:i-i -
A22U = I ((^2 ^«i J 2^2 +(^22t«a:J«2) = (^22 t«S J.7:2 >
where
^^22 — 2 L 22 V"^! J "^2 '^27 ~r '''22 V^l ) '^2)] '
From the preceding relations it seems clear tha.t Aa„ w, — L^aU = 0{h'^),
a = 1,2. Thus, the differential operator C0) is approximated to second
order by the difference operator
2
a,/3=l
meaning
Au-Lu = 0{\h\'^).
In addition to operator C2), we take into consideration one more operator
with the values
A2/= E ^0,13 y, A„/3j/= i [(^„^j/^ ),^„+ (^„^j/,^ )s„] ,
a,/3 = l
providing an approximation of 0(| h p).

Some properties of difference elliptic operators 289
Let us investigate the properties of the operator Ay = —Ay for any
y £ Qf^ = H. The operator Ay = | {A~y + A'^y) with the members
2 2
A~y=- E {k^i^y^^)_ , A+y=- E (^a/3 2/gJj,
a,/3=l ■ " a,[3 = 1
is self-adjoint: A = A* for ^^2 = ^21 ^i^^ admits both estimates B8) and
B9).
The operators A and A+ will be the subject of separate
investigations. For arbitrary elements y, v £ H the summation by parts formula
yields
2 2
C4) {A+y,v) = - Y^ {{kapys,)x^,v) = Y^ {k^^y-^^ ,v^^] ^,
a,13 = 1 a,13 = 1
2 2
C5) {A-y,v) = - J2 ((^a,8J/xv).«o .^) = J2 [^«0%;^ >^.^■J„■
a,/3=l a,[3 = 1
The interchange of j/ and v followed by that of a and /? results in the
relations
2 2
iA+v,y)= J2 (^a/3%, .2/gJ„= X] (^/3a2/s, .%.J^-
a,[3 = 1 a,[3 = 1
Putting these together with C4) we verify that (A+)* = A+ only if k^^ =
^21. In a similar manner it is plain to show that in this case {A~y = A"
and, hence, A* = A. In this regard, it is necessary to take into account
that j/j,^ = 0 for x^ = 0 and y^^ = 0 for x^ = I2, while j/^^ = 0 for Xj = 0
and j/j = 0 for x^ — /j. Passing now to the expression (A+j/, j/), which is
always representable by
2 ^2-1
a,/3=l 22 = 1
il = l
we carry the sum over a,/? under the sign of the inner product in the first
summand, leading by C1) to
2 / 2 \ 2
ci Y. i^y-^S^ 1) < E ^'"/jy-^,,feJ < C2 X] (B/-J'' 1) •
a=\ \a,/3 = l / a=l

290 Difference Schemes for Elliptic Equations
The expression for {A~y,y) can be transformed without difficulties
0
in just the same way. Recalling now that Cj < k^^ < c^ and {Ay,y) =
X]a=iB/s„ i2/s„]" we obtain c^ A < A < c^ A, yielding inequalities B8).
Inequality B9) can be deduced from the above relations by virtue of the
estimate 6E < A< AE.
4.5 HIGHER-ACCURACY SCHEMES FOR POISSON'S EQUATION
In this section we consider higher-accuracy schemes for the Dirichlet
problem A) of Section 1 in a rectangle.
1. The statement of the Dirichlet difference problem providing a higher-
order approximation. On the basis of the "cross" scheme it is possible to
construct a scheme with the error of approximation 0{\ h I"*) or 0{h^) on a
solution in the case of a square (cube) grid. In order to raise the order of
approximation, we exploit the fact that u = u{x) is a solution of Poisson's
equation
A) At« = -/(x).
Without loss of generality we may restrict ourselves to the careful analysis
of the two-dimensional case {p = 2) where
Au = Li u-I-L2 u , ^aU=^'7r~^,
oxi
a
by appeal to the difference operator
A u = (Ai -K A2) u , A„ u = u^^^^ ,
assuming u = u{x) to possess all necessary derivatives. Then
B) Ku-Lu=^ L\u+^ Llu + 0{\h\').
From the equation Li u + L2U ~ -^/{x) we find that
LI u = ~Li f - Li L2 u, LI u = -L2 f - Li L-2 u ,
so
that
K^ h"^ h"^ + K^
C) Ku^Lu-^ ^^f-U L^f-^J^LiL2U + 0{\h\')

Higher-accuracy schemes for Poisson's equation
291
7t
3'^
^6
<' 1
8 /»! 4 5
Figure 16.
We substitute here Lu = —/ and replace LiL^u by the difference operator.
The outcome of this is
Ai A2 u = u^,^,i.,^^ ~ L1L2 u =
dx\ dxl
This operator is defined on the nine-point pattern shown in Fig. 16.
The expression for Ai 7V2 u such that
Ai A2 u = Ai
i{Xj ,X2 — h^) — 2 u{x^ , x^) -\- u{x^ , X2 -|- h^)

hi hi
{u(x'i — /ij , Xj — h^) — 2 u(xj , X2 -^ h^)
-\- u(Xj -h /ij , X2 — hy) -\- 4 u(xi , X2)
— 2 u{x^ — /ij , X2) -|- u{x^ — /ij , X2 -|- /J2) " 2u(xj , X2 -|- /la)
— 2 u(xi -|- /ij , X2) -|- 'u(xi -|- /ij , X2 -|- hy)]
is needed in the estimation of the error of approximation to A^ A2 u—Li L2 u
by virtue of the well-established expansion
D)
kv = v^^
v(x + h) — 2 v{x) + v{x — h)
= v"{0, C = x + eh, \0\<l,

292 Difference Schemes for Elliptic Equations
assuming that v[x) has a continuous second derivative on the segment
[x — h,x + h];
E) Av = v,,=v"{x)+'^v(''He), e=^x + 9*h, \9*\<l,
v{x) has a continuous fourth derivative on the segment [x — h,x + h]. By
relating x^ to be fixed we might have
A2U= L2uixj,x^) +-^ -^{Xj,^, ^, = x + 6*2/i, , 1^2! <!•
Other ideas are connected with the expression
Ai A2 u(xi , X2) = Ai L2 u(xi , X2) + yI ^^ 5~^*-^' ' ^^^'
Applying formula E) with v = L2 u and x = Xj to the first summand yields
Ai L2 u{x, , X2) = Li L2 u{x, , X2) + — -g^i^i > ^2)
The same procedure works for the second summand with respect to formula
D):
what should be done is to bring together the results obtained:
(Ai A2 - Ll L2) U := 0{hl) + 0{hl) = 0A h P) .
Substituting into C) the expression for Ai A2 u in place of Li L2 u
LiL2W = Ai A2U + 0(|/ip),
and involving the equation Lu = —/(x), we finally get
K^ + K^ K^ h'^
F) Ku = Lu-^j^K,K2u-^L,f-^L2f + 0{\h\'')
( h^ hi \ hl + h^

Higher-accuracy schemes for Poisson's equation 293
Because of this, the equation
K'y = -ip, A'y = Ay+ ~r-^ Aj A2 j/,
^ = /+Y^ ^if+^ ^2/,
provides an approximation of order 4 on a solution u = u{x) of Poisson's
equation A). Indeed, formula F) gives
A'u + ^ = {A'u + ^)-{Lu + f) = 0A h H, L = L,+L2.
The operator A' is defined on the nine-point "box" pattern (see Fig. 16)
consisting of the nodes (Xi+m^ h^ , x^+m2 h^); m^ , m^ = —1,0,1, by means
of which scheme G) is representable by
(8)
5
3 \ h
6 \hl h]
12 \h\ ^ hi) ^^ ^^
where ^(^tii) _ y{^x^±h^ ,2-2), j/( + ^i'~'=) = y{^x^ -\-h^ .x^ — h^), etc. On the
square grid [h^ = /i^ = h) the final result is simple to follow:
4B/1+2/2+2/3+2/4)+ 2/5+2/6+2/7 + % , 3 2
^° 20 10 ^
(see Fig. 16).
To avoide cumbersome computations, we substitute ki f for Li f and
7V2 / for L2 f into the formula for (p having replaced ip hy 0{\ h I"*), which
does not change the order of approximation for ^ = A'u + p = 0{\ hl^), so
that
hi hi
^ = /+Y^Ai/+^A2/.

294 Difference Schemes for Elliptic Equations
2. Estimation of a solution of the difference boundary-value problem.
Consider now the difference Dirichlet problem for the scheme of accuracy
0A/i|^) in the rectangle G = {0 < x„ < /„ , a= 1,2}:
( A'y= -ip, X Eui,^, j/|^^ = ^(x),
where A'y is given by formula G). Each of the grid nodes is regular, because
the nine-point pattern belongs to the rectangle G (Fig. 16). The boundary
jf^ of the grid contains all the nodes on the boundary F including the
vertices of the rectangle. With this in mind, we set up the problem for the
error z = j/ — u:
A0) A'z =-i/>, x^ui^, z = 0 on j^ ,
where ^p = A'u + ip — 0{\ h I"*) for x G w^ if u G C'-®-'. To decide for yourself
whether the conditions of the maximum principle are satisfied, a first step
is to compare (8) with A) from Section 2. As a final result we get
A1) B{x,^)>{] for -^<^<\/5.
v5 «2
To evaluate the solution of problem A0), we should have at our disposal
the majorant of the type
Y{x) = K {ll - xl + ll - xl) .
Taking into account that AY := -4A', ki k^Y = 0, ||y || < K{ll + P)
and accepting 4A',= ||V'|lc> ^^ deduce by Theorem 3 for the solution of
problem A0) the estimate
P + P
provided that the condition
1 h, r
V5 K

Higher-accuracy schemes for Poisson's equation
295
holds. This implies that
Scheme (9) is of fourth-order accuracy when u G C^^\
f G C^"*^ and condition A1) is satisfied.
It is worth noting here that on the square grid (/i^ = h^ = h) this
condition is automatically fulfilled. A proper choice of ip guarantees the
sixth order of accuracy of scheme (9) on any such grid. Convergence of
scheme (9) with the fourth order in the space C can be established without
concern of condition A1). An alternative way of covering this is to construct
an a priori estimate for || Az |p and then apply the embedding theorem (see
Section 4).
Let fi/j be the space of all grid functions defined at the inner nodes
X E iOf^ oi the grid Of^ = {(ij h^ , i^ 112)}, 0 < i„ < Na, /i„ Nc, = /„, a = 1, 2,
0
and let fi/j be the space of all grid functions defined on the grid tUi^ and
vanishing on the boundary jf^. The accepted view is that an inner product
in the space fi/j such as
Ni-l N2-I
ij=l i2 = l
= Yl y(^)^(^)^1 ^2, y,v&^h, l|j/|| = Viy>y)-
and the operators Ai and A2 specified by the relations
Aiy = -Aiy, A2j/ = -A2j/,
where A„ y = A^ J/ for all y E ilh, will complement special investigations.
~ ...... °
Here A^ really acts from fi/j into fi/j and is identical with A^ itself in fi/j.
Therefore, Ai and A2 are linear operators defined on Hh — Oft (they can be
treated also as operators from fi/i into fi/j C fi/i)- The domain and range
of these operators coincide with fi/j = Hh,
Here the complete posing of problem A0) is concerned with an
operator equation
A2) A'z = Ai z + A2 z - (h^ + K^) Ai A2 z = tp,
VA & Hh

296 Difference Schemes for Elliptic Equations
with Xj = h'f/l2 and m:^ = h'^l^'i- incorporated. The operators Ai and A^
are self-adjoint:
{Aay,v)=z{y,Aav), a =1,2, y,veHh,
positive definite:
(A„j/,j/)>AW||j/||2, x[") = ^ dn' ^^ > ^ , a =1,2,
a ^ a
and commuting: Ai A^. = A2 Ai, In view of this,
AiA2 = (AM2)* >0.
By virtue of the relations
Aa < ||A„|| i5, II A„|| = ^ cos^ ^ < 7^' ^JM«|I < ^
we arrive at
^1 A2 < II Ai II A2, Ai A2 = A2 Ai < II A2 II Ai ,
(Xj -I- ^2)^1 A2 = Xj Ai A2 + K^ A2 Ai
<>c,\\Ai\\A2 + >c^\\A2\\Ai
< ^{Ai+A2),
yielding
lA<A' = A^+A2-{>e,+ X2) AiA2<A,
A = Ai+A2,
l\\Az\\<\\A'z\\<\\Az\\.
The operators A and A' are commuting and self-adjoint and, therefore,
for the equation A'z = ^ the estimate H^'-^H < | || V'll is certainly true.
By the embedding theorem from Section 4,
/2 3/2
Ikllr < 7=-=Az < 7?=||tM|,

Higher—accuracy schemes for Poisson's equation 297
it being understood that for any steps h^ and h^ the solution of problem
A2) can be estimated by
Ik lie = II2/-" lie <A^ II'Ml'
where
3/2
M= 7:^^ , L := max(/j ,L).
The estimate so constructed implies the uniform convergence of scheme (9)
with the rate 0{\ h I"*) for any ratio h^/h^.
3. The multidimensional case. The method for constructing a scheme of
fourth-order accuracy described in Section 1 applies equally well to the case
of several variables, making it possible to compose the difference scheme of
fourth-order approximation
A3) A'y = -ip{x), a;Gw^, y = ^.{x), a; G 7^, ,
a=l P^/La
V
A4) A J/ = V Aa y , A„j/ = yg^ ,.^ ,
a=l
^ = / + Eit^«/-
a=l
associated with problem A) of Section 1 in the p-dimensional parallelepiped
Go = {0 < x„ < /„ , a = 1, 2,... ,p} on the grid iui^ = {xj = (fj /ij ,...,
iphp),ia =0,1,2,... ,A^„,/i„A^„ =/„}.
By introducing the space Hh = fi/j and the operators A^ by analogy
with the preceding,section we draw the conclusion that the operator
A5) A'=A-^X„^ A„A/,, ^a = ^, A = J2Aa,
a=l a—i a=l
Pyta
is self-adjoint and possesses the estimates
4 — »
-—i- A<A'<A, p > 1 .

298 Difference Schemes for Elliptic Equations
These properties are an immediate implication of the chain of the relations
p 1 p 1 p 1
^^ A
cv—1 /5?^cv a = l /5^cv a = i
For p > 4 the difference operator A5) lacks the property of having fixed
sign (ellipticity). In each such case it is recommended to refer to another
operator A', which preserves the ellipticity property for any p:
V V ( /j2 \
^'2/ = E n h?+YfAjA„j/
" = 1 /3 = 1 \ /
or
A6) ^'j/=E n(i?-^/3^/3)^«j/.
It is evident that the approximation order remains unchanged and A! is
identical with operator A5) up to the terms 0(\ h I"*).
On the other hand, since E — k^ Ap > |i5,
and, hence,
.3
a=l
(-) A<A'<A.
3
With the aid of the above operator inequalities we are able to produce
the necessary a priori estimates and justify the convergence with the rate
0A/il"*) for the scheme in hand. Observe that for p = 2 operator A6)
coincides with operator A5).
We have nothing worthwhile to add to such discussions, so will leave
it at this.

Difference Schemes for
Time-Dependent Equations with
Constant Coefficients
In this chapter difference schemes for the simplest time-dependent equations
are studied, namely, for the heat conduction equation with one or more
spatial variables, the one-dimensional transfer equation and the equation
of vibrations of a string. Two-layer and three-layer schemes are designed
for the first, second and third boundary-value problems. Stability is
investigated by different methods such as the method of separation of variables
and the method of energy inequalities as well as by means of the maximum
principle. Asymptotic stability of difference schemes is discovered for the
heat conduction equation in ascertaining the viability of difference
approximations. Finally, stability theory is being used, increasingly, to help us
understand a variety of phenomena, so it seems worthwhile to discuss it in
full details.
5.1 ONE-DIMENSIONAL HEAT CONDUCTION EQUATION
WITH CONSTANT COEFFICIENTS
In this section we consider the one-dimensional heat conduction equation
with constant coefficients and difference schemes in order to develop various
methods for designing the appropriate difference schemes in the case of
time-dependent problems.
299

300 Difference Schemes with Constant Coefficients
1. The original problem. The heat diffusion process on a straight line is
described by the heat conduction equation
where u = u(x,t) is the temperature, c is the specific heat, p is the density,
k is the thermal conductivity and / is the density of heat sources, that is,
the amount of heat emitted per a unit of time on a unit of length. Thermal
conductivity and specific heat may depend not only on x and t, but also
on the temperature u. In that case the equation is said to be quasilinear.
If k and cp are constant, equation A) can be written in the form
2 7 = "^^+/' a=— , J - — ,
ot ax^ cp cp
where a^ is the thermometric conductivity (Maxwell) or diffusivity (Lord
Kelvin).
Without loss of generality we may set a = 1 and rewrite equation B)
du d'^u .
Indeed, by introducing x' = x/a and denoting once again x' by x we obtain
C). Where searching a solution to equation B) on the segment 0 < x < I,
it is sensible to pass to the dimensionless variables
, X I a^t
whose use permits us to rewrite equation B) in the form C) with 0 < x' < 1
and / = /^//a^ incorporated.
We concentrate primarily on the first boundary-value problem
associated with equation- C) in the rectangle -D = {0<;c<l,0<i<T},in
which it IS required to find a continuous in D solution u = u{x,t) of the
problem
^ = P^ + fi^'^)' 0<x<l, Q<t<T,
ot OX"
(I) u(x,0) = Uo(x), 0<x<l,
u{0,t) = u^{t), u{l,t) = u^{t), 0<t<T.

Heat conduction equation with constant coefficients 301
2. The family of six-point schemes. As before, it will be convenient to
introduce the grids ui/^ = {x^ = ih, i = 0,1,... , A'^} and w^ = {tj —
JT, j = 0,1,... , jg}. When operating in D on the grid
•^hr ='^ft X ^r = {{ih, i'r)> « = 0, 1,... ,A^, i = 0,1, ... ,io}
with steps h = l/N and r = T/jg, we denote by j// the value at the
node {Xi,tj) of the grid function y given on w^^^. The approximation here
consists of replacing the first derivative du/dt by the first order difference
derivative, the second derivative d'^u/dx'^ by the second-order difference
derivative u^^. = Au and introducing an arbitrary real parameter a. As a
final result we get a one-parameter family of difference schemes
D) ^' ^~^' = a(^j//+^ + A - ^)j//) + ^/, 0<i<N, 0 < i < io.
Sometimes scheme D) will be treated as a scheme with weights. The
supplementary boundary and initial conditions will be explicitly specified
with accurate approximations:
E) y^ = u/, ij}^ - -"^
= ui
F) y° = j/(x,,0) = Mo(a;i).
Here iff is a grid function approximating the right-hand side / of equation
C). With such a variety, we may accept, for example,
fi = f{^i^ii+ii2). ij+i/2 =h +0.5r,
Aj/i = yxx,i = (Vi-i - ^ Vi + Vi+i) / f^^ ■
When all conditions D)-F) are put together, they are constitute problem
The difference scheme D) is constructed on the six-point pattern with
the nodes
and center (Xj, ij+i) (see Fig. 4.c). The truth of equation D) is supposed
at the nodes (xj, tj_^_i), i = 1,2,... , A^ — 1; j + 1 = 1,2,... , Jq, known as
the inner nodes. The set of all inner nodes of the grid w^^^ is denoted by

302 Difference Schemes with Constant Coefficients
The boundary and initial conditions E) and F) are specified at the
boundary nodes of the grid iOf^^. All of the nodes of the grid w^^ lying
on the straight line t = t, constitute what is called a layer. Because in
scheme D) knowledge of the values of the sought function y is required on
two layers, it can be regarded as a two-layer scheme. In the sequel we
will show that the accuracy and stability of scheme D) depend on a proper
choice of the parameter a. Before proceeding to further careful analysis,
it would be prudent to examine some schemes relating to particular values
of a in light of those remarks. Foi cr = 0 we get the four-point scheme
(Fig. 4.a)
- Vi + ^
or
G) J//+' =(l~2 7)j//+7(j//_j + j//l_J + r^/, 7 = ^//i',
developed on the pattern with the nodes {x^, tjj^i), (x^, tj) and (Xjj.j, tj).
The value J//'*'"' at every point of the new layer t = i-^j is given by the
explicit formula G) through the values y^ on the previous layer t = tj. Since
the solution is prescribed by the initial condition j/9 = Uq{xi) at moment
i = i(j, it is possible to determine all the values of y on any adjacent layer
by applying successively formula G). Because of this, scheme G) is said to
be explicit.
For G 7^ 0 scheme D) refers to an implicit two-layer scheme. When
the value j//'*'' is sought on the new layer under the natural premise c 7^ 0,
we obtain the governing system of algebraic equations
(8) akyi+'---yi+' = -F^, ^^ = _ y/ + A - ^) A j// + ^/,
i= 1, 2, ...,, A^- 1
subject to the boundary conditions yl^^ = u^'^^ and y^ = u^'^^. This
system can be solved by the right elimination method (see Chapter 1,
Section 2.5). In conclusion, it is worth mentioning two other schemes. The
first one with a = I known as the forward difference scheme or pure
implicit scheme will appear as further developments occur:
(9) 2^/!^ = Aj//+^+^/.

Heat conduction equation with constant coefficients 303
The six-point symmetric scheme with a = 0.5 ascribed to Crank and
Nicolson is of the form
vi""' - vi 1
■,3
A0) "' ^ "' = 2^(j//+^+J//J+^/
3. The error of approximation. In order to evaluate the accuracy of scheme
D)-F), the solution y = j// of problem D)-F) should be compared with
the solution u = u(x,t) of problem (I). Since u = u{x,t) is the continuous
solution of problem (I), we may set uf = u^Xi^tA and deal then with the
difference zf = y? —u-. For this, the first step in the estimation of the grid
function z/ on the relevant layer is connected with norms || ■ || of proper
form, for example,
||z|| = ||z||^= max |z,|, \\ z \\ =( "e ^f h) .
Let us pass, time and again, to the notations without subscripts and
superscripts, which are good enough for our purposes:
y- = y, y/"*"^ = y, yt = (y- y)/'^.
permitting us to recast the problem we have completely posed by conditions
D)^F) as
J/i = A- (G 2/ + A - (t) y) + ^ , (x, t) e w,j^ ,
A1) y{0,t)=u,{t), ij{l,t) = u,{t), t&uj,,
j/(£, 0) = Uo(x), x&Gj^, Ay=y^^.
In trying to establish the conditions for determination of z = j/ — u we
substitute y = z + u into (II) and regard u as a known function, making it
possible to set up the problem for z:
z^ = A{az+{l~a)z) +iP, (x, i) G W;,^ ,
(III) z{OJ) = z{l,t)-=0, teuJ^,
z[x, 0) = 0 , X e iUi^ ,
where
A1) ip = A(au +(l - a)u) - Uf +If

304 Difference Schemes with Constant Coefficients
is the error of approximation (residual) for scheme (II) on the solution
u = u{x,t) to equation (I).
Recall the definition of the order of approximation from Chapter 2,
Section 1.3 saying that scheme (II) approximates equation (I) with order
[m,n) or equation (I) is approximated by scheme (II) to 0[h"^ + r") on
a solution u = u(x,t) to equation (I) if || i/'(x,i) ||(.2-, = 0{h"^ + r") or
II V' Ilc2-) < M (h"^ + t") for all i G w^, where M is a positive constant, not
depending on h and r, and || ■ IL^-, is a suitably chosen norm on the grid
We proceed to the estimation of the order of approximation for scheme
(II) under the agreement that u = u[x,t) possesses a number of derivatives
in X and t necessary in this connection for performing current and
subsequent manipulations. Within the notations
. __ du I __ du _ __ __ J
U— —, u - — , U - U[Xi,tjj^^i^) , ^j+l/2 - ^i + 2 ''' '
the development of Taylor's series for the function u = u[x,t) about the
node {Xi,tj_^_^i2) leads to the expressions:
it = 0.5 (u + u) + 0.5(u- u) = 0.5(u + u) + 0.5ruj ,
u = 0.5 (il + u) —■ 0.5 r u^ ,
a u + (I —■ a) u = 0.b (u + u) + (a —■ 0.5i) T Uf .
All this enables us to reduce ^ to ^ =. O.b A (u + u) +{a—-0.5) t A u^—u^+ip.
Substitution of the set of expansions
Au = u"+ ^ «(^) + 0(/i^)=:L«+ ^L2y + 0(/z^)
U = U+-TU+ — U+ 0{t ) ,
Z O
1 - r2 - 3
■U ~ U- - TU+ — U + 0{t ) ,
Z o
1 r^
~ {it + u) = u+ — u + 0{t^) , Uj = u + Oir"^)
Z o

Heat conduction equation with constant coefficients 305
yields
A2) 1^ = (L u - u + ^) + (cr - 0.5) tLu+ — L'^u + 0{h'^ + r^) ,
whence it follows that i/i = ("" ~ 0.5) rL« + 0{h^ + r^) for ip = f =
f(x, tj, ^12), since it — Lu + f. By virtue of the relations L ii = L'^ u + Lf =
u ("'•' + /" and L'^ u = Lii -- L f and expression A2) we are led to
A3) i; = (^--f)= [(^--i)r+i/l2]L?,-i/,2L/+0(/j4 + ^2)^
By equating the expression in the square brackets to zero we find that
A4) a=--- — = a,.
"■ ' 2 12r
For (T = G* and ip = f + -^ h"^ Lf scheme (II) generates an approximation
of 0{h'^ + r^), that is, ^ = 0{h'^ + r^). But the order of approximation of
this scheme remains unchanged if /" will be replaced by f^^. == A /, giving
p = f+^ih^ A f) 01
Dr
1 2 I Ji-l "T Ji + l
A5) p^ = y/^^i^ + - f^^i^ + i
Observe that the last formula is much more simpler for later use.
Let C!^{D) be the class of functions with m derivatives in x and n
derivatives in t, all derivatives being continuous in D. Formulae A3)-A4)
justify that scheme (II) provides approximations of
, 0{h'' + T'') for G = 0.5, ^ = / or ip = f + 0{h''+ t'') iiu&Cl,
• 0(/i^ + r) for any a ^ 0.5, <y9 = / + 0(]{^ + r), for instance, f = f
or ^ =: / if u G Ct,
• 0[h'^ + r^) for cr = cr* and (p specified by formula A5) if u G Cf.
Scheme (II) with a.= cr* and (f = f -\- ^^ h/^ K f is usually termed a higher-
accuracy scheme. The requirement of the retention of the approximation
order for a given value a guides a proper choice of the right-hand side if.
Thus, for a = 0.5 it is taken to be i^s = 0.5 (f + f), p> ~ f and more.
It is easily seen from A3) that the error 0{h''^ + r^) may be al.so
achieved for a ^ 0.5 if we keep a = 0.5-|- Iv^cy/t, where a is an arbitrary
constant independent of h and r. Certainly, in this case a depends on h
and r. The arbitrariness in the choice of a is limited by the condition of
stability of the scheme (a result is ensured under the constraint a > — 7,
for more detail see Section 4).

306 Difference Schemes with Constant Coefficients
4. Stability with respect to the initial data. Let us investigate the stability
of scheme (II) with homogeneous boundary conditions by the method of
separation of variables. For this, we proceed as usual. This amounts to
applying to scheme (II) with homogeneous boundary conditions the identities
y^y + TiJi, ay + {l~a)y = y+aTiJt
and reducing them to the following ones:
A6) y@^t) = y{l,t) = 0, tetur,
y{x, 0) = u„{z') , X eu>h-
Scheme A6) is said to be stable if for a solution of problem A6) the
estimate holds:
A7) ||j/(i)||(j)<Mi|ho||(j) + M2^ma^J|^(i')||^2). ^ ^ ^r ,
where Mi, and M2 are positive constants independent of h and r, || • |L s
and II • IL^-, are suitable norms on the grid w/j.
With (f = Q incorporated, the estimate
A8) l|j/(^)ll(i) < Milho)ll(^), iGc,,
expresses the stability of scheme A6) with respect to the initial data. When
j/(x, 0) = 0, the meaning of the stability of scheme A6) with respect to the
right-hand side is that we should have
A9) ||j/(i)||(^)<M2^maxJ|^(i')||B).
The stability of scheme A6) with respect to the initial data and the right-
hand side is ensured by estimate A7), valid for the .solution of problem
A6).
Before giving further motivations, let us represent the solution of
problem A6) as a ,sum y = y + y, where y is a solution of the homogeneous
equation
A6a) j/j-crrAj/j=Aj/, j/@, i) = j/(l, i) = 0 , y{x, 0) = u„{x) ,
and y is a solution to the nonhomogeneous equation with the initial
condition y[x, 0) = 0;
A6b) J/,-(ttAj/, = Aj/+^, j/@, i) = 2/(l,i) = 0, j/(x,0) = 0.

Heat conduction equation with constant coefficients 307
To decide for yourself whether scheme A6) is stable with respect to the
initial data, a first step is to evaluate the solution of problem A6a). This can
be done using the method of separation of variables and deriving estimate
A8) in the grid L2{i0f^)-'[ioT:m:
N-l
E
i=i
l|j/||(i) = l|j/||. where ||j/|| = \/(j/, y) , (v, v) = J2 Vi ^i >^ ■
To develop those ideas, we seek the solution to equation A6a) as a product
of two functions, one of which T = T(tj) depends only on t = tj and
the other X = X(a'^) only on x = x^ under the approved decomposition
y{x, t) = X(x) T(t). Substituting this expression into A6a) and taking
into account that
Ajz-TAX, yt = XTt,
we arrive at the relations
T ~ T K \
^ = -^ = -^' T' = T(L.,j), T = T{tA,
T[aT+{l~a)T) X ^^+^^' ^J^'
where A is a separation constant. As a final result we get
l-(l-(T)rA
T = qT ,
l + ar X
The difference eigenvalue problem for X can be viewed as the Sturm-
Liouville difference problem:
AX{x) + \X{x) = 0, 0<x = ih<l, X@) = X(l) = 0, X{x)^0,
which has been under consideration in Chapter 2, Section 3.2. As stated
therein, the problem in view has nontrivial solutions identical with the
eigenfunctions
X'-''^F{x) = V2smxkx, k = I, 2, ... , N ~ I ,
wich correspond to the eigenvalues
^"= 1,2, ... , A^-1, 0< Aj < ■■■ < A^_j,
__ 4 ^ TTh
h =
Aj =
4 . 2 ■"■^/i
/l2 2
4 . -2 T^h

308 Difference Schemes with Constant Coefficients
The eigenfunclions {X ^'^^} constitute an orthonormal system
for which Parseval's identity takes place:
B0) ll/lP = ^E/^
k = l
Here /j. are the coefficients of the expansion for an arbitrary grid function
f{x) defined on the grid ilif^ and vanishing at the points x = 0 and x — I:
fc = i
So, problem A6a) has nontrivial solutions yrj^\ = Tj, X ^''' ^ 0, where
Tj; can be recovered from the equation
B1) n = Qk n
^k - Ik ->-k - - Ik -^k ' <lk - 1 , ^^ , ■
As a matter of fact, the constant T^ is free to be chosen in any convenient
way.
A solution to equation A6a) having the form j/j-j.-) = Tk X ^''' is called
a harmonic of the attached number k. Clearly, this function satisfies
problem A6a) with the initial condition Ug(x) = T^X^''\x). Lei us find
out the conditions under which every harmonic yf^)^ ^ — 1> 2, • • • , A'^ — 1, is
stable. From the recurrence relations
\^^> y(k) "^ ^k —'Ik ^ -^k' y(k) ~ Ik y[k)
the conclusion can be drawn that for | gj. | > 1 + e, where e = const > 0
does not depend on h and r both,
l|j/gMl = k.ll|j/^.)ll>(l + e)ll2/(i)ll>(l + e)^+^ l|j/(.)ll^oo
as r -^ 0, that is, the problem concerned becomes unstable. If | 9^. | < 1 and
t zz JT is kept fixed, then ||j/(fc)|| does not increase along with increasing
i(r^O):
\\yi^;\\< \\ylk)\\ <■■■< \\ylk)\\

Heat conduction equation with constant coefficients 309
and the harmonic in question is stable. Under the choice | 9^ | < 1, we have
II J//fc-) II < II J/fj.-) II- When this is the case, the scheme is said to be stable
in every harmonic.
We now assign the values of a such that either | gj. | < 1 or —1 <
9fc < 1) which guarantees the stability of the scheme in every harmonic.
It is clear from the formula q-f. = 1 — r Aj. A + err Aj.)"'- that gj. < 1 if
1 + G r Aj. > 0, that is, a > —(r Aj.)~^. The bound g^. > —1 or
2 + Bcr-l)rA.
l + ar Af.
is ensured by the restriction 2 + B cr — 1) r Aj. > 0 or cr > | — (r Aj.)~ ^. In
that case the condition 1 + crrXj. > 0 is automatically fulfilled. Since
4
^k < ^N-i < Jp .
the following relations occur;
1 1 h^
< ^ < ~-r-
t\^ rA^_j 4r
Hence, the condition | Qf. \ < 1 holds true for all ^ = 1, 2,... , A^ — 1 under
the agreement that
1 /l2
B3) '^^2-47='^-
Thus, all of the harmonics y,f.^ = T^X^*"') are stable under one and the
same condition c > cTq. We are going to show that the stability of scheme
A6a) in every harmonic known as the spectral stability implies that in the
grid i.2-norm with respect to the initial data y(x, 0) ~ «o(^). where Uo{x)
is a grid function defined for 0 < x < 1 and vanishing at the points x ~ Q
and X = I. With this aim, the general solution of problem A6a) is sought
as the sum of particular solutions having the form B2):
N-l N-l „ N-l „
y= E'yw= EnxO'), \iyf= ^ t^
k=l k=l fc=l
Substituting here T^ = q^Ti^ and taking into account B0), we find
that
y = 'i:\kTkX('\
k = l
\\y\\' = ^E ll n < max ql ^^ T^ = max g^ || j, ||2.
k = l * fc = l *

310 Difference Schemes with Constant Coefficients
Under the constraint a > a^ it is not difficult to verify that max | 9j; | < 1
and ||y||< ||t;||or
\\y'+'\\<\\y'\\< •••<ll/ll = l|«oll-
If so, the solution of problem A6a) satisfies the estimate
B4) \\y^\\ < ||«ol|. i=l,2, ...,
which means that scheme A6) is stable in the grid L2(w;j)-iiorm with respect
to the initial data for cr > a^.
A difference scheme is said to be conditionally stable when it is
stable only if r and h are related in some way, otherwise it becomes
unconditionally stable. A scheme, stable for arbitrary values of r and h,
is said to be absolutely stable. One can encounter schemes stable for
sufficiently small h and r such that h < hg and t < t^. These schemes are
not absolutely stable, but may be unconditionally stable.
Throughout the entire chapter, a special attention is being paid to
several types of schemes.
1. The explicit scheme (a = 0). Condition B3) gives 0 > ~ - h^ {4:t)-\
that is.
B5) TI < t;
r 1
The explicit scheme is stable only under condition B5) relating the grid
steps h and r (a conditionally stable scheme).
2. For cr >  tlie implicit scheme is stable for any h and r, since a > ~ > (Tq .
Thus, the forward difference scheme (cr = 1) and the symmetric scheme
{a = ~) are stable for any h and r, what means, by definition, their absolute
stability.
3. The scheme of a higher-order approximation {cr — cr*, cr* = ~ —
/i^A2r)~^) is absolutely stable. Indeed, for any h and r
/l2 /i2 /i2
12 T At ot
4. Implicit schemes with 0 < cr < | for cr independent of 7 = r/h'^ are
conditionally stable if one imposes the constraint 7 < B— 4cr)~^.
5. Scheme A6) with cr = | + h'^ar'^ providing an approximation of
0{h'^ +i^) is stable for any h and r if a > — ^

Heat conduction equation with constant coefficients 311
So, not only the order of approximation, but also the stability of
scheme A6) depend on the parameter a.
While investigating stability we dealt actually only with two time
layers tj, tj_^_^ and the step r = tj_^_^ — tj. All the tricks and turns remain
unchanged when the grid oj^ becomes non-equidistant, that is, the step
Tj+i = ij+i — tj depends on the number of the layer. In this situation the
parameter a may depend on the number j + 1 of the layer, say a = cr•'+■',
which serves to motivate the presence of another condition a > cr^'^^ =
I — /i^/D rj_|_j) in place of B3). For instance, for the scheme of accuracy
0{h^ + Tj ) we may accept ai'^^ = | — /i^/A2 r^_,_j). The condition
a > (Tp "*"' is sufficient for the stability of the scheme with weights in the
case when the grid ui^ is non-equidistant.
5. Stability with respect to the right-hand side. As a matter of fact, if
(T > 0, condition B3) given by
_ 1 _ /l2
'^'' " 2 At'
is sufficient for the stability of scheme A6) with respect to the right-hand
side as well. It is interesting to consider problem A6b) and look for its
solution in the form
N-l 7V-1
B6) y= E nxW, ||y|p= E T^,
k=l k=l
by representing the right-hand side ip in terms of the eigenfunctions {X ^^']:
B7) ^ = ''t\kX^'\ lklP = 'EV^
J; = l k = l
Substituting B6) and B7) into A6b) and recalling that AX(*-') = -\,,X^''\
we derive the series
N-l
J2 { Tkt A + <ttK) + A, Tk - v?i } X(^-) = 0 ,
k = l
from which it follows that the expression in the curly brackets vanishes as
far as an orthogonal system of eigenfunctions is concerned:
/9Q^ T - n T ^ ^y'fc „ _ 1^ (l-(T)rAfc
B8) I,. ^ qj^lf. + — -~ , 5j. _ .
1 -I- (TtA^ 1 -I- (ttA^

312 Difference Schemes with Constant Coefficients
Putting these together with B6) we find that
/V-l N-[ N-1
With the aid of the well-known triangle inequality || v + u; || < || v || + || to ||
we establish
W iiW
w y\\
or
B9)
Further
< max] 5,1 ypjk
\\y\\ < max |5j.| • 1
k
, let
\"' , ._.,.. r
1 k \1 + <tt\ I
r
""' ' "T^ |l + <TrA, 1
/ \ J-/2
Ilvll.
1 K^
C0) cr>cro. '^o=o~7~' cr>0.
When this is the case, | 9, | < 1, 1 + o-t\ > 1 and || y || < || 2/1| + t" || V || or
II2/'^ """Ml — II ^'^ 11+'' II V'^ II- Summing over j' = 0, 1,2, ... ,j and exploiting
the fact that ||t;°|| = 0 for the solution of problem A6b), we derive the
estimate
C1) l|y^+MI< E Hlv='^"ll-
j'=o
Let us stress here that estimate C1) was obtained under condition C0). In
subsequent reasonings we get rid of the bound a > 0 and impose then the
constraint
C2) a >a,, ''^ = \ ' ^ ^'' 0 < £ < 1 ,
where e = const > 0 does not depend on h and r. True, it is to be shown
that
kfcl<l> l + crrA^ = l + ((T-(Tjr Aj.+(T, rA
k
> 1 + T, r A, = 1 + I r A, -
(l-£) h'^Xk
^ 4 ^ 4 /i2 ^ ^^

Heat conduction equation with constant coefficients 313
implying that 1 + err A,; > £ for all ^ = 1, 2,. . . , A^ — 1 and providing in
combination with B9) the validity of the following estimates:
\\y\\< l|y|l + ^lkl|£"^
and
C3) ||j/'+MI < 7 i ^ll^"ll-
£ j' = 0
For a = a^, the condition cr* < 0 ensures r < | /i^ and the value £ = |
can easily be found from the equation ~r [1 — e) = t--. Collecting estimates
B4), C1) and C3) and summarizing the preceding results, we deduce the
following assertion.
If the conditions cr > | —^ /i^ r~ ^ = (Tq and a > 0 hold simultaneously,
then scheme A6) is stable with respect to the initial data and the right-hand
side, so that the solution of problem A6) admits the estimate
ll2/''+Ml<ll«oll+ E r\\^'\\.
j'=o
When (T < 0, for the stability of scheme A6) with respect to the right-hand
side
1 (l-e)/!^
z 4 r
is a sufficient condition. Here s £ @, 1) is an arbitrary constant
independent of h and r. For such a choice, the solution of problem A6) satisfies
the estimate
l|y^'+MI<ll«oll+ ~ tr\\^'\\.
£ j' = 0
For the scheme of accuracy 0{h'^ + r^) we thus have £ = I and cr* < 0
if r < i/i2.
6. Convergence and accuracy in the space L2(wjj). We state here that the
convergence of scheme (II) follows from its stability and approximation.
The error z ~ y — u is just the solution of problem (III). Using a priori
estimate C1) behind we deduce that
C4) ||^^'+MI< E ^ll'^^'ll as a>a,, a>0.

314 Difference Schemes with Constant Coefficients
This type of situation is covered by the following assertion.
If scheme (II) is stable with respect to the right-hand side and
approximates problem (I), then it converges and the order of
accuracy coincides with the order of approximation.
0{h' + T^),
0{h''+T''),
0{h^+T),
<T= i.
a = a^, ,
^^h
n e C'l,
u e CI,
"■ 7^ cr* ,
u e cl
Upon substituting the estimates for the approximation error obtained
in Section 3 into C4) we find that
C5) \\y^
So far we have investigated stability and convergence in mean, that is, in
the grid norm of the space L2('^ft)- Meanwhile, many situations exist in
which a uniform estimate, that is, an estimate for the error of a solution
will be of practical significance with regard to the norm
II y^ — u^ \\^ := max | y-' —■!(■' | .
7. Stability and convergence in the space C. In this section we expound
certain devices for obtaining uniform estimates for the problem A6)
solution:
• the maximum principle;
• the energy method which makes it possible to establish stability
in the space C with respect to the right-hand side on account of
embedding theorems;
• the representation of a solution in integral form in terms of the grid
function of the impulse point source (Green's function).
The maximum principle and, in particular, Theorem 3 in Chapter 1,
Section 2 will be quite applicable once we rearrange problem (II) supplied
by homogeneous boundary conditions (scheme A6)) witli obvious
modifications and minor changes. The traditional tool for carrying out this work
is connected with
aTAy-y = -y-{l-a)TAy-Tifi=-F,
C6)

Heat conduction equation with constant coefficients
315
crjVi.i - {2aj + l)yi + cry y^^, = -F, , i = I, 2, . . . , N - I
C7) F, = {l-a)j y,., + A - 2A - ,7O) y, + A - a)jy,^^ + t <p, ,
i=l,2, ... ,N -I.
1
/l2
The theorem we have mentioned above asserts that for a solution to the
difference equation
Ai y,_, - d y, + B, y^^^ =-Fi, i = 1, 2, ... ,N - 1,
2/0 = 0, y/v = 0, I A, 1^0, |5,|^0,
the estimate
\\y\\c <
is valid if and only if D^ = | Cj | — | Aj- | — | 5j | > 0. For problem C6) these
conditions (| A^ | 7^ 0, | 5^ | ^ 0, A > 0) are satisfied for a > 0 and Di = 1,
so that a solution to equation C7) can be most readily evaluated as follows:
\y\\c<\\F\
for (T > 0 ,
(j/j = Fi for G = 0). A simple observation that
||^|lc<l|y|lc + ^llvllc if 1-2A-,tO>0, l-a>0.,
may be useful in the further establishment of the inequality
C8) \\y'+'\\c<\\y'\\c + ''\\'p'\\c'
which is valid for scheme C6) under the constraint
/i2
C9)
r <
2A-<t)
Summation of C8) over j' = 0,1,2,... ,j leads to the estimate for the
problem A6) solution
D0)
\\y'^'\\c<\\y"\\c+ J:r\\^'\
J 1=0

316 Difference Schemes with Constant Coefficients
Summarizing, under condition C9) scheme A6) is stable with respect to
the initial data and the right-hand side.
Applying a priori estimate D0) to problem (III) yields
ll^^'+Mlc< E ^11^^"lie-
j'=o
We give a brief survey afforded by the above results: scheme (II) converges
uniformly with the same rate as in the grid L2(w;j)-norm (see C5)) if and
only if condition C9) holds. The stability condition C9) in the space C
for the explicit scheme with cr = 0, namely r < \h'^, coincides with the
stability condition B5) in the space L2('^ft) that we have established for the
case (T < |. The forward difference scheme with cr = 1 is absolutely stable
in the space C. The symmetric difference scheme with cr = | is stable in
the space C under the constraint t < h'^.
8. The method of energy inequalities. The well-developed method of energy
inequalities from Chapter 2 seems to offer more advantages in investigating
the stability of scheme (II) with weights.
We first illustrate its employment for a differential equation in tackling
problem (I) with homogeneous boundary conditions
dii d u
1^ = 7rT + /(^>0' 0<x<l, 0<t<T,
D1) ot ox^ -
w@, t) = w(l, t) = 0 , u{x, 0) = Uq{x) .
Here the inner product and associated norm are defined by
1
(w, v) = j u{x) v{x) dx , II w 11 = \/{u, u) ,
0
where u{x) and v{x) are functions defined on the segment 0 < a; < 1 and
vanishing at the points x = 0 and x = I. Let us multiply the equation by
du/dt and integrate it with respect to x from 0 to 1. The outcome of this
is
du
Integrating the second term by parts and taking into account the equality
du du 1
dt dx
= 0.
0

Heat conduction equation with constant coefficients
317
we find that
dt
2 1 d
^2 Wt
(9w 2 f f 9u
dx \ ' dt
To majorize the right-hand side, we make use of the Cauchy-Bunyakovskii
1
4 *
inequality and the e-inequality \ab\ < e a'"^ + ^ e ^ b'^ with £ = 1, permitting
us to arrive at the relations
(-^)^
implying that
V dt
d
di
■■)
d
d
<
u
X
2
du
1
" 2
■ 11/11 <
|/(-^.^)ir
du
'+\\\f\\^
By integrating with respect to t we thus have
du{t)
dx
<
<
du{0)
dx
du{0)
dx
1
^7!
+ vf
1 wmfdt'
0
max \\f{t)\\.
0<t<T "^ ^ "
-,1/2
With the aid of the relation
II u II „ = max I u(x) \ <
"^ 0<i:<l ^ ^ I -
du
dx
we finally get
ll«Wllc<2
(9w@)
dx
\f^ o^f<^.II^WII
At the next stage a similar estimate is needed in this connection for the
difference problem A6), We proceed as usual. This amounts to introducing
the inner products and associated norms:
N-l
1=1
\\y\\ = ViV' y).
TV
(y. ■"]= Y^yiV^h.,
1=1
\\y]\ = ^Ay^]

318 Difference Schemes with Constant Coefficients
and taking into consideration the trivial identities
y = \{y + y) - \T-yt, y ^ |(y + y) + |^j/t,
(T y + A - (t) y = (cr - I) r t/i + I (j/+ y) .
Putting these together with A6b) we recast the problem concerned as
yt-if^- l)T^yt-l^{y + y) = <p,
D2)
j/(x, 0)=0, y{0,t) = y{l,t) = 0,
which reduces by multiplying equation D2) by 2Ty^h = 2{ii — y)h and
summing the resulting equality over the inner nodes x = ih of the grid (Mf^
to
D3) 2T\\yt\\^ -2{a ^ \)t^ {Ay„ y^) - {A{y + y), y - y) = 2t{<p, y,) .
By appeal to the Green difference formula derived in Chapter 2, Section
3.1
{Av, w) = {v^^ ,w) =-{vg,Wg], v„ = w„ = 0, t;/v = ic/v = 0,
with V ~ y-i, w = y^ and v = y+y,w = y — y incorporated, we find in view
of J/o = y/v = 0 that
{^yt,yt)^-\\yts]\^,
( A ( y + y), y - y) = - ( yj + % , fe - y.J = - (I I y,. ] 12 - 11 y - ] 12) ,
Substitution of these expressions into D3) leads to the energy identity
D4) 2t {\\y,f + ia - l)T\\y,,]\^) + \\y,]\^ = \\y,]\-^ + 2Ti^, y,),
which is valid for any a. Let a > a^. The accepted view is that a reasonable
form is
J = ||Hl' + ('^-|)^ll%]P with v^y,.
By virtue of the estimate emerged in Chapter 2, Section 3
D5) \\v,r < ^\\vf

Heat conduction equation with constant coefficients 319
one succeeds in showing that J > 0 for cr > (Tq. Because of D5),
>\\v\?-\h'\\v,r>Q
for (T > (Tq = \ — h'^ D'')"'^- Combination of the preceding relations reveals
a profound result known as the energy inequality:
Ilfe]|'<llfe]|' + 2r(v9, J/,), <T>a,.
If (/? = 0 and y is a solution of problem A6a), then || t;!.+-' ]| < ■ • • < || y- ]|,
meaning that for cr > cTq scheme A6) is stable with respect to the initial
0
data in the norm || 2/ IL', = || y.s ]|i which is a grid analog of the VJ/^^-norm.
No doubt, it sounds interesting, but the discovery of stability with respect
to the right-hand is of special merit in this matter. Letting
D6) <T > ,7, ,
we claim that
D7)
(^c = I
J>e\\v\\'
1 (i-£)/l
2
2 At
0 <£ < 1
for a > a.
Indeed,
J = ||,;||2 + (^_^J^||^_]|2^(^^_ l)^||^_-
2 {l-e)h^ ,,„, 1,2
>\\v\\- ' 4^ ll%]l
>\\y\\'-i^-e)\\v\\' = e\\v\\'.
Substituting D7) into D4) yields the energy inequality
D8) '2Te\\y,\\'+\\%]\' < \\y,]f + 2Ti^,y,), a > a, .
By applying successively the Cauchy-BunyakovskiT inequality and the s-
inequality we arrive at the chain of the relations
D9) 2T{<p,y,) < 2r\\<p\\ .||t;,|| < 2re\\y,\f+ f^\\<pf-

320
Difference Schemes with Constant Coefficients
Substitution of D9) into D8) gives
Summing over j' ~ 0,1,. . . ,j and keeping t;° = 0, we deduce that
ZS jl-Q
In agreement with Lemma 1 in Chapter 2, Section 3, we might have
\y\\c<i^\\ys]
thereby justifying that
y'^'\\c<
1
J' = 0
' l|2
-,1/2
a > a. .
2V2£
Applying this estimate to problem (III) we get
||z^'+ML<M2 max \\ip^'\\, M^ = —-^ , a > a,
II lie - o<^.,<^. II v- M' 2v^ - '-
thus demonstrating that scheme (III) converges uniformly, so that
\W -u^Wc < I Mih' + T^), a=\, u^Ct,
y M{h'^ + T'^), a = a^, ueCl.
A case in point is that for the explicit scheme with a = 0 the uniform
convergence does not follow from D6) under the constraint 7" < | h'"^■ But
the a priori estimate emerged in Section 5.7, namely
j-i
\\y'\\c<\\y°\\c+ E^llv^'^ lie.
j' = 0
provides support for the view that
r< |/i2.
^'■+Mlc< E^ll^^'
j'=o
"Whence it follows immediately that the explicit scheme converges uniformly
with the rate 0{t + /i^).

Heat conduction equation with constant coefficients 321
9. The third kind boundary conditions. The first kind boundary conditions
we have considered so far are satisfied on a grid exactly. In Chapter 2 we
have suggested one effective method, by means of which it is possible to
approximate the third kind boundary condition for the forward difference
scheme (cr = 1) and the explicit scheme (cr = 0) and generate an
approximation of 0(r +/i^). Here we will handle scheme (II) with weights, where
a is kept fixed. In preparation for this, the third kind boundary condition
E0) "^°' ^^ = /?! m@, t) = ^i,{t), /3j = const > 0 ,
will be imposed at the point s = 0, providing later the difference boundary
condition on the four-point pattern with the nodes (O, ij_|_i), (/i, ^j+i), @, i,')
and [h,tA. Once supplied by the difference condition
E1) cr(y^-/3, j/)o-FA-(t)(j/^.-/3, y)o = i/(.j/^o-/"i, , /}., = ^,-F|/i/o ,
where /g = f{^,ti+ii2) ^^'^ i^i — A'i(^j+i/2)i "^"^^ will show that it
approximates condition E0) on a solution u = u(x,t) to equation C) subject to
condition E0) and the order of approximation is the same as we obtained
in approximating equation C) by scheme (II) for a given value of a.
Upon substituting y = z + u into E1) we find that
E2) a {z^ - /?, f )o -k A - cr) (z^. - /?, z)o = i /i Z( 0 - i), ,
where D^ =: a (w^, — /3j w)g -k A — cr) (w,, — /3j w)g — ~ hu^ ^ + fl^ is the error
of approximation of condition E0) by the difference condition E1) on a
solution u. Developing Taylor's series for u about the node @, tj + 17")
and denoting by v^ the value of the function v at this node, we can write
down in the preceding notations u' = du/dx and u — du/dt
1^1
1 = ("o - A «o + /ii)o + (c^ - I) ^ («■' - A «)o
/i^(io + |/i< + 0(/i2 + r2).
2 ' 0 ' 2
Inserting here u' — /3j u^ — fj,^ and u'^ = u,g — /g both recovered from the
equation
/;, = (^ - 1 ) r (J'o - p, u,) + 0{K' + r2) ,
we deduce that
E3)
0(/i' + r), ai.\,
0(/i' + r2), a=\.

322 Difference Schemes with Constant Coefficients
It is straightforward to verify that at the point x = I the boundary
condition
8u(l f]
E4) ^r^—!- = P^u{l,t)-ii^{t), /Jj = const >0,
is approximated to the same order by the difference condition
E5) - [a {y,j. + C^ y)j^ + A - cr) {y^ + /Jj y)^^] = i /i y^^j^ - jl^ ,
where fi^ = Ji^ + \h]^, Ji^ = ^2(^^+1/2) and /^ = /(l.^j+i/a)- By setting
/«i = A«i + Y^ A'l + y (/o - Pi /o) + y /'i Vo >
Y^ A«2- y D+A//v)+ y
A«2 = A«2 + T?; A«2 - -^ (//v + A //v) + — ^2 V/v
and replacing |/i ?/( 0 and |/i J/f /v ^^^ E1) and E5) by ~ h p^y^ g and
^hp^yt /vj respectively, with pj, = 1 + -hPf., k = 1,2, we establish the
difference boundary conditions providing an approximation of 0{h'^ + r^)
for (T = (T* = I - -J^ /l^ 7"" ^ ■
To avoid cumbersome calculations, it will be sensible to introduce
more compact notations
A-y= y^-/^y ^ K^y^^yi±M
and rewrite the difference boundary conditions E1) and E5) in the same
forms as approved -before for scheme (II):
J/i=A ((Ty+(l-(T)t;)+v9 , j; = 0,
E6)
% = A+((Ty + (l-(T)t;)+v9+, !■=!,
with 9?" = 2/ij//i and 99+ = 2jl^/h, showing the new notations to be
sensible ones. For /3j = /Jj = 0 these can be viewed as the difference
approximations of the second kind boundary conditions. Also, the order of

Heat conduction equation with constant coefficients
323
approximation happens to be the same as we obtained for the boundary-
value problem of the third kind.
Let us reduce condition E1) to an alternative form convenient for
current manipulations. The outcome of solving E1) with respect to j/g =
y,
j+i
E7)
Vo = Xi 2/1 + 1^1 ,
"'= a:
A, =,tA + /3j/i) +
h'
-(l-a)il+p,h)
2t '
Vo + hfl, } .
Along these lines, condition E5) becomes
2//V = ^^ail-i + ' ^2 =
E8)
A, '
A^=a{l+p,h) +
2r '
If r /i^ 1
/A = ^ <! A - (t) t;A,„ 1 + A - (t) A + /?2 /i) Vn + li h
A^
2t
so that 0 < x„ < 1 for /?„ > 0, (T > 0 and a = 1,2. In the determination
of j/ on a new layer we obtain the difference equation (8) with boundary
conditions E7)~E8). This problem can be solved by the right elimination
method (see Chapter 1, Section 2.5).
Stability of scheme (II) with the third kind boundary conditions can be
discovered following established practice either by the method of separation
of variables or on account of the maximum principle.
10. Three-layer schemes for the heat conduction equation. One of the
first schemes arising in numerical solution of the heat conduction equation
du/dt = d'^u/dx'^ was the Richardson explicit three-layer scheme such as
E9)
where
/J+'
y
j-i
2t
Ay^
or
yo = A y,
yo
y-y
'2t
y = y^ ,
y-y
j-i
?/ = y\
A y = Vsx
By exactly the same reasoning as before, it is not difficult to verify that this
scheme is of order 2 with respect to r and/i both: ■0 = Aw—Wo = 0(/i^-|-r^).

324 Difference Schemes with Constant Coefficients
But it is absolutely unstable: this scheme becomes unstable for any way of
tending r and h to zero.
As further developments occur, we rewrite equation E9) as
,,i + l _ ,,i-l yi _ 2 J, J I yi
^ ' 2t /i2 ■
Upon substituting the sum j// + y/~^ for 2 j// in the right-hand side of
equation F0) the resulting three-layer "rhombus" scheme known as the
Du-Fort—Frankel scheme
^ > 2r ~ /i2
becomes explicit with respect to j// and appears to be absolutely stable
for any h and r. In giving an alternative form of the "rhombus" scheme
F2) ^? + ^2 ^" = ^^
with t/j-j = (y/ — 2y? + y/~^) T'"^ we may attempt the right-hand side
of equation F1) in the form
Vi-i -Vi -Vt +^»+i _ Vi-i -'^yj + Vi+i y» - 2yi + Vi
/l2
'2
— Vxx 7,9 lltt
Substitution of the last expression into F1) yields F2). As a matter of fact,
the "rhombus" scheme is some modification of the Richardson scheme with
the extra member on the left-hand side of F1) t' h~'^ yi^, which a.ssures
us of its stability. The proof of stability of scheme F2) is an immediate
implication of the 'general theory of Chapter 6 and is omitted here. The
error of approximation for scheme F2) is
2 2 2
iP = Aw-w„-^ W(-( = u"-u- ^ u + 0(h'^ + T'^) = -^ il + Oih'^ + T'^).
From such reasoning it seems clear that the "rhombus" scheme provides
the conditional approximation
V' = 0(r2-K/i2 + r2/i-2) =0(/i2) as t = 0{h'^) .

Heat conduction equation with constant coefficients 325
Keeping r = a/i (l + 0(h)) with a = const we draw the conclusion that
scheme F2) approximates any equation of the form
du 2 <5^w (9^w
The following implicit three-layer schemes with weights are quite applicable
in solving equation C):
• symmetric schemes
F3) yo = k{ay + {l-2a)y + ay) +^-
• nonsymmetric schemes
F4) y^ + aTy^^ = kij + ip.
Equations F3) and F4) contain the three layers (ij_i, tj, ^j+i), so that, in
what follows, tj ^ T, j >_ ]. The value y(x, 0) = Uo{x) is known in advance
and the value y(x,T) should be preassigned in addition to the available
information. For example, this value can be determined either by
yt{x, 0) = iiaix) or tj{x, t) = y{x, 0) + TU^ix)
with Ug(x) = u''(x) + f(x, 0) still subject to the condition y(x, t) — u[x, t) =
0{t'^) (see Chapter 2, Section 1).
Sometimes two-layer schemes are effectively used to assign the value
y{x,T).
By virtue of the asymptotic relation
au + (l— 2a)u + au = u + aT u^f = u + 0{t )
the symmetric scheme F3) is of order 2 in r and h for any a. The error of
approximation for scheme F4) is representable by
F5) ^ = Ku-\- Lp — [u^-\- (J T U:j;^)
= Lii-^ ip- {u- ^TU + arii) + 0{h'^ + t'^)
= (L« + /-^)+(^-/)-(^-i)r« + 0(/i2 + r2),
which serves to motivate the accuracies which interest us in the following
cases:
^=:0(/i2 + r2)
V'=:0(/i2-Fr)
for
for
(T- 2 ,
•^T^l.
V = />
^ = f-

326 Difference Schemes with Constant Coefficients
Omitting the terms 0{h?') in F5) and involving the equation ii — w" + /
insight, we deduce I
with the members
insight, we deduce that scheme F4) provides an approximation of ©(/i"*
In trying to recover y from F3)-F4) we obtain the three-point equations
F6) Ay,_,~Cyi + By,+, = -Fi
with the right-hand sides —Fi depending on y, y, (p and the usual boundary
conditions for i = 0 and i = N. This problem can be solved by the right
elimination method. In the process of computing the values oft; and y on
two previous layers should be placed in the storage. In the case of two-layer
schemes it suffices to save merely one preceding layer.
Stability of three-layer schemes will be established in Chapter 6. For
the purposes of current section we confine ourselves to sufficient stability
conditions for the symmetric scheme F3) and scheme F4), respectively:
(T > J and cr > — |. In just the same way as we did for two-layer schemes
the difference boundary conditions with a highly accurate approximation
can be developed for the third kind boundary conditions E0) and E4). For
the symmetric scheme F3) the boundary conditions providing an
approximation of 0{h^ + r^) reduce to
t;o=A {a y + {1 - 2 a) y + a y) + ifi , i = 0,
yo =A+{ay + {l-2a)y + ay)+<p+, i = N,
with
A-y^ y^-J^y, A+y=^y^y^y,
2/1 ^h
showing the new notations to be sensible ones.
On the same grounds as before, the third kind difference boundary
conditions for the nonsymmetric scheme F4) are specified by the relations
Piiyt + '^Tyit) = ^~y + 'fi~' * = o.
F7)
^2 («/i+ '^^y«) = A+t/ +V+, i = N.

Heat conduction equation with constant coefficients 327
These boundary conditions generate an approximation of 0(/i^ + r) for any
(T if we agree to consider Pj = /52 = 1 and
0.5/1 "^ .V ' . . Q 5^
But the same procedure works for cr = | and the boundary conditions F7),
making it possible to achieve 0(h^ + r^). Eventually, by successive use of
the members
1 h' ,^hp, hp,
-=2 + T27' ^^ = ^ + —' ^= = ^ + —'
/l2
<p = fix, t + r) + -^ {/"{x, t + T) + fix, t + T)),
/j2
,^, =f,^it + T)+j {^i,{t + r) + /'(O, t + T)- pj{0, t + T)),
/j2
ly, = 11., {t + 'r) + — {^,it + r) - /'(I, t + T)- pj{l, t + T))
we arrive at scheme F4) of accuracy 0{h'^ + r^) and the error of
approximation 0{{h'^ + T'^)/h) at the nodes s = 0 and s = 1,
5.2 ASYMPTOTIC STABILITY
1. p-stability. In studying various schemes for the heat conduction
equation we have already mentioned that the simplest way of relaxation of the
stability condition connected with upper bounds on the step in time r is
the transition to implicit schemes. All of the implicit schemes with a weight
of the upper layer a > a^, cTq = | — ^ h'^ t~^ , are stable. In particular, the
schemes with cr = | and a = 1 are absolutely stable, At the final stage
of research the scientists are interested in the accuracy with which an
approximate solution to a differential equation could be found. It is natural
to try to reach a prescribed accuracy by carrying out the minimal
possible number of computer operations (arithmetic and logic), that is, with

328 Difference Schemes with Constant Coefficients
the minimal expenses of execution time. A number of operations may be
minimized by improvements of difference schemes as a result of modeling
the basic properties of a differential equation in the space of grid functions
as well as possible. One of such typical properties is the true asymptotic
behavior of its solution as i —> oo.
To illustrate our approach, let us consider, for example, the heat
conduction equation
A) Yt=l^^^ °<^<^' ^>°'
w(j;,0) = «(,(«), Q<x<\, w@,O = mA,0 = 0, i>0.
The problem concerned can be solved by the method of separation of
variables, within the framework of which we seek its solution as a sum
u{x, t) = J2 Cj; e ^''^ ^kix) , ^kix) = \/2 sin nkx ,
k = l
where
1
Aj, = Pn^, Cf, = (wq, Xk) = / u^{x)Xkix) dx ,
0
oo CO
ll«(^>Oll'=E c2e-2A.'||X,|p= Y. c^-2A.^
k=l k=l
since ||Xj; || = 1. As Aj. increases along with the subscript k, we thus have
X^. > Aj and
oo
k = l
which means that the solution of problem A) admits the estimate
B) l|w(OII < e~^i' ||m@)|| for any t>0.
With increasing t the harmonics W/j,-) = Cj, e"'^*;* Xk{x), k > 1, damp
more rapidly than the first one, so that for sufficiently large t
C) u{x,t) « c^e-^i' Xi{x), Cj 7^0.

Asymptotic stability 329
This stage of the process refers to the regular behavior. When the
solution of a difference scheme for problem A) also possesses the properties
similar to B) and C), the scheme is said to be asymptotically stable.
We now deal with the scheme with weights
yt = A{ay+{l-a)y) , xeui,,, t = JT>0,
D) yo=y/v=0, y{x, 0) = Ua{x), «ewft,
^y = Vxx ■
The solution of the problem under consideration has been obtained in
Section 1.4 by the method of separation of variables and can be expressed by
the following series
N-l
yi = Vi^r 'ij)= E Cfc ql Xkixi) ,
k = l
where
_ l-il^a)r\l __ 4 7:hk
''' - 1 + ^rA^ ' ^^^ - /? '^" ^ '
Xk{xi) = V2 sm -Kkxi, cj. = i%,Xk), (y, v) = J2 Vi'^i^-
The orthonormality of the system {Xj;} gives
\\y'\\'= "y: ^^ < p'^''Ecl = p'^\\y°r,
k=l k=l
which assures us of the validity of the estimate
E) \\y'\\<p'\\y°\\, P = .^^B"^ 1 \ik\-
l<k<.N—l
Scheme D) is said to be p-stable with respect to the initial data if estimate
E) is valid. An estimate of the form C) is admissible for the solution of
problem D) when p < 1.

330
Difference Schemes with Constant Coefficients
2. Asymptotic stability. Let us describe rathei- mild conditions under which
the behavior of the solution y^ of the difference problem D) becomes regular
if one makes t: = jr large enough:
F)
y- ^yi^i 'tj)~ Ci5/ Xi{xi).
Clearly, it is possible only in the case when the first harmonic dominates,
that is, max | ^j, | is attained for ^ = 1, so that
G)
max Qj. = p z=
l<k<N-l
l-(l-(T)rAf
1 + (Tr AJ*
What is available to find out the conditions ensuring the validity of formula
G) is the function
m =
I - {I - t) ^,,
1 + cr^i
I + all
where ^j = r Aj* and ji = t X'l > ji^ under the natural premise ./'(^) > 0,
^ > ^j, Alternative forms of the function /(^t) may be chosen in a number
of different ways. In subsequent reasonings we prefer one of them:
/(^) = /,(^) Uiii) K-\ K = [l^aii,f{\ + aiif>Q,
= fi- fi^ > 0 ,
=:2 + Bcr-l)(^ + ^l)-2(l-(T)(T^^l,
implying for t < f^ that
r 2 + ^t + ^j > 0 if ,7 = 1,
'^^^ 1 2-(^ + ^j) >2-r(A + .5) >0 if a = 0,
where fg = 2F + A)~^, E = Aj* and A = A'?,_ . In particular, we have

Asymptotic stability
331
for the symmetric scheme with o" = | under the constraint r < 2 (E A) ^/^.
We bring together the results obtained and state that
1
1 + tS
p= I l-^'^
for a = 1 and any r > 0 ,
for a = 0 and r < r^ = ,
0 + A
for cr = 2 ^^'^ '' S ''o "
/6A
In each such case the estimate || y^ \\ < p^ \\ y'^ \\ is certainly true. Moreover,
/O-' —> 0 occurs as i^ = j t —^ oo. As we will see a little later, it will be
convenient to represent p by
p = e-^^+^\ T<p = ^^,
1^1^ = t6.
It is straightforward to verify that for c = |
Tip= p.^-\
1- 1^1
12 80
It follows from the foregoing that
p^ = exp < -8tj -
12 80
12 80
tj } < e"**J,
The regular behavior for the symmetric scheme with cr = | can be
described as follows:
y{x,,t,) « c,e-''^+<'''X^{x,),
where
,-{
tH^ tH^
= 0{i
12 80
In dealing with the pure implicit scheme we might have
1
6 = X
P =
1 + rE
> e
p-i = e 3 3

332 Difference Schemes with Constant Coefficients
with the asymptotic expansion C = |''"^^ ■" \t'^6^ + ■■■ = 0[t) . It
is evident that p^ approximates e~ *J poorly as compared with another
scheme we have mentioned above.
Our approach is especially clear in the forthcoming example with
t8 = 2-
For the value 6 / A = 0,01 the quantities
/'l.=o,5 = Po.s =—1= °-^^^2, H,=i =Pi = Y^ = 0-8333
need to be considered with more care than usual. Their comparison with
g-ri _ g-0.2 _^ 0.8187 gives
p^^ = A - 0.0006) £-"*, Pi = A + 0.018) e-'" .
To be more specific, for 6t. = l{j = 5) we have
pI^ Ki 0.997 e-**i, /)/ « 1.09e-*'j,
thereby clarifying that for Stj = 1 the quantity p^ differs from e~ i by
0.3% and pI - by 9%.
The above example shows that although the scheme with cr = 1 is
absolutely stable and, in principle, may be used for any r, it is not accurate
enough at the stage of the regular behavior when r grows. In order to retain
a prescribed accuracy (here it is meaninful to speak only about the relative
accuracy), we should refine successively the step r = r- with increasing tj.
So, we are much disappointed by the main advantage of the scheme with
G=1 stipulated by its stability for any r > 0.
The symmetric scheme with cr = |, which is absolutely stable in
the usual sense: \\y^ \\ < \\y° ||, is conditionally asymptotically stable for
T < Tq. Being concerned with the explicit scheme for cr = 0, we observe that
the condition of asymptotic stability r < 2 (E + A)~ practically coincides
with the usual stability condition r < 2 A~^ for small values of E/A.
For the heat conduction equation
4 . 2 tt/i . 4 2 Tr/i c A 4

Asymptotic stability 333
and the stability condition r < /j^/2 also implies that t < 2 {6 + A)"-'. In
the case of a symmetric scheme
2 h'- h
\/E A sin tt/j tt
and the condition r < h/ir is not burdensome. Thus emerged the typical
relation 7 = r//i^ < l/Gr/i), so that 7 < 10/7rfor h — 0.1 and 7 < 100/7rfor
h = 0.01. The first eigenvalue A^ = E of the difference problem concerned
is
V e y V 3 ^ 45 ^
where ,J = 'irh/2. Plain calculations show that
I 0.97 A, for /i = 0.1,
\ — < A — TT
' ~ I 0.9997 Aj for /i = 0.01, ' "^
It is interesting to learn what may happen in dealing with the symmetric
scheme with a = ^ and one of the possible steps t > t,^, say r = mT,^,
m > 1? Then maxjt | rjf, \ is attained for k = N — 1:
irA-1 1-2t-'A-'
P= kiv-il ~ ~
irA+1 l + 2r-
Upon substituting here
1 _ 1 _ 1 /^ _
T A ruT^A 2 m \l A
we get /> = e"'' ^+'' ?(''' *) with
."- ^
<5
4 m
2
\/^
fl — n
6 To
Am
t-^^-\
t6
4m2
4??j2 ' ' V /
Thus, in the symmetric scheme with a = ^ the improper asymptotics is
revealed for m > 1:
yi^^i , <j) ~ cyv_i exp I - ^—^ <^. I XAr_i(xJ ,

334 Difference Schemes with Constant Coefficients
where
Xj\[-i(x^) = sin 7rGV — 1) X; = sin 7rGV — 1) ih
= sin 7r(l — /i) i = (—1)'~ sin ttx^
so that
y{xi, tj) K, cj^^^ expl -^^h \ i~^y~^ ^li^i), ^lixi) = sin ttx^ .
This provides support for the view that the solution is completely
distorted. From such reasoning it seems clear that asymptotic stability of a
given scheme is intimately connected with its accuracy. When asymptotic
stability is disturbed, accuracy losses may occur for large values of time.
On the other hand, the forward difference scheme with cr = 1 is
asymptotically stable for any r and its accuracy becomes worse with increasing
tj, because its order in t is equal to 1. In practical implementations the
further retention of a prescribed accuracy is possible to the same value for
which the explicit scheme is applicable. Hence, it is not expedient to use
the forward difference scheme for solving problem A) on the large time
intervals.
3. The scheme of second-order accuracy (unconditionally stable in the
asymptotic sense). Before taking up the general case, our starting point is
the existing scheme of order 2 for the heat conduction equation possessing
the unconditional asymptotic stability and having the form
,,7+1/2 _ j
{E-\arK) ^—^ = A - ^ .) A^/^
(8)
j/7 + 1 _ j/7+1/2
~y'
where A j/ = y^x^ ^ 's the identity operator, yi'^'^l'^ is the intermediate
value and G = 2 — v2. For i = 0 and i = N the homogeneous boundary
conditions are specified by
(9) y^+'^' = y}/''' = 0 , yi = j/^ = 0 .

Asymptotic stability 335
The usual trick amounts to eliminating y^+^l^ from (8) so as to reduce the
governing difference equations to the following ones:
[e- ^A^ y^+'l' ={E + r{l^a)K)y\
E- ^A).y+^:=.y+^/^
As a corollary to this fact, we see that
A0) (e-''-^k\ y^+' ={E+T{l-<T)K)yK
Having substituted into A0) the identity of artificial character
1+1 1 , y y
or y = y + Ty-(, we recast the scheme at hand as
A1) (^~l^^j yt=[^+^A']y-
The accurate account of the approximation error of scheme A1) on a
solution u = u(x, i) to equation A) can be done using
2 _ 2 2
A2) V = A w — —— yV" w — Wj + or A Wj — A^ Wj .
Upon substituting into A2) the expansions
7 2
Ku = Lu + — L2«+0(/i4),
h~ ^2 /' .^^ , f du h~ d'u
= L2w + 0(/i2),

336 Difference Schemes with Constant Coefficients
where
9^w du „
Lu = ^^^ = -^:— . « = «!,,
we find that
ij.
L u =
- L u —
du 1
57-M
du
'a'-
a
+ 1) L'-u + Oih'- + T'-).
Hence, i/" = 0{h'^ + T^) foi a = 2~a/2, satisfying the equation ^ cr^ —cr+i =
0. Other ideas are connected with the method of separation of variables,
whose use permits us to look for a solution to equation A0) with zero
boundary conditions j/g = j/^y = 0 as a sum
N-l
yj _ J2 T/ Xkix), Xkix) - y2 sin irkx .
k = l
By inserting this expression in A0) we obtain the equation related to T^ :
or, what amounts to the same, T^ — qj. T^ , where
\~{\-a)T\l
(Ik =
l+io-rA
Furthermore, adopting arguments similar to those being used for the scheme
with weights we arrive at the chain of the inequalities
\W^'\\ < Hh/ll < ■■■ ^p'^'Wy'W
with p = maxfc | g^ |. In the further analysis we shall need yet, among other
things, the formula
l-(l~^)rA;'
P
(l + iGrA-/i)"
wr
here A^' = Ah '^ sm'{irh/2). In giving it we will show that max^ | g^ | is
attained for ^ = 1. With this aim, we have occasion to use the function
m
1+ ^a p,

Asymptotic stability 337
for which
I /(/^) ! - /(/^i) < 0 foi- ^> fH if A'l < 1 •
It is necessary to consider merely the case A—cr) /.i > 1, since otherwise
I /(a*) I = /(aO < /(A'i) for A* > A'l- In th^^t case the difference in question
becomes
= ;^ { 2 + B o- ~ 1) //j + - o-Vi
where D=n + |o-/ij M+^d/iij .
Recalling that G = 2 — a/2, ct^ = 4G — 2 and /.ij < 1 we get the
lower estimate which will be the subject of special investigations in the
near future:
/(Mj-i/aoi
> ^[2- [2a{l- a)+^^a'- {I- a) - {2<T -\)) ^^+\<T^ ^i'-]
= -^{2- F.5- 10.5 0-)// +C.5 f7-2)^^*2} .
The square trinomial in the curly brackets equals 0.05/i^ — 0.349/i+ 2 and
has the negative discriminant d = 0.349^ — 0.402 < 0. Because of this, we
deduce that /(/Xj) — | f{n) | > 0, meaning
1^A-a)fi, _ l-(l-f7)-
l + if7//j) (^l + |f7r<5y
under the constraint /ij = t6 < 1, which is valid if being replaced by the
condition r Aj = ttt^ < 1.

338 Difference Schemes with Constant Coefficients
With this relation in view, it is not difficult to derive the asymiptotic
expansion
with the miemibers
p=er''+''', 0 =
24.6 271.7
which serve to motivate the double inequality />•' < e~**J+^*J < e''^*'
and the a priori estimate \\y^ \\ < e'^^*' \\y° \\ for scheme (8) under the
constraint t 6 < 1.
4. Asymptotic stability of the three-layer scheme. The object of
investigation here is the three-layer schemie
A3) I j/t-|j/f + ^y=0, A = A\ A>6E, 6>0,
which is unconditionally asymptotically stable for r < 1/BE). To make
sure of it, an alternative form j/o -|- r j/j-j + Ay = 0, which provides an
approximation of order 2 for the heat conduction equation with constant
coefficients A), will appear in the further developmient:
4> = Uc + TUff + Ail. = Uo + TUf^t — Au — T AUo — i r^ AMjj
= u + Tu — Lu — tLu— ^t'^Lu + 0{h'^ + r^)
= (w - Lu) + t{u- Lu) + Oih^- + T^-) = 0{h^- + t^) .
Here we used also the initial equation u = Lu and its corollary ii — Lit.
In conformiity with the method of separation of variables,
yl=T,{t^)X,,
where Xk refers to an eigenfunction of the operator A. Recall that, by
definition,
AXk=\lXk, k = l,2,...,N,
With this in mind, we initiate the derivation of the difference equation
I T^+' - 2 T/ + I T/-^ + A[! T Tl+' = 0

Asymptotic stability
339
C + 2T\1)Ti+'-4T^+Tr'=0,
whose solution is sought in the form Tj^ = q^ . Omitting the subscript k for
a while, we investigate the quadratic equation related to g^:
C + 2//)g2-4g+l = 0, /i = rA ,
with discriminant D = 4A — 2fi). For D < 0 its roots gC^'^) are complex-
conjugate and gf^^gf^^ = | g p = C -|- 2/i)""', that is,
(J.2)
1
<
1
Let now D > 0. In that case
^i
A-2)
3 + 2/i
A new function
2-7r^^
3 + 2//
with the derivative
1 d<p
2 + v/r^^^
3 + 2//
/^i < /i < /^iv .
< 0
<^ 5// B + VI - 2//) yr=T7I 3 + 2//
is aimed at establishing that g^ < g^ ^ for fcj > fcg- Hence, max fin)
is attained for /t = /tj = rAj, giving
l-i'l<t^<t^N
2 + Vl - 2//j
max <^(//) = <^(//j) == —— if l-2//j>0.
It may happen that i D^ = 1 — 2/tj. < 0 for some k > 1, thus causing the
occurrence of the event that
Ik
A.2I
1
3 + 2//,.
Since C + 2/t,) "^ < <^^(/tj), the maximum value max I g^
k
always be attainable for k = 1, is equal to
2 + V1-2//J
''' which
3 + 2//J
if /^. < 2

340 Difference Schemes with Constant Coefficients
For large j the first harmonic y^ k. a-^p^X^ dominates in the solution
k = l ^ '
which has been constructed before.
By means of the quantity />•' = e~^ •°g(i//') with
1
2 ii.
3 1 + VI - 2^ij
1 + =//
3 /"l
it is plain to calculate
logl=log(l + ^0-log(l ^ ^'
p ^v ' 3^v ^v 3 1 + yr^^
The expansion in powers of p,^ yields log {\/p) = p^ -\- t: p^ -\- • • •, so that
p^ =exp{-.5<^. - ir2,53^^.}.
Thus, the three-layer scheme A3) of accuracy 0{h~ -\- r~) possesses the
proper asymptotics as t: -^ oo under the unique restriction t6 < 1/2,
which is not burdensome. Comparison of the final results with the two-
layer scheme D) reveals some formal advantage of the three-layer scheme
over the symmetric two-layer scheme with cr = -1, which is conditionally
asymptotically stable if we imposed the extra constraint r < Tq = l/vE A,
Tg = Tg(/i), in addition to the usual one t6 < 1. However, this restriction
is sufficiently weak in real-life situations and, therefore, it is meaningless
to speak about any practical advantage of the three-layer scheme. For
this reason the two-layer scheme is quite applicable and more efficient in
practical implementations.
5.3 SCHEMES FOR THE HEAT CONDUCTION EQUATION
WITH SEVERAL SPATIAI VARIABLES
1. The explicit difference scheme. The schemes considered in Section 1
ma}' be generalized to the case of the heat conduction equation with several
spatial variables.

Heat conduction equation witli several spatial variables 341
In a common setting, let G = G + F be a p-dimensional domain with
the boundary F and let x = (xj, x^, • ■ ■, Xp) denote an arbitrary point in
it. It is required to find in the cylinder Qrp =:Gx[0<t<T]a. continuous
function u(x,i), x G G, satisfying the governing equation
A) ^ = A« + /(x, 0, xeG, t>0,
and the supplementary conditions
w(x, 0) = Wo(x) , xGG, u{x, t) = ij,(x, t) , xGF, t>0,
wh
ere
P a2
v-^ a u
dx'-
« = ! «
is the well-known Laplace operator.
For the purposes of the present section, let us introduce the grid uif^ =
{ a;; e G} in G and denote by jj^ the set of all nodes of w^j belonging to
F and by uif^ the set of all inner nodes x^ £ G, so that uij^ = uij^ + -y^^.
The starting point in the further development of the difference scheme is
the approximation of the elliptic operator Aw. We learn from Section 1 of
Chapter 4 that at all of the inner nodes A u ~ A u for x £ w^.
By inserting in A) the difference operator A in place of the Laplace
operator we are led to the system of differential-difference equations
B) ^ = A^y-f <^(x, 0, xew;,,
where i'(x,<) is defined on the grid w^ for every < > 0. The size of the
system B) is equal to the total number of the inner nodes TV of the grid
w^j. Here the function <^(x,<) approximates /(x,<) on the grid w^j.
At the next stage we introduce the grid in t
C^r = {tj = JT, i = 0,1, 2,... , Jo , ioT = T]
with step r. In order to pass to the difference scheme for a new function
y{x,t) defined on the grid uij^^ — uij^ x uir = {i^it tj)) x G lO}^, t £ w^},
it is necessary to replace the system of differential equations B) by some
difference scheme written in terms of t. Choosing, for instance, Euler's
scheme we obtain the explicit scheme
j/7+1 _ yj
C)
= Ay' +f', i = 0,l,2,
/ = j/(x, O) = wo(x), j/^'l =A*^ J = 0,1,2
ih

342 Difference Schemes with Constant Coefficients
The value j/-'"'""' on the relevant layer is to be determined by the recursion
y^+' = y' +T{ky^ +^^), i = 0, 1, ....
For the sake of simplicity we are working in a parallelepiped G such
that { 0 < a;„ S ^a) Q^ = 1) 2, • • • , p } and on a certain grid w^j = | x^ =
(ij /ij,. . . ,ih),i^ = 0,1,... , 7V„, /i„ = /„ N~^} equidistant in each of
the directions x^. We refer to the Bp-|-l)-point operator A of second-order
approximation with the values
D) Aj/=EA„J/, A„ J/= j/_^ ^^^ = ,
a = l "„
where y is the value of the function j/(x, t) at a fixed node x = (ij /ij, . . .,
iph^), y^-^'^^ = J/(x(±i"),0, x(±i") = (i, ft,,..., i„_,A„_,,(i„± !)/»„,
i„_l_j /i„_|_i, ... ,«' ftp) is the node adjacent to x (x'-"'""'"-' stands on the right
from X and x'-""'"-' stands on the left from x). With the aid of the well-
established relations
a-l
12
I h P = /j2 + /j2 + . . . + /j2 _
we calculate the residual
ij = Au^+ip^^^ ^ = 0(\h
T
provided the asymptotic expansion (p^ = /(x, ^.)-|-0( | h p-|-r ) holds.
Scheme C) is conditionally stable in the space C with respect to the
initial data, the right-hand side and the boundary data. The maximum
principle for the difference problem C) may be of help in establishing the
indicated properties with further reference to the canonical form
E) J/^+'= 1-2E/I^ U
^ a = I «
P
3
+ E /^ (j/^-(x^+'^^)+J/^'(x^"'''^))+^^^'-

Heat conduction equation witli several spatial variables 343
Comparison with the general difference equation
F) A{P)y{P)= J2 BiP,Q)yiQ) + FiP),
QePatt'(P)
which has been considered in Section 2 of Chapter 4, shows that in the case
of interest
P = ix,t^^,), Q=ix,t^),(^x(+'^\t^),(^x(-'''\tj), a=l,2,...,p,
A{P) = 1 , B{P, Q) = 1 - 2 X] ^ •
a=l «
The boundary 7 of the grid ui for equation E) consists of the nodes (x, 0),
X e W;j, and (x,<-'), x £ 7^^, t^i < ^,-_|_i- It is straightforward to check that
DiP) = AiP)^ J2 BiP,Q) = 0, B{P,Q)>Q
QePatt'(P)
if
a=l a
Common practice involves the decomposition for the solution of the problem
concerned as a sum y = y + y, where j/ is a solution to the homogeneous
equation
yt=^y, j/|^„=/^'> 2/(x, 0) = wo(x),
and y is a solution'to the nonhomogeneous equation
yt = Ay + (fi, y\-y,=^' y(x, 0) = 0.
As a matter of fact, we obtain for y equation F) with F = 0. On the
strength of the maximum principle (see corollaries to Theorem 2, Chapter
4, Section 2.3) we have
l|j/(x, tOllc < nrax( max max |/i(x, ^./) I, [[ Wo(x) || ) ,
■* \ X67), t ■i<tj -^ /

344 Difference Schemes with Constant Coefficients
provided condition G) holds. With this relation in view, we derive the
inequality
(8) \W\\c < „m£«^. 11/^'" lie, + ll«ollc;.
where ||/.i|L; := maXxg^i |/.i(x,<)|, which expresses the stability of the
explicit scheme C) with respect to boundary conditions and initial data for
(9) r<r,, r,= l(^±^
By analogy with the one-dimensional case in the estimation of y (see Section
1.7), let us recast the difference equation as j/-''''"' = F^, where
^ a=l a / a=l «
and II F^ W^-, < \\ y\\^. -|- r || <^ H^^ for t <Tq, thereby justifying that
l|j^"+Mlc = ll^''llc<li^"ilc + ^li^''llc
for the same values of r < Tq. Summation over j' = 0,1,... ,j yields the
inequality
A0) Wy'^'Wc < E r\\^ 11^, T<T,,
j'=o
characterizing the stability of the explicit scheme with respect to the right-
hand side. For r < Tq relations (8) and A0) together imply the estimate
A1) II y'^' \\c < II «o lie + „ max II ^^^' II + E r II ^' \\^ .
With the notation h = min /i„, the stability condition (9) takes the form
e-
2p
thus demonstrating that the admissible step r of the explicit scheme
decreases with increasing the dimension p.

Heat conduction equation witli several spatial variables 345
2. The explicit three-layer scheme. We now turn to the simplest explicit
three-layer scheme known as the Richardson scheme and being an analog
of scheme E9) from Section 1:
A2) y -^ = Ay^+<pK
I T
However, it is absolutely unstable. Substitution of the half-sum
i (yr' + vi^') = \{yr-^yi + vD+yi = \^yL^yl
for yl, where
y'+'-2y^ +y^-'
Vtt — n '
T
into the right-hand side yields the p-dimensional analog of the Du-Fort-
Frankel scheme. The forthcoming substitution
a=I « a=l "
leads to an alternative form of writing
P 1
which
A3)
where
will be
a-1 a
involved further in the explicit scheme
y= + ^ yii = -^y + 'P,
'^ 1
In the case of a cube grid with h^ ■= h.^ = ■ ■ ■ = h = h one succeeds in
designing one more scheme instead of A3):
2
A3') ^? + 1^ J/,-, = Aj/+^
and in establishing as its immediate implication the useful formula for
determining ii = J/■''*'"':
(l-F2 7)y=(l-2 7)j/-F47J/-F2rAj/-F2r<^,

346 Difference Schemes with Constant Coefficients
where y = y^, y = y-'"^, 7 = pr/ h~ and A = Aph~~ Also, we point out
here without proving that scheme A3) is stable for any r and h. This fact
follows from the general stability theory developed in Chapter 6. Also, this
scheme generates conditional approximation, since its residual behaves for
r = 0(/i2) like
9
T) T
ip = Au + ip- uo - -p- Wj-j = 0(/j^)
if we accept <p = f and h^ = h^ = ■ ■ ■ = h = h.
3. Schemes with weights. \'Vhen discretizing equation B) in t, the scheme
with weights arises naturally in one or another form:
A4)
yt = A{ai/ + {'l-a)y)+(p, x e w^ , t = JT>0,
J/L, = M . J/(x, 0) = Wo(x) , X e w,, .
In preparation for this, we agree to consider (p = f ~ /(x, l_,_j ,3). As
before, we suppose once again that G is a parallelepiped and A is specified
by formula D). We investigate the order of approximation by appeal to the
expression
/1 \ u-\- u ( 1 \
au + {\-a)u= -——+ 0-- - ruj
2
for GW + A — cr) « and, after this, touch upon the residual
V' = A((TW+A — G)w)+<^ — Uj
w + w
A
2
2
+ (o-- 2 ) ^^"* + '^~"i
L«+ (f7-i)rL« + /^«.+ (^-/) + 0(|/i|2 + r2)
2
where Lu — Au, u = u{x,tj_^_^|2), f = f and w =r du/dt. It is therefore
concluded that
Oi\h\'- + T'-) if
a
1
^ 0(|/^P + r) if ^T^i

Heat conduction equation with several spatial variables 347
The maximum principle can be applied to any such scheme with
weights under the constraint t < t^, where
1 / '^ 1 \ "'
by means of which it is not hard to show that for cr = 1 there is no limitation
on r. Because of this, a priori estimate A1) for the problem solution is
certainly true. Indeed, it is possible to reduce A4) to F), by means of
which
A(/^) = l+2crf]~, .r>0,
a a a=1 «
The restriction r < Tq follows from the initial assumption concerning the
nonnegativity of the coefficients -S(-P, Q). It is clear that D(P) = 0.
Instead of A4) it is reasonable to deal with the scheme with different
weights (T„ related to the directions x^:
p
1
a=l
for which
yt= E Aa (o-ai/+(l -^a)j/) .
Pi X -1
r<ro, ^0 = 2 ( E
4. The scheme with increased accuracy. For the stationary problem
Aw=-/(x), w|p = y^i(x),
in a parallelepiped the scheme of accuracy 0[\h |^) has been constructed
in Chapter 4. In the two-dimensional case p = 2 this scheme reduces to
A'y= -(fi,
where
7 2 I l2 l2 l2
A'y = A,y+A,y+ ^-^ Ai A2 J/ , ^ = / + ^ Al / + ^ A^ /.

348
Difference Schemes with Constant Coefficients
We are going to show that the scheme
/J? + hi
A5) yt = Y^ ^«(^«y + (^-^«)y)+ '|2 ' ^1^2 j/ + ^,
y\.. = A*,
j/(x, 0) = Wo(x) ,
with the ingredients
A6)
1 h'-
/ = /
i+i/2
provides an approximation of 0(| /i j'^ + r^) in light of the representation of
the residual
u + w / 1 \ ^ \ h- + K . .
o-„ - - I r A„ w J + Ai A2 w + (^ - Wj
^ = E A
2 \ " 2 1'"" "V ■ 12
This can be done by inserting the well-established expansions
w + u
u + 0(r2) ,
/j2
A„ w = L„ w + yI L; u ,
L'^U — La U — Li L2U — La f,
The outcome of this
^ = ^
/l2
a_
12
u^ = u -|- 0(r") ,
A„ Wj = La it + 0{t^ + hi),
a = 1, 2.
LaU+Oi\h\^ + T'-) = Oi\h\^ + T'-)
with G„ still subject to the first condition A6). In order to prove the
convergence of this scheme with the rate 0(\h \'^ + r^), it is necessary to
obtain an a priori estimate for a solution of the problem with zero initial
and boundary conditions. Having no opportunity to touch upon this topic

Heat conduction equation with several spatial variables 349
we address the readers to Chapter 6 and Concluding remarks therein. We
note in concluding that scheme A5) is stable for any r and /i„ and do not
pursue here analysis of this: the ideas needed to do so have been covered.
The second, no less importance, question is: how to solve the system
of difference equations
2
E Ta„A„y-y= -F .
a=l
At first glance, the matrix elimination method suits us perfectly. But
0(N~) operations are required in its application, where TV is the total
number of the nodes of the grid uJj^. Just for this reason scheme A5) is
unacceptable in practical implementations. We will show later that it may
be replaced by some schemes of the same order 0{\ h j'^ + r^) and 0{N)
arithmetic operations are required in this connection for determination of y
by applying successively the scalar elimination for a three-point equation.
The resulting schemes are said to be economical, so there is some reason
to be concerned about this. One needs to exercise good judgment in
deciding which to consider. The scheme we have constructed above is aimed
at designing other schemes with the indicated property. Here we will not
elaborate on these matters. If you wish to explore this more deeply, you
might find it helpful to refer to Concluding Remarks at the very end of the
present chapter and references therein.
5.4 SCHRODINGER TIME-DEPENDENT EQUATION
1. Two-layer scheme with weights. We are now interested in various
difference scliemes for the Schrodinger equation
/'^\ 01 ox-
w(x-, 0) = Wo(«)> w@, <) =u(l, <) = 0, i = yri.
By analogy with the heat conduction equation we employ the method of
separation of variables, in the framework of which a solution of this problem
is sought as the series
k = l

350
Difference Schemes with Constant Coefficients
where
Cfc = ("o. ^k) = / Ua{x) Xkix) dx ,
0
A^ = A;^ TT^, Xk{x) = v2 sin wkx.
Granted the grid a);^^ = ui,^ x CJr with uif^ = {x^ = s /i, s = 0,1,.. . , TV,
hN — 1} and uir = {tj = j t, j = 0,1,...}, the difference scheme with
weights
B) tyt = A{ay + {l-~ a)y) , y{x, 0) = u^ix) , j/^ = j/^ = 0 ,
where Ay = y^^. and a = a^ -\- i a_^ is a complex number, comes first.
Of our initial concern is the residual
l/) = A ( G M + A — d) W ) — i Wj
A
u + u
+ T\^(r-~ ^jAUf-Ut
Au+[a--]TAu-iu + Oi/'^)
{Lu - ill) + -- L^ u + {a - ^_) T Li + 0{h'^ + T^-)
^f+(^^-)r)L« + 0(/.^+r^),
where u = u(x,tj + ^r), w = du/dt and Lu = d'^u/dx'^. It follows frc
the foregoing that
0{h'- + T'-) if f7=i ,
^ = <( 0(ft^ + r2) if .= 1^^ = .,
0(/i2 + r) rf a- 7^ i o" 7^ o", .
The scheme so constructed is of increased accuracy for the parameters
1 ih'-
Ut '
1
2 '
12r

Shrodinger time—dependent equation 351
The metodology of the method of separation of variables guides the
choice of y(x, t):
k = i
where Xk{x^) refer to the eigenfunctions of the operator A, meaning, by
definition,
AX,+ A^X, = 0, fc = 1,2, ... , TV^l, X,@) = X,A) = 0,
4 irkh
Xu{x) = \rl sin T^kx , Aj^^ = — sin^ —- ,
On the strength of the preceding decomposition we establish the recurrence
relation for the coefficients
where
z-(l- <j)t\ ^ -A -o-o)rAfc+f(l + o-jrAfc)
and the useful relations for j/-'"'""'.'
k = l
k = l
Simple algebra gives
(l + ^^rA,)^ + (l^^o)^r^A^^ B^o ^ 1) r^A^
'''' ' A + ^, rX,r- + <^l r'-K A + ^: r\,y- + ^^l r'K '
implying that | g^ | < 1 and thereby justifying that this scheme is stable:
Wy'^'W < \\y'\\< ■■■ < 11/11 for ^o>|-
If (Tg < i, then I gj, I > 1 and the scheme becomes unstable, thus
causing some difficulties. In particular, the explicit scheme with cr = 0 is

352 Difference Schemes with Constant Coefficients
unstable for any r//i~ = const. In contrast to the approved schemes for the
heat conduction equation, there are no conditionally stable schemes among
ones with r//i^ = const : all the schemes with cr^ > ^ are stable, while
others turn out to be unstable.
We have found earlier for the scheme of accuracy 0{h'^ + r^) that
1 ih'-
The same is still valid for the scheme at hand. From what has been said
above it is clear that for cr^ = ^ this scheme is unconditionally stable.
The meaning of stability \\y^ || < M\\y° || with constant M > 1 is
that we should have
rnax I gfc I < 1 + Co r < e'^o '', Cq > 0 .
A similar estimate for | g^ | can be derived also for ctq < | by merely setting
r= Oih^) or
1 Co 4
"° = 2 " 7a^ ' "^ = /? •
Indeed, in the case of interest
< A ^ 2f7o) (tA') t<c,t
if
T <
(l-2f7o)A2 (l-2f7oI6
or, what amounts to the same,
r- ^ ' CqT 1 Co
J^ < -r, 7T^ ^—^nr^ < CqT tor o- = -^ .
This provides support for the view that the explicit scheme with cr = 0 is
stable:
\\y'\\ < e'^'niy^W
if r < CgA"^ = T^Cg/i"', where Cg > 0 is an arbitrary number. The
condition r < t^ Cg/i"* is very tough and unnatural. Here r//i^ (but not
r/h*) is a dimensionless quantity. Because of this, the explicit scheme is
unacceptable for the Schrodinger equation.

Shrodinger time—dependent equation 353
2. Three-layer schemes. Of special interest is the three-layer scheme with
weights
iyo = A {a y + {1 - 2 a) y + a y) ,
C)
J/o = J/iV = 0. y°ix) = Uoix) , y^{x) = Uoix) ,
where y = j/-'"'""', y = y-'~^, y = y-' and cr = (Tq is a real number. For any
a the scheme is of second-approximation order with V" = 0(h~ + r~). For
later use, we seek a particular solution in the form yl{Xg) = Cf. ql Xk{x^).
Substituting this expression into equation C), recalling the definition of
eigenfunctions AXk = —X/^Xk and omitting the subscript k of q/, and A^,
we eventually get the quadratic equation for q:
{i +2 ficr) q'^ - 2fiBa - 1) q + 2 fiq - i = 0 , f.i = t \ ,
with discriminant
-- = ^^ {2 a - if - 1 - 4 ^^- a^ = {I - 4a) 1^1^- - I .
Plain calculations show that
1 1
D < 0 for a>
^4 4r2A2
and give the roots of the quadratic equation
t + Z jj, a
Under the constraint
11 A 4
",^ 4 - iT^A^ ■ "^ = /^ '
the particular solutions yl do not increase with increasing j:
||,,J+i II < 11,,J II
\\yk II ^ 11 y/k 11 •
Being concerned with q^'^ = e^^^k^ we look for the general solution of the
problem in view as a sum
y^ = E (ttfc cos jip^ + 13f, sin jV^ ) Xk{x)
k-l

354 Difference Schemes with Constant Coefficients
with a/, and P/, arising from the initial conditions
y° = "o . J/' = Wo
0 y' - y°
Vt = —^ = «i .
/ -.^ N r, / / ^r ^ 2ai. sin^(<^j./2)
sm ip^
No wishing to load the book down with more a detailed derivation of
possible estimates we quote here only the final result
\\y'\\<M{\\y°\\ + \\y°\\) for a> ^ ^'
4 16r2 •
Careful explorations of this sort will appear in Section 6 in trying to
establish stability of schemes for the equation of vibrations of a string.
5.5 THE TRANSFER EQUATION
1. Explicit schemes for the Cauchy problem. The first-order equation
du du
-TT + a 7^
ot ox
= A
known as the transfer equation, describes, for instance, the behavior of
the density p = p{x,t) of incompressible liquid moving along the Ox-axis
with velocity v.
dp dp
dt dx
Here we treat it as a model one. However, the arguments about this
matter can result in the design of interesting experiments, whose aims and
scope are to test and improve admisssible schemes for rather complicated
equations of acoustics, kinematic integrocIiiTerential equations of neutron
transfer, nonlinear equations of gas dynamics, etc. Because of the
enormous range and variety of problems dealt with by mathematical physics,
the contents of this section would be of the methodological merit.

The transfer equation 355
The problem we must solve is the Cauchy problem
, du du
A) -X—h a — = 0 , - oo < x < oo , t>0, u[x,0) = Uq[x) ,
under the natural premise a = const ^ 0. The solution of problem A) is a
"travelling wave":
u{x, t) — Ug{x — at) ,
where a is the wave velocity and Wo(^) is a differentiable function,
For the purposes of the present section we introduce in the plane (x, t)
the grid
^h = { a-'i =ih, i= 0, ±1, ±2, ...} ,
^r = [ij = JT , i = 0, 1, 2, ..,}
with steps h and r in x and t, respectively, and begin by constructing one
of the well-known explicit schemes for the Cauchy problem
B) y^^'-y^ +a ^'>'~^'' =Q, j/; = «o(^.)>
or j/j -\- ay^ = 0. The pattern of this scheme consists of the three nodes
{x,, tj), (Xi_^j, t^) and (Xi, tj_^_,) (see Fig, 17a),
Obviously, scheme B) is of first-approximation order with respect to
T and h, since it,s residual behaves as follows;
ip = Uf+au^ = {u + a u') + ^tu+ \ah u" + 0{h^ + r^) = 0{t + h) ,
For a > 0 the sche^me at hand turns out to be absolutely unstable. Now
what we must do is to discover that some particular solution becomes
unstable albeit with obvious modifications of equation B):
C) ^/^' = -7^/.^+l + (l + 7)^/^ ^= T'
With this aim, we seek a particular solution to this equation in harmonic
form. This amounts to
D) yl^qU^'^ i = y^, ^7^0,

356
Difference Schemes with Constant Coefficients
(Xj-, Ij_|_i)
[Xj, Ij_|_j j
(^8 1 S'+l)
i-^'ii^jj [■'^'i + li^j) [•'^■i-li^j) [•''ii^j) [•'^i-lt^j) (■''ii^jj I'^'i + l
(Xj_ J , Ij_l_j j
(S^ijlj+l)
(•^i-l 1 '^j+1 j
( Xj _ J , I j j
(Xjjljij)
[•''ii^j )
Figure 17.
Having substituted D) into C), we obtain
q =; —7 e^'^+-j+l = l + (l — cos (p)-j — ij sin <^
and find, by simple algebra, that
|g|2= (i + (l-cos <^) 7)^+72 sin^ <^= 1+47G+1) sin2(<^/2).
Hence, | g | > 1 for any fixed 7, bounded from below as r ^ 00, if sin (p/2 ^
0. It is worth noting here that the case where sin <^/2 =; 0 corresponds to
the values j/^ = 1 = const , Then
Vk
00 as J
00 .

The transfer equation 357
Remark Keeping 7 = ar/h = 0(h), that is, r = 0{h~), we might have
I g I < 1 +Cg r, where Cg > 0 does not depend on r and h, and the harmonic
in question remains bounded:
\yl\< ef''^ ''i = e'o 'j < e=o ^ = M for Q<tj<T .
This means that
\yi\<M\yl\
w
ith the constraint ar/h' < Cj (in our example j/^ | = 1), where Cj =
const > 0 does not depend on h and r.
We may recommend one more explicit scheme on the pattern
consisting of the three nodes (x^,tj), (Xj-, <j_|_i) and (Xj_j,<j) (see Fig. 17b):
E) -'' + a = 0 , a > U ,
r h
or y-t + ay^ = 0. This scheme also generates an approximation of order 1,
since i/" = Uf + au^. = 0(/i-|-r). After scrutinising the available information
on the difference equation
d f
y/"^'= A-7I//+7y/_i > '^^~h^^'
we deduce for 7 < 1 that
||j/^'+Mlc<(i-T)l|y^'llc + 7l|j/^'llc = lh/llc.
meaning the stability of the scheme in the space C:
F) Ifj/^+Mlc < Ih/llc fo'- 0<7<1.
By analogy with scheme B) for a > 0, it is straightforward to verify that
scheme E) is unstable for a < 0, while scheme B) is stable under the
constraint | 7 | = | a | r /i"-' < 1, so that the inequality
lh/+Mlc < Wy'Wc
holds true for all j = 0, 1, 2,.

358 Difference Schemes with Constant Coefficients
2. Explicit schemes of a higher-order approximation. Of major importance
is the explicit scheme of accuracy 0(/i^ + r) having the form
j-\-i j j j
G) '^ Z^ + a l^i+iJui = 0
T 2 h
or j/j + aj/o = 0, where j/o . = (j/j_|_i — yi_i)/{2h). This scheme is
constructed on the four-point pattern (Fig. 17c) consisting of the four nodes
(«i, ^j+i)i (^i, ^j)i (^8-11 ^j) ^iid {x^J^^, tj). Obviously, scheme G) is
unstable for every fixed 7 =: arh'^ and arbitrary sign of the coefficient a.
Indeed, upon substituting harmonic D) into equation G) we get the
equation for q:
g-l+O ^ = 0, f=V-l, '^ ^ ~r ' q=l-iysm(p.
Whence it follows that | g P = 1 -|- 7^ sin" (p > 1, so that \y^ \ = \q\^ ^ 00
ELS j ^ CO. Adopting the arguments similar to those used for the scheme of
Section 1.10 and replacing y^ by the half-sum |(j/^_|_j -|-J//_i), we obtain
the stable scheme on the same pattern:
2h
for which
||j/^+Mlc<lhyilc<---<lh/llc
for any I7I < 1 and arbitrary sign of the coefficient a. More specifically,
for a < 0 we thus have 1-|-7=1— |7|>0 and 1—7=1-|-|7|>0.
In the estimation of the residual for scheme (8) it is possible to rewrite
it as
O.bh'-
Vt Vxx- + a J/s =0
r ^
in light of the trivial relations
Hyk+i + yk-i) = hiyk+i +yk-i -'iyk) + yk = Vk + \h''-y^^j..

The transfer equation 359
The residual on a solution u(x,t) is
0.5/i^ 1 .. 1 /i^ „ ^,,9 ,,
With the members recovered from the equation w = a^«", the preceding
becomes
which provides support for the view that scheme (8) generates a conditional
approximation, because it approximates the equation only if h~/t ^ 0 as
h ^ 0 and r ^ 0. In my view, two important things are that for r = 0(h)
we might achieve ^ = 0(t + h) and the second order of approximation
i/i = 0{h~) is attained for r = h/\ a |.
To make our exposition more complete, we involve the four-point
scheme of second-order accuracy
(9) Vt + ayo - \Ta''-y^^. = 0
on the same pattern as was done for scheme G) (see Fig. 17c). Plain
calculations of the residual with ii-\- au' = 0 and w = —aii' — a~ u"
incorporated give
i> = Uf + auo - T^ra u^^
= ti + i r« -F 0{t''-) + au' + 0{h'-) -~a'-Tu" + 0{h'- + r)
= (u + a u') + i r (M - a^ m") + 0{h'- + r^) = 0{h'- + r^) ,
so that ip = 0(/j2 -\- r^).
Let us investigate the stability of scheme (9) by the spectral method
having represented it beforehand in the form
27G- l)j/fc+i+ 2
yt" = A - 7') J// + ;^ 7 G - 1) yi+^ + ;^ 7 G + 1) yi-^ ■
Via transform yl = q^e'''^'^ we derive the useful expression for q:
g = 1 — 7" A — cos (f) — iy s'm (f , | g |- = 1 — -y- A — -y-) A — cos (f)',
by means of which we establish that | 7 | < 1 is a necessary stability
condition, whereas ly] > 1 is a sufficient unstability condition in the situations
when I7I is fixed and I7I = const with varying parameters r and h.

360 Difference Schemes with Constant Coefficients
3. The boundary-value problem. We are interested in learning more about
the boundary-value problem when the boundary value /.i(<) is specified at
the point x = 0 and a solution is sought for x > 0 and t > 0:
du du
-T—h a 7— = 0, t > 0 , 0<x<oo, a = const > 0 ,
A0) 9i ox
u{x, 0) = Wg(x-) , X >Q , w@, t) = ii{t) , t > 0 ,
under the natural premise Wo@) = /i@). In the case of the differentiable
members Uq(x) and ij,(t) the function
f ii-q{x — at) for t < x/a ,
u(x, t) = <
[ l.i(t — x/a) for t > x/a ,
is just the solution of the problem under consideration.
In working on the grid cOj^ = { x^ = ih, f = 0,1,2, ... } with spacing h
and the grid uj^ = {tj = jr, j = 0, 1,. .. } with spacing r we construct the
implicit scheme on the pattern depicted in Fig. 17d in the usual way:
7+1 7 j+1 j + 1
A1) '^ -y^ +a'^ -'^-^ = 0.
T h
Alternative forms of this scheme
% + a i/g = 0
and
7/^'+' - "^ v^+^ + -i— v^ .^ - ^ > 0
7+1 7+1 n
are more convenient for later use. With these, the computational procedure
may be carried out with the starting point fc = 1, j = 0. Then
1 1
7+1^ 7 + 1-' 7+1 7+1
With knowledge of y\ we are able to calculate the remaining values y^ up
to some 2 = j^. After that, we will find y^ for 0 < i < ig, etc. by setting
k = 2.
We now deal with the family of schemes on the four-point pattern (see
Fig. 17e)
J/t + o- i/s + A ~ 0-) J/f = 0 . « e w,, , < e w^ ,
A2)
j/(x, 0) = Wo(a;), j/@, ^■) = 0, a=l.

The transfer equation 361
Observe that scheme A1) belongs to this family, corresponds to the case
G=1 and has the residual i/" = "j + f Wj + A — cr) Wg. By inserting here
the asymptotic expansions
Mj = ii + 0(t'^) , Wj = w' - i /l U" + 0{h^-) ,
V = V + ^ TV + 0{t') , V = V — ^ TV + 0{t'^) ,
where v = v\^ , ,„ and v = w,-., we arrive at
V- = i r B 0- - 1) «' - I /i «" + 0{h'- + t'-)
= ^{2aT + h-T)u + 0{h'- + t'-).
which serves to motivate that the scheme with weights is of order 2: ip =
0[h'^ + r^) if G = ^ — ^ /i r""' = ""o- In all other cases (f 7^ cTq) it is of order
1, since i/i = 0{t + h).
In order to demonstrate that scheme A2) is stable with respect to
initial data for
1 h
" ^ 2 - 27 = "° '
we proceed as usual. This amounts to forming the grid w^ = {x^ — ih,
i = 0, 1, .. . , TV, hN = / } on the segment 0 < x < / and introducing the
inner product and associated norm by
N
(y.'^l := E J/i'^i^. II J/II ~ \/(y. y]-
8 = 1
Other ideas are connected with the well-established expressions
{,= _ (^ + ^)+-(,}_ ^)
and
and forthcoming substitution v = y^.. Upon receipt of the available
information we reduce the scheme concerned to

362 Difference Schemes with Constant Coefficients
The next step is to multiply the resulting equation by 2r j/^j = 2 (j/j — j/g)
and take into account that
2 Vt Vxt^^V Vg = {v'^)g + h (Vgf = (j/2)_ + /( (y_j2
with 1" = j/j. The outcome of this is
r (y2)^ + 2 (( ^ - 1) r + i ft) (j/,,)^ + (y^y- - (j/,)^ = 0 .
With the aid of the relations
N N
Yl{y1)s,ih = E [{ytfi -(j/t)?-J = {yt?N-ivt)]
i-l i=l
and j/j g = Vti^^t) = 0, since y{0,t) = 0, we get
A3) riy,)% + 2{i<r^l)r+'^h)\\y,,\\^- + \\y,\\^- = \\y,r-
as a final result of multiplying once again by h and summing over all grid
nodes x = i/i, i = 1, 2,. ., , TV. It follows from the foregoing that
\\yt'\\<\\y^\\< ■■■<\\yl\\
if (cr — i) r + i /i > 0, giving a > a^. This scheme is stable in the energy
norm \\y\\^^-^ = \\ys\\-
In Chapter 6 we will deal with the two-layer scheme of general form
A4) By, + Ay = 0,
where both operators A and B really act in a prescribed Euclidean space
H, A = A* > 0 and fl > 0. A necessary and sufficient stability condition
will be established in the form
A5) B > \tA.
Moreover, || y^+^ \\^ < \\ y" ||^ in the norm || y ||^ = ^/{Ay, y).
In studying problem A2) we refer to the space H of all grid functions
defined on the grid w^ and vanishing for i = 0 and the operator Ay = y^,
for which the estimate
(Ay, y) ~\yl, + \h\\yg |p > 0

The transfer equation 363
holds true due to the relation yy^^ = |(j/^)g + \h{yj;)'^. An alternative
form of scheme A2) may be useful in the further development:
A6) {E+aTA)y,+Ay^Q.
Since A > 0, there always exists the inverse A""^ > 0, by means of which
we recast A6) as
A7) (A-i + f7r£)j/,+ j/ = 0.
Comparison of A7) with A4) shows that B = A~^ + ctt E, A = E and
condition A5) signifies that
(a ^ + aTE)x,x] — ^t{x,x)
= {A ^x, x) +{a- -) T (x, x)>0,
yielding (Ay, y)+(a—^)T\\Ay\\'^ > 0 for the substitution A~^x = y and
providing the relation [Ay, yj = |j/^ + ^/j||Aj/|pin the case of interest.
The stability condition A5) taking now the form
A8) i4 + (i/. + (^-i)r)||Aj/|P>0
is valid for cr > cTq. This provides enough reason to conclude that the a
priori estimate \\y^ \\ < \\y°\\ is certainly true. Likewise, a scheme with
weights may be designed for the system
du dv dv du
which is equivalent to the equation of vibrations of a string
Tip ~ 5x2
and the stability conditions for such a scheme may be established in a
similar way as we did before.

364 Difference Schemes with Constant Coefficients
5.6 DIFFERENCE SCHEMES FOR THE EQUATION
OF VIBRATIONS OF A STRING
1. The statement of the difference problem and calculations of the
approximation error. In this section we study the equation of vibrations of a
string
In this view, it seems reasonable to pass to the dimensionless variables
X = Xj/l and t = at^/l, due to which the initial equation is representable
by
A) ^ = 3^ + /(^.0. 0<x-<l, Q<t<T.
At the initial moment the supplementary conditions are specified by
, / X / N du{x, 0) - r \
B) W(X, 0) - Wo(.'c) , j^ =Wg(;c),
where Uq{x) is the initial deviation and Uq[x) is the initial velocity. The
string's ends move in accordance with the known laws
C) u{Q,t)=n,{t), «A,0=A'2@-
By analogy with Section 1.2 we introduce in the domain D = [Q < x <
\,Q < t < T] the rectangular grid w^^. As equation A) contains the
second derivative in t, the number of layers cannot be any smaller than 3.
Retaining the preceding notations, we have
,■ . ,-+1 - i-i y-y y-y
y = y, y = y- , y = y' , yt = . %- = >
r r
A J/ = yg:, ,
yt - yt _ y- '2y + y _ Ut + Vt _ v- y
^" " r " r2 ' y\- 2 " 2 r •
As before, we replace the derivatives built into equation A) by the formulae
-Q^ ^ '"■it' 1^ ~ A « = «5.,. , / ~ <^ .

Difference schemes for the equation of vibrations of a string 365
A key role in subsequent discussions is played by a family of schemes with
weights
J/«=A(o-j/+(l-2o-)j/+f7J/)+<^, <^ = f{x, tA,
D)
%=/-'i(^). J//v = /^2@. y{x,Q) = u^{x), yj(x',0) = Wo(x),
where Uq{x) will be specified below.
The boundary conditions and the first initial condition u{x, 0) = Uq[x)
on the grid Co^^ are satisfied exactly. A choice of Wq(x') is stipulated by the
wishes that the approximation error u{x) — du(x,0)/dt = u{x) — Uq{x)
would be a quantity of order 0{t~). From the chain of the relations
u,{x, 0) = iiix, 0) + -Tu{x, 0) + 0{t'-)
= u,ix) +^-t{ u"{x, 0) + fix, 0)) + Oir'-)
= U,ix)+^-T{u';ix) + f(x,0))+OiT')
it is readily seen that u{x) — Uf{x, 0) = 0{t'^) if we accept
E) u,{x) = u,{x) + i r {u'^{x) + f(x, 0)) .
Thus, the difference problem D)-E) is completely posed. With regard to
y — y^'^^ we set up on the basis of D) the boundary-value problem
^ t' (J/4Y + J/tY ) - A + 2 tr 7^) J//+' = -f, , 0 < f < iV ,
r
F, = B J// - j//-^ ) + r^ A - 2 fi) Aj/J +GT'-Ky'-' + r^ ^ ,
which can be solved by the right elimination method, stable for any cr > 0
(see Chapter 1, Section 2.6).
The next step is to calculate the approximation error of D) in the case
if = f{x,tj) by investigating the difference between a solution y of problem
D)-E) and a solution M = w(a;,<) of problem (l)-C). Substituting j/= z + u
into D) yields
Zji = A ( (TZ + A — 2 cr) Z + GZ ) + l/) ,
F)
^0 - ^N = 0 1 •^(*' 0) = 0 1 ^ti^, 0) = ^(*) 1

366 Difference Schemes with Constant Coefficients
where i/" = Ai^cr u + A — 2 a) u + a ii) + (p — u^^ is the approximation
error of scheme D) on the solution u = u(x,t) and u = Uq{x) — Uf{x, 0)
is the approximation error for the second initial condition j/j = Uq(x). In
accordance with what has been said above, p = 0(t'^). The well-established
expressions u — u + t U-^ and u = u — t u^ allow us to deduce that
a u + (I — 2 a) u + a u = u + a t~ u^^ ,
implying that
ip = Au + aT~Auff + ip — Uf-t
G) = Lw + f7r'^Lw + /-«+0(/i2 + r2),
Moreover, i/" = 0(/i^ + t^) for an arbitrary constant a, which does not
depend on r and h.
Let a = a — h~ /A2 r~), where the constant a independent of h and
r is so chosen as to provide the stability of scheme D). As can readily be
observed, it suffices to take (t > \ {1 — £)~^, since scheme D) is stable for
a > 5A— £)~^ — 5 7~^, 7 = T/h, £ > 0. Scheme D) with the member
(8) ^ = /+^r
permits us to make the order of approximation more higher and achieve
The boundary conditions of the third kind
are approximated by the following difference equations:
Pi y« = A" ( o- {/ + A - 2 o-) y + G 2/) + <^" , i = 0,
/32J/« = A+ (o-y+(l-2f7)y + f7y) +<^+, i = N ,
whe
^ h/2 ' ^ h/2

Difference schemes for the equation of vibrations of a string 367
Here the approximation error for the boundary conditions is a quantity
0(/j2 + r2) if
a — const ,
The
<^^
forth(
/(^
;,0.
Pi = P'l
coming substitutions
G =
Pi =
1
12r2+'
hp,
3 '
/>. = !+ ^,
-^W =/^. W + y ( ^ -/'A> 0 -/?./A> O)
are best suited for the design of the scheme of accuracy 0{h'^ + r^)
approximating the initial equation at the nodes x = Q and x = 1 to 0{{h'^ + T'^)/h)
and the boundary conditions to 0{h'^ + r^).
2. Stability analysis. We now investigate the stability of scheme D) with
respect to initial data in the case of homogeneous boundary conditions and
zero right-hand side of the equation. A reasonable statement of the problem
is
y^, = K{Gy + {\-2G)y + Gy) = Ay^''\
Da)
J/o=J/iV=0. J/(«, 0) = Wo(«) . %(«, 0) = Wo(«)-
On the same grounds as before, we look for its solution by the method
of separation of variables, still using the framework of Section 1.4 where
particular solutions are sought as a product y(x,t) = X(x)T(t) ^ 0.
Substitution of y = X T into equation Da) gives
Putting these together with the boundary conditions j/^ = j/yy = 0 we set
up the eigenvalue problem for X{x):
AX-FAX:=0, x-ewft, X@) = XA) = 0, X{x)^(],

368 Difference Schemes with Constant Coefficients
whose solutions
are
X^ = — sin^ —— , X'^''\x) = a/2 sin wkx .
From (9) we get the difference equation of second order for Tk{t):
{Tk),^ + A, Ti"^ = 0
or the equation
A + tr r2 A,) t, -- 2 ( 1 + (tr - i) r2 A, ) Tfc + A + tr t2 A,) tfc = 0 ,
which can be rewritten as
A0) t,-2(l-a,)T,+t, = 0, a,
\ + aT'-\k
We may attempt a solution of the preceding equation in the form
Tk — Tk{tj) = ql. Thus the quadratic equation for q arises from A0):
g^ — 2A--Q;)g + 1 = 0 (the subscript k is omitted for a while). Careful
analysis of its roots gj j = 1 — a ± \/a^ — 2 a shows that for 0 < a < 2,
the values q^ ^ — 1 — a ± i \/a B — a) are complex with | gj ^1 = 1- It is
sensible to pass to a new variable (pf. for which
cos ipf,=l-ak, sin iff, = x/a^ B - a J ,
making it possible to get qi ' — e^fk and q^ — e~^^^, due to which the
general solution to equation A0) is representable by
Tk{t,) = Ck{q^^yy + Dk {qi'^y = Ak cos JV, + B,. sin jV, ,
where Ak and Bk stand for arbitrary constants.
After that, wejook for a solution of problem Da) as a sum of particular
solutions
N-l
A1) y' = J2 {^k cos jipk+Bk sin jipk)X(''\x).
k-l
Let Uq), and Wg^ be the appropriate coefficients in the relevant expansions
of Uq(x) and Uq{x):
A2) n,ix) = 'Y:uokX^'Hx), «„(x-) = 'e'«o.-Y(^)(x-).
k=l k=l

Difference schemes for the equation of vibrations of a string 369
By relating the initial conditions j/° = Wg and y° = [y^ — y°)/T = Uq{x)
to hold for sum A1) we establish the relationships for determining the
coefficients Ak and Bk:
Ak = UQf. , Ak h Bk = u^k .
T T
making it possible to find that
, 1 — cos (fif. T ^
13 Ak = Uok , Bk = : Uok + -. Uok ■
sm ipk sm ipk
Having substituted Ak and Bk into A1), we are led by minor changes to
n< V cos ^ipk sm (fik )
k-\
After the first stage the estimate of \\y^ || for scheme Da) will be
derived for G = 0 relating to the scheme
A5) ytt = -i'^y, % = J/iv = 0. J/(«,0) = tio(«). J/t(«>0) = Wo(«)-
In that case
1 2
Oft = i r
•^k - 2 ' ' -^k = ^ik , COS (fk = 1 - ^ik , sin ipk = yA^7B~^-7«fcy
When the steps of the grid UJ|^^ are related by
where £ > 0 is an arbitrary number, we find that
A7) ^k < j^^, k = l,2,...,N-l,
yielding
f^k
A8) cosi,, = ^l-^> ,^^^^
By virtue of the relations
sin <^fe __ 2 sin i<^jt __i.. ^ 2 sin i (^^ / T
2 sin i <^fe f
cos ^fk > \J
1+e

370
Difference Schemes with Constant Coefficients
and
2 sin 5 <^fc __ v^2(l-cos ipk) __ ^/^^^
the estimate occurs:
A9)
""^^ > JX, '
1+e
Observe that expansion A4) implies that
N-l
lll/^'ll <
^^litllk±u,,x('^
y
N-i
k-\
sm <^^
Using estimates A8) and A9) behind we finally establish
(«ofc)'
WW < V ^ ii«oii+| E
/k = l
A,
We note in passing that the expression
[KkY
E
k-\
A.
is nothing than one admissible norm on the space -ff^-i known as the
negative norm:
where Ay ~ —ky — —J/j^. in the space of all functions y defined on the
grid w^ and vanishing at the points s; = 0 and x = \. Indeed,
N-\
A-«„ = E«o.^-^^^') = E T^^^'^>
k-\
k-\
implying that
N-\
[A ^ Up , «o) = ^
k-\
Thus, we have proved that under condition A6) scheme A5) admits the
estimate
B0)
\W\\< /^(ll«oll + ll«olU-0

Difference schemes for the equation of vibrations of a string 371
This estimate remains valid for scheme Da) under the constraint
1 + £ h'^
B1) a >
4 At
2 '
where £ > 0 is an arbitrary number. To make sure of it, we should replace
everywhere in the above proof fi). by a^ = (| r^ A^) A + cr r^ A^)~^.
The superposition principle unveils its potential in investigating the
stability of scheme D) with respect to the right-hand side by considering
the problem
Db) J/,-, = A £/('')+ ^, y,=y^, = 0, y{x,0) = 0, j/,(x, 0) = 0,
whose solution is sought as a sum
B2) y^ = E tY^'^',
i'=o
where Y^'^ as a function of j solves for fixed j' the homogeneous equation
B3) Y^i'^' =A{aY^+''^'+il-2a)Y^'^'+aY^-''^')^, 0 < j'< j ,
supplied by the boundary conditions
B4) Y,''^'= Y^'^'^0
and the initial conditions
, , ., ., yi'+i.i'_ yi'.i' y^'+i.i' .,
B5) y^ '^ = 0 , Y/ -^ = = = $J ,
T T
where ^-^ is so chosen as to satisfy the nonhomogeneous equation Db).
A similar problem arises naturally for the function $■'. By the
definition of y^''^".
4= lY^+^-^+ZrYi'^\
7-i
Aj/(") =Gr Ay^'+^'^' + ^r A(yj'-^")(").
j' = 0

372
Difference Schemes with Constant Coeffieients
Putting these together with Db) we find that
B6) Y^+^-i -ar'- KY^+^'i = Tipi,
which allows us to derive the equation for $ = $•'= Y^'^^'^ t~^:
B7)
$-o-r-A<I' = <^,
$0 = $yv = 0 .
For the purposes of the present section, let us estimate a solution y^ of
problem Db) in terms of the right-hand side <^, provided the stability condition
B1) holds. Having stipulated this condition, estimate B0) is certainly true
for a solution of problem B3) and takes for now the form
y7.7
<
l+£
Yt'''
r
yi +1.7
By the triangle inequality we deduce from B2) that
7-1 „
\w\\ <
l+e
y^ y>'+''.'"
j'=o
A~'
Equation B7) is needed in obtaining a bound of
yi'+i.i'
, m wnic
hich
$ and (f should be expanded in the series with respect to the eigenfunctions
of the system {X (*')};
B8)
Af-l
k = l
N-1
k = L
Substituting B8) into B7) gives $fc = (^^ A -^ or^Aj,)-'-. With these
relations in view, we establish for a > 0
^-1 ^9
N-l
'^ii^= E? = E
9
k = l
. = 1 (^ + -r^-',r-h ,- ^k
N-l 9
meaning
y7 +1.7
< T \\V^'
U~i
Thus, if G > 0 and condition B1) is fulfilled, the estimate holds for
scheme Db):
7-1
\\y'\\ <
l+e
E
^ W'P' \\a~'

Difference schemes for the equation of vibrations of a string 373
By exactly the same reasoning as before we deduce for problem D) with
zero boundary conditions y^ = yj^ = 0 that
llj/^'ll < V^ (|Ij/°II + IIj/?IL- + EHI^'"IU-)
under condition B1) and the restriction a > 0. Intuition suggests and in
this case it does not deceive us that for a special choice j/' = j/(r) the
stability of scheme A5) under the constraint t < h may be proved in the
space ^2-
We learn from Section 5.1 that the difference scheme
B9) y^t = yL' j = i,2,...,
C0) / = «o- yt=< + -2^yL-
provides an approximation of 0(/i^ + r^) to equation (l)-B). Other ideas
are connected with expressions of the solution j/-' in terms of j/° =:; Wg and
Wg. Following established practice, we find that
N-i , „ ,
C1) y' = Y. ("ofc ^°^ ^^k + ^p^ ^i^ ^^k \x^^\
k=l ^ rk /
where Wg^ are Fourier coefficients of Uq(x), the sense of the quantities Wgj,
and (fj, being reserved. Squaring C1) and applying the estimate
-,2
2 -: COS jipf. u^f. sm jipf. < T —^ COS jipk + M sm jip,. ,
Wy^r < ll«olP +
we obtain
To evaluate the lower bound of the expression
r2 1
sinVi h{^-kT^K)'
we take 7 = r/h <'\. Along these lines, it is evident that
A.(l-^j=^sin^^^l-,^sin^ —
4 . , vkh / . , irkh \
4 o Trkh r, Trkh
= _ sm — cos —
4 . 2 irh 2 ""^ sm 7r/i

374 Difference Schemes with Constant Coefficients
In Chapter 2 it has been already shown that 4/i~^ sin~(|7r/ij ) > 8 if
/ij < |. Therefore, keeping /ij = 2h we find that
sin Trh 4 . ^ tt/i, „ , 1
sin^ -r^ > 8 , h<
/i2 h? 2 - ' - 4
Thus, if T < h and /i < 1/4 the proper estimate for the solution of problem
B9)"C0) is
\\yj ||2 < |U ||2_|_ ilU ||2
II y II :^ II "oil ^ s II 0II •
3. The energy inequality method. An investigation of difference schemes
for the string vibration equation may be carried out by means of the energy
inequality method (see Section 1). Here we restrict ourselves to stability
with respect to the initial data with regard to the problem
Vit = A{(Ty+il-2a)y+ay) ,
C2)
%=yiv = 0> y(x,0) = ug(x), y^(x,0) = ua(x).
Bearing in mind that a y + A — 2 a)y -\- ay = y -\- u t'^ y^^ and attempting
equation C2) in the form
C3) {E-aT^K)yj, = Ky
where E is the identity operator, we take the inner product of the resulting
equation C3) and the quantity j/o = (j/j + j/f)/2. The outcome of this is
C4) ((£-t7r2A)j/,-,,j/o) = (Aj/, j/o).
The trivial identities
1 1
(j/« , J/=) = 2 (II J/flH J . - (Aj/«. J/j) = (J/5« . J/j ° ] = 2 (I I yst]?)^
help rearrange the left-hand side of equality C4) as
C5) {{E-<TT'K)y,,.,yo)^\{\\y,f + aT'\\y,,][')^.
Further, we will show that for any function y = y{x^ t^) vanishing at
the points x — Q and x = 1
C6) -{Ay,yo) = l{\\y, + ik]\n,-Y^\\yi.]\'),-

Difference schemes for the equation of vibrations of a string 375
Indeed, from the first Green formula (see Chapter 2, Section 3) it follows
that
"(Ay, y=) = {v, vo ]
with V = y^. Having involved in the further development
we arrive at identity C6).
Substituting C5) and C6) into C4) yields the energy identity
C7) {\\yfr+{^-\)r'\\yf,]\' + k\\y, + y,]\')^ = o
or
£j+^ = £i ^ where
C8) £^ := IIj// \?+{a-\) r^ Wy^,]^ + \ Wyl+yft-
Let us find the values of cr, for which the quantity 8^ is nonnegative when
y^ and y^~^ are arbitrarily taken, by simple observations (see Chapter 2,
Section 3) that
4
ft
and
fe]P < T2\\y-i\?
\y,r
+ {--\)r'\\y,,]['>{\l{' + {<r-\)r^-)\\y,,_
With this in mind, we conclude that the right-hand side of C8) is
nonnegative if
('"> .'^i-??^ -'=1-
Here the expression (^■') = \\y^ \\^ can be viewed as a norm (or, more
exactly, as a seminorm), permitting us to write down
D0) ^^ = ||j/^||^ = ||j/ni' + (^-i)^'ll4]l' + illJ/| + J/i"]P-
Note that such combined norms depending on values of y on several layers
are typical for multilayer schemes. This is especially true for three-layer
schemes.

376 Difference Schemes with Constant Coefficients
The identity C7) implies the stability with respect to the initial data
in the norm D0): || y^+^ \l = || y" \l for all i = 0,1,....
So, condition C9) is sufficient for the stability of scheme C2) with
respect to the initial data in the norm D0). In particular, scheme C2) with
G = 0 is stable with respect to the initial data under the condition
D1) T < h.
Often this stability condition is named the Courant condition because
it has been proved for the first time by R. Courant, C. Friedrichs and G.
Levy in 1928.
For the equation in the dimensional variables
condition D1) takes the form r < h/a, where a is the sound velocity.
4. Determination of nonsmooth solutions by the difference method.
Numerous problems of mathematical physics describing shocl< processes in
gases, liquids and solids lead to the problem of determining nonsmooth
solutions of second-order hyperbolic equations, the simplest of which is
equation D2) of vibrations of a string. Since those solutions do not
possess the second-order derivatives involved in the equation, the words "a
solution satisfies the equation" should be understood in some generalized
sense. One of the possible definitions of generalized solutions is due to the
fact that the differential equation follows from an integral conservation law
if continuous derivatives emerged in equation D2) exist. In this case the
generalized solution is meant as a function u(x,t) having in the domain
G = {0 < X < 1, 0 < t < T} the bounded piecewise continuous derivatives
du/dx, du/dt and satisfying the integral equation
du , ■■) du
c
where C is an arbitrary closed curve in the domain G. If the first derivatives
are discontinuous, then for the characteristics x i. at = const the jump
conditions should be satisired:
du
Ik
= ±a
du
dx
[.f]:=.f(e + 0)-/(e-0) as C = x±at

Difference schemes for the equation of vibrations of a string 377
In giving a generalized solution of the problem
—^dx = a-^ ——0 , G<x<l, t>0,
D3) ^' ^'
du
u{x,Q) — Uq{x), ^(x', 0) = Wj(x), w@,<) = 0, u{l,t)—Q,
we rely on the scheme with weights
D4) J/,-, = A(a-y + (l-2a-)j/ + a-j/), Ky^a^y^^,
with the appropriate supplementary conditions. When studying the
convergence of the scheme with weights, it is supposed that a solution of problem
D3) exists and is smooth enough. It is possible under certain conditions
relating to the smoothness of the initial data.
Does the same scheme converge if m = m(x', t) is a generalized solution?
The grid solution of problem D4) turns out to converge to the generalized
solution with the rate 0{\/\/h-\- r). We do not dwell on confirming this
statement.
In trying to find a generalized solution of problem D3) one can come
across oscillations of the grid solution and its derivatives ("ripple"), which
essentially reduce the accuracy of a scheme. What is more, the lines of
discontinuity of derivatives spread over several grid intervals, thus causing
the difficulties in determination of proper discontinuity propagation
velocity. This is the result of introducing the fictitious friction (dissipation) in
the difference approximation.
The ripple is stipulated by the fact that difference harmonics reveal
the dispersion, that is, determination of a harmonic velocity depends on
its number, whereas for the differential equation all harmonics have the
same velocity a. In order to improve the quality of a scheme, one needs
to minimize the dispersion. Among various schemes D4) with weights the
scheme relating to~
1 / 1 \ UT
has the minimal dispersion and allows us to overcome the obstacles
mentioned above. It is of order 4, that is, 0{t'^ -\- h'^) on sufficiently smooth
solutions u = u(Xjt). On nonsiiiooth generalized solutions the
approximation error for scheme D4) with the weight cr = cr* is as large as for schemes
with any weight cr ^ cr^. However, due to reduced dispersion the scheme

378 Difference Schemes with Constant Coefficients
with the weight cr = cr* is more precise and reproduces much more better
the characteristic features of generalized solutions.
A stability condition established for scheme D4) such as
1
u > - -
- 4
1
_
4^2
= i('-
1
'7'
is certainly true for the scheme with the weight cr = cr* if 7 < 1 or r < /i/a,
that is, under the same condition as for the explicit scheme. In this context.
Let us stress here, that in an attempt to relax the "ripple" by
introducing the viscosity, the distortion of the solution profile and accuracy losses
occurred.
5.7 SELECTED PROBLEMS
1. For the heat conduction equation the difference scheme is suggested:
t/' . = - ('i/+'-. + t/' .) i= 1,2,... ,7V-1,
Prove its absolute stability, find the order of approximation and point out
the method for solving the problem.
2. Find the order of approximation for the difference scheme
2 y^xx,i ' ^xx\i J 10
i=l,2,...,7V-l, j/^=j^=0, y° = u,{x,),
prove its absolute stability and investigate the stability of the elimination
method being used for determination of
iJ + \ i= 1,2,... ,7V-1.

Selected problems 379
3. The statement of the problem is
du d'^u
Ik
du
dx
U (x, 0) =: Uq{x) .
- , 0 < X < 1 , t>0,
g^iO,t) = ^,it), uil,t)^^,it),
Construct a perfect difference scheme of accuracy 0( r^ + /i^).
4. Find stability conditions for the difference scheme
^yi,j= ^j/^+■ + A - ^)J/L., - 9(K^' + A -/^)J^) •
i = l,2,...,7V-l, y'^^y:=0, y° = u,ix.),
a = a^ + ia^ , /i = /ip + i //j ,
approximating the equation
. du d'^u
t -KT = jr^ -qu, q = const > 0 ,
w@, ^) = w(l, ^) = 0,
u{x, 0) = Ug(x).
5. Determine the exact solutions of the boundary-value problem
du du
1^+1^ = 0, 0<x<X, 0<t<T,
dt dx
u{x, 0) = Uq(x) , u@, t) — u^(t),
and of the difference scheme approximating this problem:
yt,^+yl^'^0, i = l,2,..., n= 0,1,2,...,
x^ — ih , t„ = nr.
Proceeding from the explicit representation of solutions, show that if the
condition y = r/h = const is fulfilled as r ^ 0, /i ^ 0, then convergence
occurs only for 7 < 1 and the difference problem solution coincides with
the differential problem solution for 7 = 1.

380 Difference Schemes with Constant Coefficients
6. The problem
du d'^u d'^u
dt dxf dxl
w(x,0) = 0, X e G = { 0 < Xi </i, 0 < ^2 <'2}.
w(x,<) = 0, X e 5G, <>0,
is approximated by the explicit difference scheme on the grid
^ft T = ^ft ^ ^T, ^T = {^„ ^ nT, r > 0} ,
i= 1,2,... ,7Vi-l, i= 1,2,... ,7V2-1, h,Ni = I,, h^Ni = k-
What maximal step r in time should be taken to provide stability of the
scheme when I, = 1, /j = 10, TVj = 10, N^ = 100?
7. Prove for the problem
du du
-7:-- T-, t>0, -00 < x < 0 ,
ot ox
u (x, 0) = Ug(x), ^ < 0 ,
w@,0 = 0, <>0,
that the difference scheme
l/j^' - l/j ^ yj -Vi-i
T h
is absolutely unstable and the scheme
ui^' - y'i ^ i4+i - yj
r h
IS stable under the condition t < h .
8. Show that for any r and /i a pure implicit difference scheme (a forward
difference scheme) approximating the problem
du d'^u
it(x-,0) = Uq{x) ,
du , , 9w ,
IS not asymptotically stable.

Selected problems 381
9. For the three-layer scheme
Vii = ^1 vlt' + A - ^1 - ^2) J/L + ^2 y7~'
a) investigate the approximation error, b) find a stability condition, c)
impose the initial conditions y°, y^'-, i = 0,1, . .. , N, not changing the order
of approximation for this proposal.
10. What differential equation is approximated by the difference equation
il+2j)y'^+'=il-2y)yr'+2j{y^^^+yl^),
where 7 = r//i^. Find the order of approximation for this.

stability Theory of
Difference Schemes
In this chapter we study the stability with respect to the initial data and
the right-hand side of two-layer and three-layer difference schemes that
are treated as operator-difference schemes with operators in Hilbert space.
Necessary and sufficient stability conditions are discovered and then the
corresponding a priori estimates are obtained through such an analysis by
means of the energy inequality method. A regularization method for the
further development of various difference schemes of a desired quality (in
accuracy and economy) in the class of stability schemes is well-established.
Numerous concrete schemes for equations of parabolic and hyperbolic types
are available as possible applications, bring out the indisputable merit of
these methods and unveil their potential.
6.1 OPERATOR-DIFFERENCE SCHEMES
1. Introduction. In Section 4 of Chapter 2 the boundary-value problems
for the differential equations Lu = —f{x) have been treated as the operator
equations Au = /, where A is a linear operator in a Banach space B.
In the study of nonstationary processes described by partial
differential equations of parabolic and hyperbolic types
du d u
-^ = Lu + f(x, t), — = Lu + f{x, t), 0 <t <to,
383

384 Stability Theory of Difference Schemes
the variable t (time) plays a key role and, following established practice,
should be marked out throughout the entire chapter. Here L is a differential
operator acting on u(x,t) as a function of a point x = (xj, x^, ... , x ) in
some p-dimensional domain G. For any fixed t, the function u(x,t) is an
element of the Banach space B. Therefore, instead of u(x,t) we obtain an
abstract function u(t) of the variable t, 0 < t < t^, with the values in the
space B; meaning u(t) £ B for all t G [0, t^]. The operator L acting on
u(x,t) as a function of the variable x is replaced by an operator A acting
in the space B. The operator A generally acts from a space Bi into a
space B'z (its domain 'V{A) C Bi is everywhere dense in Bi, while its range
TZ{A) C B2)- In this regard, we take for granted that Bi = B2 = B, making
it possible to set up the abstract Cauchy problem
du
dt
+ AU = f{t), 0<t<tg, W@) = Wg
where Wg is a given element from the domain 'D{A).
The above reasoning is of a heuristic nature and is aimed at carrying
out some analogy between the methods of the general theory of differential
equations and those of the theory of difference schemes, the framework and
methodology of which are outlined in this chapter.
A Cauchy problem is said to be stable with respect to the initial
data and right-hand side if
t
\\u{t)\\<M,\\u,\\ + M2 J WmWdt',
0
where Mi = const > 0 and M2 = const > 0.
In conformity with the superposition principle (A is a linear operator),
the stability of the Cauchy problem with respect to the right-hand side
follows from the uniform stability with respect to the initial data
II "(Oil < A^i|l«(^')ll. t>t' >0,
where u(t) is a solution of the homogeneous equation.
2. Operator-difference schemes. We now consider a linear system Bh
depending on a parameter /i as a vector of some normed space equipped with
the norm \h\. With regard to the linear system Bh, it is reasonable to
introduce a collection of norms II • 11^, || • |L -,, II • Ho )> •••, thus causing
linear normed spaces Bh , Bh , Bh , ■ ■ ■ ■ For the sake of simplicity, we

Operator—difference schemes 385
involve in further development the norms || ■ |L s, (( ■ (Lj ■,, ■•• on the
space Bh, assuming || ■ ||, to be a basic one on Bh-
At the next stage we introduce on the interval 0 < t < t^ & r-step grid
^T = {tj = n, i = 0,1,... ,io, T = tjjo}, iOr = [tj = n, o < i < jj
and pass to abstract Z?ft-valued functions y)J, fhr' '^^^- °^ °^'^ discrete
argument t = jr (^ cOj, so that yf^J £ Bh for all i = jr £ w^. Let Aj^^,
Bf^J, Cfj^j, ... be linear operators dependent on the parameters h and r
and acting from Bh into Bh for every t ^ ujj. For the moment, omitting the
subscripts h and r we can write down y„ = y{nT) = y{t„) = y, A(t), B{t)
and C{t). This should cause no confusion.
We call a family of difference equations of order r — 1
r-l
Boitn)yn+i = Xl*^»(^")^"+i-'' + ^'"' n = r-2,r-l,r, ... ,
5 = 1
depending on the parameters h and r with operator coefficients B^, C^, . . . ,
Cr-i, which are linear operators acting in the space Bh and dependent on
h and r, an r-layer operator-difference scheme or simply r-layer scheme.
If the inverse operator B~^ exists, a solution j/„+i of this problem can
be expressed in terms of the initial vectors J/q > ?/i > • • ■ ^^^'^^ '^^^ right-hand
side /. As usual, we assuriie all the vectors to be given and consider only
two-layer and three-layer schemes
(f) B^y„^^ +B^y„ = Tip„, n = 0, 1,..., % given,
B) B^y„^^ +B,y„ +B^y„__i = Tip„, % and y^ given,
3. The canonical form of two-layer schemes. Any two-layer scheme A) can
be written in the form
C) B(tJ y^+'-y- +A(t„)y„ =y>„,
n = 0, 1, . .. , Vo^Bh given.
Indeed, by comparing A) with C) we see that B — Bq and A = (-Sg -|-
flj )/r. Within more compact notations
y = yn= y{in). ii = yn+i = yi^n+i) = yi^n + ^).
y-y y-y
j/ = j/„-i. % = —^ . yi=—^'

386 Stability Theory of Difference Schemes
equation C) can be recast as
D) Byt + Ay = ip{t) , t = t„ = nT e co^ , ?/@) = y^ ^ Bh .
We call both equations C) and D) the canonical form of two-layer
schemes. Equation D) is similar to the differential equation
„ du . „, ^
Example 1 For the heat conduction equation
du ^ . ^ d f du
at aX \ aX
we made in Chapter 5 the design of the two-layer scheme with weights
yf = K{ay+{\- o-)y) +ip , Av = {a{x) v^)^^ .
With the aid of the the identity
y — y
y = y + T = y+ry^
T
we reduce it to
yt- (TTAyf- Ay = ip .
Comparison of this equation with equation D) gives
B = E+(ttA, A = ~A.
This form of writing reveals what is meant by the canonical form of a
weighted two-layer scheme.
At the next stage we proceed to solve equation D) with respect to
y = J/n+i- K' the inverse operator B"^ exists, one can write down
E) y=Sy + Tip, S=E-tB"^A, ip=B~^ip.
The operator S is called the transition operator (from one layer to
another). In addition to the canonical form D), alternative forms of
writing will appear in the sequel for two-layer schemes; By = C y + t ip ot
-S j/„_|_i = C y„ + T(p^j where C = E — t A, E being the identity operator.
In the case B = E, scheme C) is called an explicit two-layer scheme:
T
permitting us to determine the value y„_^_i on the upper layer by the formula
yn + i — yn ~ ''' ^yn'^''' fn- ^^ B ^ E, then scheme C) is called an implicit
two-layer scheme.

Operator-difference schemes 387
4. The canonical form of three-layer schemes. We may attempt the three-
layer scheme B) in the canonical form
F) B y^^'^y--^ + j;(y^^^^-2y„+y„,i) + Ay„ = y.
By comparing F) with B) we see that such a writing is always possible if
we agree to consider
Also, it will be sensible to introduce the notations
ii-y ,,, _ Vt-Vt _ y-'^y + ii
2r '
and regard the equation
J/= = ^^ , Vu
(8)
Byo +T'Ry^^+Ay = ip{t) .
0 <t= nreco^ , y{0) = y^ , j/(r) = y^ ,
to the canonical form of a three-layer scheme together with F).
Example 2 We now turn to the weighted three-layer scheme
(9) yo + A(^aiy+ {l- a^- a2)y + o-^yj = if
and try to reduce it to the canonical form. With this aim, we make use of
the formulae
, y-y , y-'2y + y , t"^
y = y+ —^—\- 2 ~ ^^ + ^ ^t + Y y^'''
y-y,y-'2y + y r^
y = y ^ + 2 = y-'^y- + Y ^^^ •
2
ytt
Upon substituting these expressions into (9) we write the weighted scheme
in the canonical form (8), where
A0) B = E + t{(t,-(T2) A, R = ^{(T,+(T2)A.

388 Stability Theory of Difference Schemes
The passage from F) or (8) to B) leads to the equation for determining
the value j/„_|_i on the upper layer t = t„_^_l^.
(fl + 2r72)j/„+i = 2rB72-A)j/„ + (fl-2r72)j/„_i+2r^„,
thereby clarifying that problem (8) is solvable if the inverse operator ( -S +
2t7?) exists. Moreover, the value y„,i can be expressed in terms of
the values j/„ and j/„_i on the preceding two layers. Therefore, two initial
vectors j/g and j/j (or j/g = j/@) and y^ — J/t@)) are required to be given
in such matters. If flg = fl + 2 r 7? = £ is the identity operator, then the
three-layer scheme (8) is said to be explicit; we thus have for it
J/„+i = 2rB72-A)j/„ + (fl-2r72)j/„_i+2r^„.
Ill the case when B -\- 2t R ^^ E, scheme (8) is said to be implicit.
Sometimes it may be useful to have at our disposal together with the
canonical form (8) the three-layer scheme in the form B) or
A1) By.+{E + T^R)y-,,+Ay r.<p.
This equation is obtained from equation (8) by the formal substitution of
E/t'^ -F R for R.
5. The notion of stability. The notion of stability for three-layer schemes
is of our initial concern. By a two-layer scheme we mean a set of operator-
difference equations D) depending on the parameters h and r. We pre-
assumed here that the operators A and B are given on the entire space
Bh-
As a matter of fact, we will consider the set of solutions {y}ij{t)}
of Cauchy problem D) dependent on the input data {ip-i^^{t)] and { j/gft}-
Scheme D) is said to be well-posed if for all sufficiently small t < t^ and
\h\<K
A) a solution of problem D) exists and is unique for any initial data j/g^ G
Bh and right-hand sides ^p}ij{t) G Bh for all t ^w^;
B) there are positive constants Mj and Mg independent of h and r and
disregarding to the choice of j/g^j G Bh and ip^^{t) G Bh such that for any
t ^ Cbj for a solution of problem D) the estimate holds;
A2) II y^,{t + r) 11^^^^ < M, \\ j/g, ||^^o^ + M, ^max^ || ^^,{t') \\^^^^ ,
where || • |L s, || • |Lo-, and || • \\,r^ . are suitable norms on the space Bh-

Operator-difference schemes 389
Inequality A2) expresses the property of continuous dependence which
is uniform in h and r of the Cauchy problem D) upon the input data. Here
and below the meaning of this property is stability. A difference scheme
is said to be absolutely stable if it is stable for any r and h (not only for
all sufficiently small ones). It is fairly common to distinguish the notion
of stability with respect to the initial data and that with respect to the
right-hand side. Scheme D) is said to be stable with respect to the
initial data if a solution to the homogeneous equation
Da) By,+Ay = 0, t = nT>0, 1/@) = j/o .
satisfies the estimate
A2a) II l/ftr(^ +011A,) < A^il|yoJI(iOj.
Scheme D) is said to be stable with respect to the right-hand side if
for a solution to equation D) with the zero initial conditon j/@) = 0
Db) By, + Ay=^, 1/@) = 0,
the inequality holds:
A2b) ||l/ftr(i + OII(u) < M, ^max^ || ^,,(t') ||B,) ■
For later use, we approve for a solution y of problem D) the decomposition
y = y(^> + y^''^>, where yC is a solution of problem Da) and y^'"^' is a solution
of problem Db). On the strength of the triangular inequality
||l/;.r(^ + OII(U) < l|l/it'(^ + OII(u) + lll/ir'(^ + OII(U)
the combination of estimates A2a) and A2b) gives estimate A2). In a
similar way it makes sense to introduce the notion of stability for three-
layer schemes. However, in this case we are to consider the pair of vectors
Yn + i = {y„, Vn+i } with the norm of the special type
A3) llYn + i II = ||l/n +l/„ + l IL., + WVn + l " Vn L... .
U'»J y'-h) y'-h I
where || ■ |L.-, and || • |L»»-, are suitable norms on the space Bh. Various
norms of the type A3) appear, time and again, in the stability analysis of
three-layer schemes by means of the energy inequality method (for more
detail see Chapter 5, Section 6).

390 Stability Theory of Difference Schemes
Thus, the three-layer scheme (8) is said to be stable if for any initial
data j/o, j/j and for any right-hand side if{t) its solution satisfies the estimate
A4) \\Ynr{t + r)\\^^^^ < M, || y,,(r)||^^„^ + M, max^ II ^..@11B,) ■
where Mj and M^ are positive constants independent of h and r and
disregarding to the choice of j/q, j/j and <f{t).
The main problem under consideration amounts to the following one.
Suppose that equation D) is uniquely solvable with respect to y^^^ for
any j/„ and <f{t). What are conditions on the operators A and B for a
scheme to be stable in the sense of the above definition? In other words, we
wish to establish sufficient stability conditions for scheme D) and obtain
a priori estimates of the form A2), Moreover, sufficient conditions should
be convenient for practical verifications in the case of concrete difference
schemes associated with equations of mathematical physics.
In what follows stability of differential schemes will be given special
investigation irrelevant to approximation and convergence.
6. Sufficient stability conditions for two-layer schemes in linear normed
spaces. We now raise the question concerning sufficient stability
conditions for two-layer schemes in linear normed spaces. In full details these
investigations will be carried out in Section 2 for the case when Bh = Hh
is a real Hilbert space.
In what follows the Cauchy problem A) is supposed to be solvable,
that is, the inverse operator B~^ exists. Therefore, scheme D) can be
written in the form
A5) j/„ + i = Snyn + Tfr,, f„ = B'^ <p„ , Ti = 0, 1, . .. , yo ^ Bh ,
where
A6) - Sn = E^tB-^ An
is the transition operator. The operator ,?„ depends on t„ = nr, h and r,
however, neither for Sn nor for 5„, yl„, j/„, ipn and y^ the dependence on
h and r is explicitly indicated.
By virtue of the recurrence relation A5) we find that
n
A7) yn + l = Tn + l^oyo+Y^T-Tn + lJ + i fj,
J=0

Operator—difference schemes 391
where
-'n + ljj — ^n ^n — l • • • '-^j+i ^j j
Tn + l, 0 = Sn Sn-l ■ ■ ■ Si Sq , Tn-\-l^n + l = E .
The operator Tn + ij is called the transitiozi operator from the layer j to
the layer n-\-\, while the operator Tn+i, o refers to the resolving operator.
The triangular inequality yields
n
A8) ||l/„+il|(i) < ||T„+i,n|| ■\\yo\,-^+Y.j\\Tn+i,ni\\ ■ 11/, 11A).
where || • |L, is any suitable norm on the space Bh, making it possible to
arrive at the following assertion.
Theorem 1 For the stability of scheme A5) it is sufficient that for any
A9) l|7;^,,|| < M,.
Moreover, Aavin^ stipulated this condition, the solution of the difference
problem C) satisfies the a priori estimate
B0) II Vn + i 11A) < M, (^11 y, ll(^) + E HI B-' ^, \,) )
for all 0 < n < Tig.
Note that estimate B0) implies A2) with constant Mj = Mj t^ and
II f] 11B) = II ^i^V] 11A) incorporated.
Theorem 2 For the stability of scheme C) it is sufficient that for the norm
of its transition operator S'- the estimate
B1) ||S'^||<l + c„r
is valid for ail j = 0, 1, .. . , rig — 1, where c^ > 0 js a constant independent
of T and h. Moreover, under condition B1) a priori estimate B0) holds
with constant Mi = exp {cgtg}.

392 Stability Theory of Difference Schemes
To prove this assertion, it suffices to verify that estimate B1) implies
estimate A9):
11 ^ )J II "" II ^n—l ^n — 2 ' ' ' ^j-\-l ^j II
<||,S„_i|| •||S;._2||---||,5- + i|| .||,S-||<(l + c„r)"-^-
< A + Co r)" < A + Co r)"° < e''°"°'' = e'°*° =: Mi ,
Quite often, the reader can encounter the statement that "stability
with respect to the initial data implies stability with respect to the right-
hand side". How is one to understand the nature of this assertion?
We say that scheme D) is uniformly stable with respect to the
initial data if the Cauchy problem is stable:
B2) y„^i = Sny„,n = j,j +1,... , j/^-, j = 0,1,... , n given,
for any j = 0, 1,. .. , Uq — 1, that is,
B3) lll/n 11A) ^^■^lill/y 11A) for an 0 < i < n < Uo ,
where Mj is a constant independent of r and h both.
If the condition of uniform stability is satisfied, then estimate A9)
holds true for the resolving operator T„ a. Therefore, Theorem 1 asserts
that estimate B0) is valid for a solution of problem D). This type of
situation is covered by the following results.
Theorem 3 If scheme D) is uniformly stable with respect to the initial
data, then it is also stable with respect to the right-hand side under the
condition of the norm concordance
B4) 11^11B) = ||5-Vll(i).
Moreover, condition B4) assures us of the validity of a priori estimate B0).
It is worth noting here that condition B1) is sufficient for uniform
stability with respect to the initial data.
The object of investigation is the two-layer scheme with constant
operators A and B, not depending on t„ = nr:
yn+i= Sy„+Tf„, /„ = 5-Vn. n = 0,l,...,
B5)
5"=: E-tB-^A.

Operator-difference schemes 393
where y^ is given. If a scheme with a constant transition operator 5* is stable
with respect to the initial data, then it is uniformly stable with respect to
the initial data, since
B6) T„,, =r„_,,o = ,S"-^
Theorem 4 The stability with respect to the initial data of scheme B5)
with constant operators is necessary and sufficient for the stability with
respect to the right-hand side, provided condition B4) of the norm of
concordance holds. Moreover, in that case a priori estimate B0) is valid.
Sufficiency. The stability with respect to the initial data means the
boundedness of the resolving operator
B7) ||7;,o||<Mi.
In accordance with what has been said above, this implies condition A9)
and so it remains to use Theorem 1.
Necessity. Let scheme B5) be stable with respect to the right-hand
side, meaning the validity of the inequality for the solution of problem Db)
B8) lll/n + ill(i)<Mi^r||5-V,||(i), n = 0,l,....
3 = 0
This estimate holds true for any right-hand side /'■ = B~ ^if-. This provides
enough reason to conclude from A7) that
B9) y„ + i = Y^TTn + ij+ifj.
i=o
The assertion will be proved if we succeed in showing that property
B7) is true. Choosing r/- = E-q we deduce from B9) that
Vn + l = Tn + l^lf = Tnflf
and
lll/n + lll(l)<l|7;.,0|| • ||/||(i).
On the other hand, B8) yields ||j/„ + i|L-| 1^ ^i ll/llfiv Comparison of
these inequalities gives B7), from which the stability with respect to the
initial data immediately follows.

394 Stability Theory of Difference Schemes
Summarizing, we draw some conclusions which will be needed in the
sequel.
1. When the transition operator happens to be constant, the stability
analysis with respect to the initial data is mostly based on estimates of the
norms of the transition operator.
2. The condition of concordance between the norms of the right-hand
side and the solution such as
11^11B) = l|5~Vll(i)
is a severe constraint. If || B~ |[ < Cj, where Cj > 0 is a constant
independent of h and r, then || f \\,^s < c^\\if |L , and instead of B0) we obtain the
estimate
t
C0) \\y{t + r)||(^^ < Ml ||j/@)||(^) + M, J^ ^ Mi%,), ^2 = M, c, .
t'-O
In Section 2 a priori estimates will be obtained for which condition B4) of
the norm concordance is not required.
3. Scheme D) is stable if || Sj || < 1 + c^t for all j = 0,1,... , n^ — 1.
In practical applications of this sufficient stability criterion one needs to
reveal some properties of the operators A and B ensuring condition B1).
Such conditions are established in Section 2 of the present chapter. They
asquire the form of linear operator inequalities for the operators A and B
acting in the Hilbert space H„ = Bf^.
7, Approximation and convergence. The notions of approximation,
convergence and accuracy for operator-difference schemes are introduced by
analogy with the corresponding notions for the operator schemes A^y^ = ip^
arising earlier in Chapter 2, Section 4. Only a few editorial changes will
appear in this matter. For instance, together with the norms || • |L , and
II • ||,2 N it will be sensible to introduce the additional norms
ll^''r(^n)ll(U.)=0<*S,^r<*„ Il^'-(^"')IIA,0'
which complement further stability analysis.
Thus, let Bh be a linear space with norms || • |L , and || • |L^ , on it.
' " " "(ifcj " "BfcJ
We denote by B^^ and Bj^ , respectively, the resulting normed spaces and
assume in the sequel that
y,,{tjEBi'\ fuAtn) ^ B^u^

Operator-difference schemes 395
for all t„ = nr and that the operators of scheme D) carry out B/^ into
B'
B).
An,Bn : 4^^ ^ fif
Let Bq and Bq be normed spaces with norms || • jL , and || • \\,^ ,, u{t) be
an abstract Bq ^-valued function of the argument t E [0,T] and let f(t) be
an abstract Bq -valued function of the argument t E [0,T]. Furthermore,
we refer to linear operators Vj^°' , a = 1,2, projecting Bq"'' onto Bj^ :
u,=vi'KEBi'^ if uEBi'l
A=pf^/e4'^ if /eBp.
We taliie for granted the condition of the norm concordance
,l™Jl^i"^IU) = ll^^"'lU)' -here ^(") G B^ , a = 1, 2 .
Let y}i^{tj) be a solution of problem D) and u[t) be a continuous
function of the argument t, so that
In such a setting the error z^^ = j/j^^ — m]^ needs investigation. We say that
scheme D) converges on an abstract function u{t) E Bq li
lim max ||vL"m/|L s=0.
Scheme D) converges with the rate Od/il™ + t^) or is of accuracy
0(\h\"^ + r*-') on an abstract function u{t) E Bq' ii
max ||j;^;_„,^|| <M(|M'" + r'=),
where M = const > 0 is independent of h and r both.
An a priori characteristic of a scheme is the error of approximation.
The approximation error on a function u[t) for scheme D) is liinown as the
residual
u
i + i
i'L = Bur " ^ " + Mr K " Wr. Hr = i^i^ ' ^^ ^)

396 Stability Theory of Difference Schemes
Scheme D) provides an approximation on a function u(t) if
C1) max ||'0^|L, . —^0 as \h\^0, r ^ 0.
Scheme D) provides an approximation of 0[\h\"^ + t^j on a function
u{t) E B^^^ if
^maxJ|^^||(,^^<M(|/.r+r*),
where M = const > 0 is independent of h and r both.
Along these Imes, we may set up the problem for the error z^ = y^ —uj^;
C2) B— ZlL+Az^ ='<P\ i = 0,l,..., z° = y„~ul.
If scheme D) is stable, then for a solution of problem C2) the estimate
holds:
Ik^'ll(U) < A^i l|j/o " <II(U) + ^2 ^max J|^^'||(,^j ,
thereby justifying the next assertions.
Scheme D) converges on
erates
an approximation
approximates the element
WVohT " "ft (U)
a function u[t) if
on u[t)
«@);
+ 0 as
and the
\h\^0,
it is stable,
initial val
r^O.
geii-
Je Vo
Scheme D) is of accuracy 0{\h\"^ + r*"') on a function u[t)
if it is stable, provides an approximation of 0A/11™ +r*'') on
u[t) and
l|yo"<ll(i,) = 0(|ftr + r'=)
In particular, u(t) may be a solution of a certain differential equation.
In that case we say that the difference scheme approximates the difference
equation, provided condition C1) holds, etc.
We note in passing that one is to understand the statement "if a
scheme is stable and provides an approximation, then it is convergent"
given in Chapter 2, Section 2 as follows: both the difference equation and
the initial value generate an approximation (if we accept j/^ = V^^ "@),
then ||j/oft, " M;,@)||^^^j = 0).

Classes of stable two-layer schemes 397
6.2 CLASSES OF STABLE TWO LAYER SCHEMES
1. The problem statement. We pursue the stability analysis of two-layer
schemes by having recourse to their canonical form
A) But + Ay = ^p{t), t^nT^Lo^, y{0) = y„ .
Let Bj^ = Hj^ be a finite-dimensional real space, ( , ) be an inner
product and || a; || = y^B', x) be the associated norm in the space Hj^. The
operators A and B of scheme A) depend, in general, on h, r and t. Here
and below, we agree not to indicate explicitly the dependence on t.
The main goal of our studies is to find out sufficient conditions for
the stability of scheme A) and obtain a priori estimates for a solution
of problem A) expressing the stability of this scheme with respect to the
right-hand side and the initial data. In preparation for this, a solution of
problem A) can be written as a sum y = y + y, where y is a solution to the
homogeneous equation with the initial condition j/@) = j/@) = y^:
But+Ay = 0, t=eojr< 1/@) = %.
and y IS a solution to the nonhomogeneous equation with the zero initial
condition;
Bu^+Ay = ^{t), t=(^Lo,, 1/@) = 0.
The meaning of the estimate
B) lll/(i + r)ll(i)<Mi||i/@)||(^,
of a solution of problem (la) is that scheme A) is stable with respect to
the initial data, while the estimate
C) ||l/(t + r)||(^^<M2^ma|J|^(OI|B)
of a solution of problem (lb) expresses the stability of scheme A) with
respect to the right-hand side.
We will also use an alternative definition of stability of a scheme with
respect to the right-hand side:
D) m + r)||(^^ < M, ^max (||^(t')llB) + II^K^')IIB-))
where
^,it')={^it')~^it'~T))/T.

398 Stability Theory of Difference Schemes
When the triangle inequality
is put together with estimates B), C) and D), we derive the a priori
estimates
E) Mt + r)||(i, < Ml ||j/J|(^, + M2 ^max^ lk(i')llB)
and
F) m + r)ll(i) < Ml IIj/JI^^j + M2 ^max (||^(t')llB) + ll^*(OIIB-)) •
The forthcoming energy norms will be taken as the norm [| • |L, in
addition to the approved basic norm \\y\\ = \/{y, y)'-
G) \\y\\, = ^{Ay,y) for A = A* > 0,
(8) \\y\\^ = ^{By,y) for B = B* >0.
Scheme A) is said to be stable in the space -ff^ (or in Hb) if estimate
E) is- valid in the norm || • ||^^j = || • ||^ (or [[ • ||i = || • ||^).
2. A primary family of schemes. We will discover stability in a certain
primary family of difference schemes. Before going further, we regard
operators A and B to be bounded linear operators defined on the entire space
-ffft, 'D{A) ~ V{B) = Hh- In what follows the difference problem A) is
presupposed to be solvable for any input data y^ and f{t), that is, there
exists a bounded operator 5~^ with the domain V[B~^) = Hh- For the
sake of simplicity, we take for granted in the detailed account below that
A) the operators A and B are independent oft, that is, are constant
operators;
B) B is a positive operator: 5 > 0;
C) A is a self-adjoint positive operator: A ~ A* > Q.
Conditions (l)-C) in combination with the solvability requirement
single out a family of admissible schemes known as a primary family from
the set of all possible schemes A). Observe that condition 1) can be
weakened in a number of different ways. Sometimes we will deal with operators
A and 5, which are dependent on t, that is, withA = A[t) and B = B[t).
In the weighted scheme for the heat conduction equation in Example
1 of Section 3 the operators A and B are specified by A = —A and B = E +

Classes of stable two-layer schemes 399
arA. Smce A < || A||^and^ > A/|| A ||, we have 5 > A/|| A ||+crr)A > 0
under the restriction a > —l/(r||A||). The operator A = A* > 0 is
independent oft, so that conditions A)~C) are satisfied and the scheme
concerned belongs to the primary family for a > —h^/^'ic^T), where Cj =
max a[x).
x£0Jh
3. The energy identity. We study the problem of stability of scheme A) by
the method of the energy inequalities involving as the necessary
manipulations the inner product of both sides of equation A) with 2rj/j = 2(j/ — y):
(9) 2T{By^ , y) + 2r(Aj/, y^) = 2r(^ , y^) .
Using the formula
, y ~\- y y "^ y ^ / \ t
A0) ^~"~2 2" "^ 2 ^^"^^^ "" 2 ^*'
we rewrite (9) in the form
A1) 2r(E^0.5rA)j/,,j/,) + (A(y + y),y^j/) =2r(^,j,,).
Lemma 1 Let A be a self-adjoint operator, then
A2) {A{y+y),y^y) = {Ay,y) - {Ay,y) .
Indeed, the chain of the relations
{Ay,y) = {y,Ay) = {Ay,y)
occurs due to the fact that A is self-adjoint operator and, therefore,
{A{y + y),y~-y) = {Ay, y) + {Ay, y) - {Ay, y) - {Ay, y)
= {Ay,y) " {Ay,y).
Substituting A2) into A1) we obtain the energy identity for scheme A):
A3) 2r(E ~ 0.5rA)j/, , y,) + {Ay, y) = {Ay, y) + 2r(^ ,y^) .
4. Stability with respect to the initial data in Ha- Stability of scheme A)
with respect to the initial data is investigated with further estimation of a
solution of problem (la).

400 Stability Theory of Difference Schemes
Theorem 1 The condition
A4) B>^A
is necessary and sufficient for the stability in Ha of scheme A) from the
primary family with respect to the initial data with constant Mj = 1, that
IS, condition A4) is necessary and sufEcient for the validity of the estimate
A5) lll/nlU<lll/olU, n=l,2,...,
wAere y„ is a solution of problem (la).
Sufficiency. Granted condition A4), the energy identity for problem
(la) (with <^ = 0)
A6) 2r(E -0.5rA)y, ,j/,) + {Ay,y) = {Ay,y)
implies the inequality {Ay, y) < {Ay, y) or || y |R < || J/1|^ , yielding
WVu + iWa < l|j/nlU < ••■ < WVoWa-
Necessity. Suppose that scheme (la) is stable and estimate A5) is
satisfied. We are going to show that this leads to the operator inequality
A4), that is,
A7) {Bv,v) >0.5t{Av,v) for any v^H.
We begin by placing identity A6) on the first layer (n = 0):
2t{{B - 0.5rA)j/,@) ,j/,@)) + {Ay, , y,) = {Ay, ,y,) .
By virtue of A5) this identity can be satisfied only for
2t{{B - 0:5rA)j/,@), %@)) = {Ay, , j/J - {Ay, ,y,)>0,
meaning
(E-0.5rA)j/,@),%@))>0.
If J/o G -ff is an arbitrary element, then so is the element v = j/j@) =
—B~^Ayg G H. Indeed, taking any element v = j/j@) ^ H we find y^ =
—A~^Bv G H for A~^ does exist. Thus, the inequality is met for any
V = j/j@) G H, that is, the operator inequality A4) takes place, thereby
completing the proof.

Classes of stable two-layer schemes
401
Remark 1 Condition A4) is sufficient for stability in the sense of A5) of
scheme A) if _8 = B{t) > 0 is a non-self-adjoint variable operator.
Remark 2 The fact that condition A4) is, in a certain sense, natural can
be clarified a little bit by considering the simplest example relating to the
difference scheme
Vn + l " Vn
aj/„=0, n = 0,l.
yo = "o
where a and b are positive numbers corresponding to the differential
equation
du , ,
fe----|-aM = 0, t>0, u@) = Mo ,
The difference equation yields
r a^
1 - -r- I l/„
l/n + l I <
The stability requirement
ll/n + l I < l%l
is obviously satisfied if jl — ra/fe| <lor"l<l— rajh < 1, that is, for
b > 7:Ta. The similarity with the operator equation B > T:Ta is clear at
the first glance.
Example We now consider tlie weighted scheme
y, + A{ay+{l^a)ij)=0,
whose use permits us to illustrate the effectiveness of the stability condition
A4). For this, we _write down this scheme in canonical form accepted in
Section 1;
A8)
{E+aTA)y^ + Ay = 0, B = E + a t A .
If A = A* > 0 is independent of t and a > —1/G'|| A ||), then the weighted
scheme belongs to the primary family (see Section 2). The necessary and
sufficient stability condition A4) is of the form
5-0.5rA = E + {a-0.5)TA> 0.

402 Stability Theory of Difference Schemes
With the relations A < || A \\E and E > A/\\ A || in view, we obtain
B -0.5tA> (l/ll a II + (tj-0.5) r) a.
The preceding inequality implies that condition A4) is equivalent to the
constraint
This condition is necessary and sufficient for the stability of the weighted
scheme, which interests us.
In the case of a model heat conduction scheme
yj + A (cry + A - tj) J/) = 0 , Ay = y^^, 0 < x = ih < I, hN - I,
J/@.^n) = 0. l/(l.'^n) = 0. l/(a-'>0) =: Mo(a;), a; = i/i e [0, 1],
corresponding to the first boundary-value problem for the heat conduction
equation
du d'^u
~di ^ ~d^ ' 0<a;<l, t>0,
M@,t) = i<(l,t) = 0, t>0, u{x,Q)=^Uo{x),
we deduce that (see Chapter 5, Section 1)
A = - A , Ay = Ay for y e fi ^ = i/^ ,
/i2 2 - /22 ' » 2 4 r cos2 ^
2
,2,
The condition cr > a^, a^ = /i /Dr) > a^, has been found in Chapter
5 by the method of separation of variables.
Suppose now the operator A > 0 not to be self-adjoint. Then scheme
A8) does not belong to the primary family. However, it can be replaced by
an equivalent scheme from the primary family. Since A > 0, there exists an
inverse operator A~^ > 0, whose use with regard to equation A8) permits
us to confine ourselves to
By^+Ayz^O, B = A-^ + aTE, A = E.

Classes of stable two-layer schemes 403
The operator A — E = A* > 0 is independent of t and the operator B is
positive for tj > 0. Stability condition A4) in the space H^ = // takes for
now the form B - \tA = A"^ + (tj - \)tE > 0, valid with a '^ ^^. The
condition tj > ^ is sufficient for the occurrence of the relation
II l/„ II < Ill/oil for G > 0.5, A^A*, A>0.
5. Stability with respect to the initial data in Hb- Let us write down the
second energy identity for scheme (la) assuming B also to be a self-adjoint
operator: 5 = 5* > 0. At the first stage we take the inner product of (la)
and 2rj/:
A9) 2rEj/,,y) + 2r(Aj/,y) = 0.
Relying on the formulae
y = 7^{y + y) + T^Vt ^ y = 7^{y + y) - r^yt
and making use of Lemma 1, we find that
2T{By^,y)= {B {ij - y),y + y) + t^B y^,y^)
= \\y\\l-\\y\\l + r^\\yt\\l.
'2TiAy,y)= ^ {A{y + y - T y.t),y + y + T y^)
= ^lll/ + l/||^-Ylll/JI^
Substitution of these expressions into A3) leads to
B0) \\y\\l+rm\y,\\l-0-5r\\y,\\l)+0.5r\\y + y\\l = \\y\\l.
Theorem 2 Let the operators A and B involved in scheme A) be
independent oft,A* = A>0 and B* = B > 0. Condition A4) is sufficient
for scheme A) to be stable with respect to the initial data in the space Hb
with constant Mj = 1,
Indeed, let B > |tA, then
lll/JI«~0.5r||j,JP =(E~0,5rA)j,,,yJ>0

404 Stability Theory of Difference Schemes
and relation B0) yields ||j/||g < ||j/||g, meaning
\\yit)\\B < \\yio)h-
Remark If the operators A and B are commuting, then condition A4) is
necessary and sufficient for the stability of scheme (la) in the space Ho-
WynWo < WyoWo
where D = D* > 0 is any operator commuting with A and B. This is
acceptable if we involve, for instance, D = E, D = A'^ oi D = B'^ for
B = B* ,so that II j/„ II < II j/o II, A|| j/„ || < A|| y„ ||, 5|| y„ \\ < B\\ y„ ||, etc.
6. Estimates of the norm of the transition operator. Stability
considerations are connected witli the use of a new method based on the estimation
of the norm for the operator of transition from one layer to another. This
method actually falls within the category of energy methods.
We may attempt the difference scheme (la) in the form
B1) y = Sy, S = E~tB~^A,
where 5" is the transition operator. Let D = D* be an arbitrary constant
operator in the space H. With these, it is plain to show that
B2) ll^llz; = lh5y|L<||S'||^ • llyjl^,
where || 5* jj?, is adopted as the smallest constant M subject to the inequality
{DSy,Sy)<M{Dy,y).
As can readily be observed from B2), scheme B1) is stable in the space Ho,
that is, II y„ ||^ < || j/q ||^ if the norm of the transition operator does not
exceed unity: || 5" ||^ < 1. This condition is equivalent to being nonnegative
of the functional (M = 1)
JobA ^ {Dy,y) ~ {DSy,Sy) >Q .
Of special interest is the case D = A, for which || 5" || . < 1 for 5 >
\:tA. Indeed, upon substituting the expression for 5* into the functional J^i
we obtain
JA[y] = {Ay,y)-{ASy,Sy)
= (Ay, y)-{A{E-TB-' A) y, (E ~ tB''A) y)
= 2T{Ay,B-^ Ay) -/■'{AB-^ Ay,B-^ Ay).

Classes of stable two-layer schemes 405
The outcome of further substitutions x = B~^Ay or y = A'~^Bx is the
same:
B3) Ja [y] =Ja[x] = 2t [{B x,x)~^t {Ax, x)] .
This Implies that the conditions IIS*!!^ < 1 and B > \tA are equivalent,
providing support for the view that Theorem 1 holds.
Since A = A* > 0, there exists a square root A^l"^ = (A^/^)* > 0, by
means of which the expression for J/i [y] can be recast as
JA[y]^{A"^y.A'l^y)~{A^I^Sy,A"'Sy)
^\\A'lM?~\\A'"Syf
= \\A^r'yf~\\{E-rC)A'r'y\\\
where C = A^f^B-^A^f-. Substitution u = A^l'^y gives
JA[y] = \\uf~-\\{E~TC)uf.
This implies that the condition
B4) II i? ~rC II <1
is equivalent to the relation B > \tA. No assumption is made here that B
is a self-adjoint operator.
It seems worthwhile giving the special case when
5 = 5* > 0, A = A* >0,
for which the equivalence of the conditions
B5) J'b[i/]>0 and 5 > 0,5 rA
is certainly true. Since B = B* > 0, there exists 5^/" = (S^/^)* > 0 and
this property is valid for the operator
C = B^'^A-^ B^'^, C = C*>0.
By applying successively the substitutions B~^''^Ay = x, C^l'^x ~ u, C =
B^l"^A'^B^I''^ and B~^''^u = v we arrive at the chain of the relations
JB[y] = 2T{Ay,y) ~ T^ {Ay, B-' Ay)
= 2t{B^'^x,A-'B''''')~t^{x,x)
— 2t{Cx,x)—t [x,x) ~ 2 t{u,u) — t {C~ u,u)
= 2t{u,u) ~ t\B-^I'^ AB-^l'^u,u)
= 2T(^{Bv,v)~^{Av,v)y

406 Stability Theory of Difference Schemes
Thus, the functional JbIv] is transformed into the same form B3) as was
done before for J/i[j/]:
JB[y] = 2T(^{Bv,v)~^{Av,v)) ,
where
V = 5-1/2 ^1/2 5-1/2 ^ ^ _ y^A'^ 5^/2 ^,-1/2 ^1/2 ^
This implies the equivalence of conditions B5), which assures us of the
validity of Theorem 2.
Therefore, the method of estimation of the transition operator norm
permits us to prove that condition A4) is necessary and sufficient for the
stability of scheme A) with respect to the initial data in the space H/i (for
B ^ B*) and in the space Hb (for B = B* > 0) with constant Mi - I.
7. The method of separation of variables. If the operators A and B are
self-adjoint
B6) A = A*>0, B = B* >0,
then the stability of scheme (la) in the spaces H^i and Hb for
B > LA
can be proved by means of the method of separation of variables following
established practice (for more detail see Chapter 5),
Let A^ be the dimension of a finite-dimensional space H, A^. be
eigenvalues and <^j. be orthonormal eigenfunctions of the problem (see Chapter
1, Section 1 and Chapter 2, Section 1)
B7) A4 = A,54, k=l,2,...,N,
where, in addition, E,^j,,,^„) = Ej,,„ (S).,. = 1 and S,.„^ = 0 for k 7^ m). All
the eigenvalues Aj. of problem B7) are positive, because A > 0.
In such a setting a solution of problem (la) is sought in the form
B8) yit)=j:ckit)Ck-
By virtue of the relations
k=\ k=\

Classes of stable two-layer schemes 407
we find upon substituting B8) into (la) that
' c,{t + t) - c,{t) ,,„,,,,„, „Q
i:{^^^^^i^P^+A,e,@}5 4
k = l
Putting this together with the orthogonality of the system {S.^} we arrive
at
T
N N
J/(i + r) = I] c,(t + r) 4 = ^ A - r A,) c,{t) ^ .
k=l k=l
The norm || y{t + T) ||^ = (Aj/(t + r), y{t + T)) can be most readily evaluated
by observing that
Ill/Will = (E c.(t)A,54,E c.(tL) = E ^clit).
^k=l *=! ^ k=l
All this enables us to deduce that
\\yit + r)\\l= E A,c2(t + r)< max (l-rA,J £ X,cl{t)
yielding
\y{t + r)\\^<^umx^\l~TX,\\\y{t)\\^.
Whence it follows that
B9) \\y{t + T)\\^<\\y{t)\\^<\\y{0)\\^
if
\1-tX^\<1, k=l,2,...N.
This condition provides support for the view that ~1 < 1 ~ tX,, < 1 or
C0) 0<Aj<2/r, k=l,2,...N.

408 Stability Theory of Difference Schemes
Let us show that inequalities A4) and C0) are equivalent. To this
end, we operate with the expression
N
5j/~0.5rAj/ = ^c,.(t)E4--0.5rA4)
k = l
N
^c,(t)(l-0.5rA,M4,
k = l
by means of which it is plain to calculate the functional
N
{By,y) - 0.5 r(Aj/, J/) = ^ c2(t) A - 0.5 r A,).
Because of this form, the equivalence of A4) and C0) is obvious.
We thus have shown that under conditions B6) inequality A4) is
sufficient for the stability of scheme (la) in the space H/i, that is, relation
B9) occurs. Let us stress that the requirement of self-adjointness of the
operator B is necessary here, while the energy method demands only the
positivity of B and no more.
One can prove in a similar way the stability of scheme (la) in the
space Hg, provided condition C0) holds.
8. The p-stability condition. A more general definition of stability with
respect to the initial data became rather urgent and extremely important.
Let D = D* > 0 be a constant operator. Scheme A) is said to be
p-stable with respect to the initial data if for a solution of problem (la) the
inequality holds for any y^ £ H:
C1) \\yn\\u<p''\\yo\\D^
where p = exp{cor}, c^ is a constant independent of ft, r and irrelevant
to the choice of j/q.- If scheme (la) is p-stable in the space -ffu, then it is
stable in the same space Ho '■
Ill/„llz5 <Mi||%||^, t,,:^nT<i,
with constants Mi = exp {c^io} for c^ > 0 and Mi = 1 for c^ < 0.
The two-layer scheme (la) with constant operators A and B can be
reduced to the explicit scheme
C2) ^!i±iZ:^ + Cx„ = 0 or x^^,={E~tC)-x^,

Classes of stable two-layer schemes 409
provided that the forthcoming substitutions are carried out before going
further:
1) a;„ = B^l'^y^ for 5 = 5* > 0, where C = Ci = B-^l'^AB-^I'^\
2) a;„ = Ai/2j/„ for A = A* > 0, where C = C2 = A^I'^B-^A^l"^.
Definitions 1) and 2) together imply that
l|a;„|| = lll/nllB for C = Ci , x„=B^/'-'y„,
ll^nll = iil/nlU forC = C2, x„ = Ay'y^.
The condition of p-stability of the implicit scheme (la) in the space Hjj for
the choices D — B and D = A is equivalent to the condition of p-stability
of the explicit scheme C2) in the space H: || a;„ || < />"|| x^ ||, n = 1, 2,. ...
Lemma 2 Let scheme C2) be given with a constant operator C. The
condition of p-stabiiity of this scheme is equivalent to the boundedness of
the norm of the transition operator
\\S\\ = \\E-tC\\<p.
Indeed, we have Xj =; Sx^ and \\x^\\ < \\ S \\ ■ \\xg\\ for n = 1. By
comparing this inequality with C1) for n = 1 we finish the proof of Lemma
2.
Lemma 3 If A = A* > 0 and B = B* > 0, then the inequalities
C3) 7i B < Aj2 < B liE <Cf^< E
are equivalent for the operators C = B'^l'^AB-^I'^ and C = A^l"^B'^A^l"^.
Proof Let C ~ B'^'"^AB'^'"^ and 7 be an arbitrarily taken number. The
difference
{Cx,x) - 7 {x,xY= {B-^l'^AB-^l'^x,x) - 7 {x,x) = {Ay, j/) - 7 {By, y)
with regard to y = B~^''^x shows that the signs of the operators C — fE
and A — 7S coincide. No property of positiveness of the operator A is
required here.
Let now C = A^/^S"^A^/^. First, we are going to establish the
equivalence of the inequalities C > j E {(J < j E) and E > fC'^ {E <
fC"^). Upon substituting y for C^'^x we obtain
{Cx,x)~^{x,x) = {C^I''x,C^I''x) -'i{x,x) = {y,y)~i{C-^y,y),

410 Stability Theory of Difference Schemes
thereby justifying coinciding of the signs of the operators C — j E and
E ~ jC'^. We now insert A"^/^5A~^/^ m place of the inverse C~^ and
accept V = A~^''^y. The outcome of this is
{Cx,x) - 7 (a;, x) = (j/, y) - 7 {A-^l''BA-^l''y, y) = {Av, v) - 7 {Bv, v) ,
meaning that the operators C — jE and A — jB possess the same signs, By
merely setting 7 = 7i and 7 = 72 we draw the conclusion that inequalities
C3) are equivalent.
Lemma 4 If the operator C = C* > 0 and r > 0, then the conditions
C4) ||S'|| = ||i;~rC||<p,
C5) Ijlf i, < C < 1±^ i?
are equivalent.
Indeed, since the operator S = E — tC ~ S* is self-adjoint,
||S'||= sup \{Sx,x)\= sup \(y{E ~ tC)x,x)\,
11^11 = 1 ll-^ll = i
so that
~\\S\\E<S<\\S\\E
~PE <S<pE
~pE < E ~tC < pE ,
or
and
we deduce that
^.ji1e<c<'-±J^e.
In view of this, condition C4) implies condition C5). The converse
implication is simple to follow.
Theorem 3 Let A and B be positive operators and A = A*, B = B* > 0.
Then the conditions
C6) IJIZ 5 < A < i±^ 5

Classes of stable two—layer schemes 411
are necessary and sufficient for the p-stability in the space Hb of scheme
(la);
\\yn\\B<p"\\yo\\B-
When, in addition, A is a positive operator, these are necessary and
sufficient for the p-stability in the space Hj\:
IIj/JUSp'MIi/oIU.
In order to prove this theorem, we beforehand reduce the implicit
scheme (la) to the explicit scheme C2) with the operator C = B~^''^AB~^''^
(or C = A^I'^B-^A^I'^ for A > 0) and then collect the results of Lemmas
3, 2 and 4.
Remark One fails to prove Theorem 3 on account of the energy identity
B0). However, the method developed in Section 6 may be of assistance in
achieving this aim. In the case D = B conditions C6) are equivalent to the
condition \\ E — tB~^A ||g < p or to being nonnegative of the functional
Jb [y] = p' {By, y) - [B {E -rB'^A) y,{B-T B~'A) y) .
Suppose now that i7 is a finite-dimensional space of the dimension A^.
Substituting y = X]i: = i '^k^k^ where <^j, is an eigenelement of problem B7),
into the expression for the fuctional Jb[j/] yields
N
I-/' ^ ^ ^ l + Z'
JB[y] = Y.'lip^-^^-^^k?)>^ foi-^<Afc<
k = \
which is equivalent to condition C6),
9. Stability with respect to the right-hand side. Recall that in Section 1 we
have established a Theorem 3. This is a way of saying that the stability in
the norm || • |L, with respect to the initial data implies the stability with
respect to the right-hand side taken in the norm || 'p\\,2) — II -^"''"'i^ llnv ^^
immediate implication of this is covered by the followmg assertion.
Theorem 4 If condition A4) is satisfied, then scheme A) from the primary
family of schemes is stable with respect to the right-hand side and for a
solution of problem A) the a. priori estimate holds:
n
lli/n+ilL<llj/olU + EHI^"VfclU-

412 Stability Theory of Difference Schemes
If, in addition, the operator B is self-adjoint, then
n
II Vn + l \\b < II Vo Wb + YI'^ II "Pk llfi-i •
What is more, a priori estimates C) and D) are valid with the
members ||<^||B) = ll<^IU-i and ||<^||B) = Ikll-
Theorem 5 Under the condition B > \tA, scheme A) from the primary
family of schemes is stable with respect to the right-hand side and for a
solution of problem A) the a priori estimate holds:
n
C7) II Vn + l lU < II % \\a + II ^0 lU- + E HI ^k lU- •
k = l
Proof In preparation for this, a solution of problem A) can be arranged
as a sum
C8) j/„ = t;„ + «;„ ,
where w^ is a solution of the equation (of the so-called "stationary"
problem)
C9) Aw;„=<^„_i, n=l,2,..,, «;« = Wj.
Upon substituting C8) and C9) into equation A) a new problem arises for
D0) Bvt+Av=^^, v^-y^~w^,
where <^„ = —[B — TA)Wf „ and (p^ = 0. In the estimation of v Theorem 4
gives
n
D1) ll^n + llU<ll^olU+EHI^"'^HU-
k = 0
A simple observation that
«;, = A-Vf, ||A'/^5-Vll = ll(S-^C)A-'/Vf||

Classes of stable two-layer schemes 413
with C = A''^5"'A^'^ enables us to evaluate the summand
in inequality D1). Stability condition A4) has been used in Section 2 to
obtain through such an analysis estimate B4)
\\E~tC\\<1,
implying that
||A'/2S-V||<||A-'/Vf||<lkflU —
where the norm ||<|C'f||^-i = J{A'^ipf^ipj). Consequently, a solution of
problem D0) satisfies the estimate
n
k = \
which in combination with the inequality
ll'^olU < llJ/olU + lh^olU = llj/olU + lkolU-i
gives
fc = l
Finally, by appeal to the triangular inequality from C8) we obtain C7).
Thus, Theorem 5 is completely proved.
Theorem 6 If the conditions
D2) ■ B > ^-^ tA, B = B*
are satisfied, then for scheme A) from the primary family of schemes the a
priori estimate holds:
1 J- "
D3) lll/n + llll<l|j/oll^ + ^EHI^*llB-"
*: = !
where e > 0 is a constant independent of h and r both.

414 Stability Theory of Difference Schemes
Proof The energy identity A3) is involved at the first stage, Estimation of
its right-hand ,side 2r((|£3, j/j) is stipulated by successive use of the generalized
Cauchy-Bunyakovskil inequality and the e-inequality
2r (^, Vt) < 2r II ^11^-, • || ?/, Ij^ < Sre^ || j/, ||| + ^ || ^|||_, ,
Upon substituting this estimate into A3) we obtain
2r ((A - £j S - f A) j/„ J/,) + {Ay, y) < {Ay, j,) + ^ || ^ |^_, .
If condition D2) is satisfied, the number e^ may be chosen so that 1/A ~
£j) = 1 -|- £, that is, £j = e/{\ -\-e). Under such an approach we arrive at
(l~£jS-^A = (l~£j(S-i±^rA)>0,
(Aj/^+i,l/fc+i) < (Aj/j,,j/fc) + ^—r||<^J||_, .
Summing up the latter inequality over A; = 0, 1, 2, . . . , leads to estimate
D3).
What are the conditions under which the stability in the norm
II f \\B^ ~ II 'i^ II I'f^veals itself? The following assertion answers this question.
Theorem 7 Let the condition
D4) B > eE + O.brA
be satisfied, where s is a positive number, and scheme A) belongs to the
primary family of schemes. Then for a solution of problem A) the a priori
estimate holds:
II l|2
^ll<^ill •
1 "
D5) lll/n + llll<l|l/oil^ + ^E
Proof With regard to identity A3) the Cauchy-Bunyakovskii inequality
and the e-inequaJity together give
2r(^,j/,)<2r||^|| ■ || j/J| < 2r £ || j/, |p + ^ || ^ Ip,

Classes of stable two-layer schemes 415
Furthermore, we substitute this estimate into A3). Having stipulated
condition D4), it follows from the foregoing that
or
Summing up then over j = 0,1,, ., , n leads to D5). Thus, Theorem 7 is
completely proved.
Remark 1 Theorem 7 remains valid in the case of a variable operator
B ~ B{t) and Theorem 2 continues to hold for a variable operator A = A[t).
The reader is invited to verify these facts on his/iier own with the aid of
the proofs of the aforementioned theorems.
Remark 2 If A = A*, 5 = 5* > 0 and conditions C6) are satisfied, then
scheme A) is p-stable in the space Hg with respect to the initial data, that
is, a solution of problem (la) satisfies the inequality
WVuWb < /'"IIj/oIIb-
On the strength of Theorem 3 in Section 1 this implies the following
estimate for a solution of problem A):
n
k = 0
It is worth noting here that the positiveness of the operator A is unnecessary
in this matter. Apparently, this remark needs certain clarification. Let, for
instance, A > —c^E and c^ > 0. When this is the case, the condition
A > -^^ B or A+ ^^^- 5 > 0
r r
can be met only for p = exp {c^t} > 1, that is, for c^ > 0. Under the
agreement B > sE, £ > 0, we find that
and, therefore.
A+ B> A + CoB > (-C, +0^6) E
T
A>'-^B
T
if we accept c^ > c^/e. To make sure of it, we may choose as S, for example,
the operator
B:^E+tR, i? = 0.5A', A' = A + c,^>0.
Then £ = 1, c^ = c^ and p = exp {c^t^.

416 Stability Theory of Difference Schemes
10. Stability of the weighted scheme. As an example of applying the
theorems just established the weighted schemie comes first
D6) yt + A{ay + {l-a)y) ^^, y{0) = Vo ■
Recall that in Section 1 scheme D6) has been already reduced to the
canonical form
D7) {E+aTA)yt + Ay = ip, 1/@) = j/o ■
By comparing D6) with A) we see that B = E + arA. Suppose there
exists an inverse operator A~^. Applying A"^ to D7) reveals the second
canonical form of the weighted scheme:
Byt + Ay = ip, j/@) = j/q ,
D8)
B=A-^ + aTE, A = E, ^-A-^tp.
Ill the sequel the form D7) is more convenient for the case of self-adjoint
operators A and the form D8) - for the case of non-self-adjoint positive
definite operators A = A{t). We will elaborate on this later.
The first analysis is connected with the case when A is a constant
self-adjoint positive operator A = A* > 0. As we have shown in Section 2,
a necessary and sufficient condition for the stability of the weighted scheme
D7) with respect to the initial data is
Under this condition estimate A5) holds true for a solution of problem
D7). In particular, for an explicit scheme (for tj = 0) the condition a y^a^
implies r < 2/|| A||, that is, an explicit scheme is stable in the space Ha
for r < 2/11 A ||, A scheme with r > ^ is unconditionally stable, that is,
for any r. In Section 2.4 a model example with Ay = —Ay = —j/gj, for
y ^ ilh = -ffA has been considered in full details, in which || A || < 4//i^ and
the appropriate explicit scheme is stable for r < hh'^ ■
For the heat conduction equation with a variable coefficient k{x) we
might have
Lu= ^— (k(x) ^ ) , 0 < k < c^,
dx\ ox I ~
-2 )
At/ - {a{x)y^)^ , 0 < a < C2
and Uq < \- ~ K^/{Ac^t), while an explicit scheme is stable for r < ^/i^/cg

Classes of stable two-layer schemes 417
Theorem 8 Let A be a self-adjoint positive operator independent oft = nr;
A = A* > 0. Then for the weighted scheme D7) estimate C7) is valid for
(T > (Tg, estimate D3) for a > ^i^ — tott, £ > 0, and estimate D5)
for cr > tjj, tj, = ^ — A — £)/(r|| A||), 0 < £ < 1, where a number e is
independent of h and t both.
The above statements will be proved if we succeed in verifying the
fulfilment of the conditions of Theorems 5, 6, 7 with the aid of the inequality
B
rA = E+{.-\)rA>{'+ia-\)r)A.
Suppose now that A = A{t) > 0 and A* ^ A, that is, A = A{t) is a
positive non-self-adjoint variable operator.
Theorem 9 Let A — A{t) > 0 be a positive non-self-adjoint variable
operator. If a > i, then for scheme D6) the estimate holds:
D9) II l/„ + i II < II I/O II + II (^" V)o II + II (^" V)n II + E HI (^" Vk k II ■
k = l
Proof Consider scheme D8) with the right-hand side (p = A~^ip. When
c >^ t:, the conditions of Theorem 5 are satisfied for this scheme and
inequality C7) applies here together with the established relations: ||j/||j = ||y||
and
ll^f.fclU-' =\\hk\\ = \\iA~'^)i.k\\'
As a final result we obtain D9).
Remark If A* = A* (t) is a self-adjoint operator, then estimate D9) holds
true for cr >^ag.
Theorem 10 Let'A{t) = A*{t) > 0 and a > a„. Then the foUowmg
estimate is valid:
E0) lll/n+ilP<lll/olP + +;^Er||^,|p^_..
^^ k = i
Proof The energy identity A3) for scheme D8) takes for now the form
E1) 2r((A-i + (G-0.5)ri?)j/„j/J+||2/|p = ||j,|p+2r(A-V,l/t).

418 Stability Theory of Difference Schemes
The condition a y^ (t^ implies that
E2) B-'^A>eA-^.
Indeed, for a >^ a^
B - ^1 = A-^+(G-0.5) r^
= eA-^ +A ~e)A-^ + (cr-0.5)r^
\\M\
= e A-^ + {a - a^) tE >_ e A-^ .
We have taken into account here that A"^ > i?/||A||. Substituting E2)
into E1) we obtain
E3) 2r£||j/,||^_,+ ||y|p<||j,||2 + 2r(A-V,l/*).
The generalized Cauchy-Bunyakovskii inequality and the £-inequality
together yield
E4) 2r(.4-V,l/*)<2r||^||^_. ■ llyJU- < 2 re ||y,||^-. + ^^ |MI^-. .
Substituting E4) into E3) we arrive at the relation
\\y\?<\\y\? + ^^M\l-.
or
U/fc+ilP < l|j/fcll^ + ^ll<^JI^-
Summing the preceding over fc = 0, 1,2, . . . leads to estimate E0).
Lemma 5 Let A be a positive operator for which the inequality
E5) WAxW'-' <A{Ax,x),
is valid with A = const > 0. Then
E6) A-'^ > -r E and A<AE.

Classes of stable two-layer schemes 419
Indeed, putting x = A~^y we obtain from E5) the inequahty
{y,y)< A{A-^y,y),
meaning E < AA~^ or A~^ > E/A. By virtue of the inequahties
{Ax, xf < II Ax f ■ II X IP < A{Ax, x) \\ x f
we deduce that {Ax, x) < A\\ x |p, giving A < AE. Lemma 5 imphes that
B = A-'^ +aTE >{l/A + aT)E , 5-0.5rI>0 for tj > o-^
C.. '^O 1
where A = E, a^ = ^ - ^/{tA).
Lemma 6 Let A be a positive definite operator and inequality E5) hold.
Then
E7) ||(^ + GrA)-i(^-(l-G)rA)||< 1 for a > a„ = ^--~,
A tA
E8) \\{E + aTA)-^\\<l fora>0,
E9) l\^E + aTA)-'\\<- fora>a, = l--^^-=^,
e 2 tA
0 < £ < 1.
Proof 1) Since B > \tA for a > a-g, by applying Theorem 1 to scheme
D8) we deduce that for a solution of problem D8) the estimate is valid for
^ny y„ G H and if = 0:
F0) II 1/n + l II < Ill/nil-
As can readily be observed, scheme D7) for ip = 0 can be rewritten as
Vn+i =Sy^, S = {E+ arA)-' (i? - A - a) tA) .
From here and relation F0) we obtain estimate E7).
2) In order to estimate the norm H-S""' ||, where B ~ E -\- arA, it
suffices to establish an inequality of the form B > aE, tj > 0. Then
EII a; IP < {Bx,x) < \\Bx\\ ■ \\x\\, \\Bx\\ >6\\x\\,
and, consequently, il 5-1 || < 1/S. If tJ > 0, then 5 > _£; and \\B-^\\ < 1.
If tj > (Tj, then
B > E+a^TA = E + 0.5tA '—^ A .
In conformity with E6), A < AE, making it possible to conclude that
B>E+0.5TA-{l-e)E>eE + 0.5TA>eE
and, therefore, ll-B'^ || < l/s. Observe that estimate E8) holds true also
for any non-self-adjoint operator A > 0, thereby completing the proof of
the lemma.

420 Stability Theory of Difference Schemes
Joint use of Lemma 6 and inequality F0) permits us to state the
following.
Theorem 11 Let A — A(t) be a positive operator and condition E5) hold.
Then for scheme D6) with a > a^ the a priori estimate
F1) \\y{t + r)\\<\\y{m + \JZ^\\Vt'\\
t' = 0
is true. If the following conditions
1 1
tJ > 0 , ^ 'iL^a ■, ^a~
2 rA
are simultaneously satisfied, then estimate F1) is attained for £ = 1.
To prove this theorem, let us write down scheme D6) in the form
Vn + i = Sy„ +TB-^ip„ ,
where 5" = (^ + aTA)-^{E - A - cr)rA) and 5 = ^ + arA. Using the
triangular Inequality and estimates E7)-E9) behind, we obtain the relation
WVn + l II < \\yn\\+ -Iknil.
from which estimate F1) Immediately follows.
11. A priori estimates in the case of a variable operator A. So far we have
established stability in i/^i under the agreement that operator A is constant,
that is, independent oft. In the case when A(t) — A*(t) > 0 depends on
t, this obstacle necessitates imposing the Lipschitz continuity of the
operator A(t) in the variable t
F2) 1 {{A{t) - A{t - r)) a;, a;| < r C3 {A{t - r) x, x)
for all a; G -ff, 0 < t < UgT, where c^ is a positive constant independent of
h and r both.
A primary family of describing schemes Is specified by the following
restrictions:
A{t) = A*{t)>0 for all t e w^ ,
F3) A[t) is Lipschitz continuous in t,
B{t) > 0 for all t e w^ .

Classes of stable two-layer schemes 421
As before we assume the existence of an inverse operator B~^{t), which
assures us of solvability of problem A) for any Input data j/g and (p{t). This
family obviously contains the primary family Introduced in Section 2.2.
The method of energy inequalities shows in such a setting that the
conditions
F4) B{t) >0.5TA{t) for all t^oj^,
F5) B{t)>eE + 0.5TA{t) for all tEtUr, 0<£<1,
turn out to be sufficient for the stability of scheme A) with variable
operators A{t) and B{t). The norms || • ||^, || • ||^_i themselves happen to be
dependent on the variable t:
Therefore, it makes sense to speak about stability in the space Hj^n-j
(instead of Fl/i) and Hg^^ty
The energy identity A3) with A = A{t) is the starting point in special
investigations. To obtain a recurrence inequality, we should modify the
expression
{Ay, y) = {A{t) y{t), y{t)) = {A{t - r) y{t), y{t))
+ {(Ait)-A{t-T))yit),yit))
and estimate the second sumniand on the right-hand side with the aid of
inequality F2):
{A{t)y{t),y{t)) < A + TC,){A{t ~ T)y{t),y{t)) .
Upon substituting this estimate into A3) we obtain the energy inequality
F6) 2r {{B{t) - 0,5 r A(t)) y{t),y{t)) + E{t + r)
<{\ + TC,)£{t) + 2T{<p{t),y{t)),
where
£{t + r) = {A{t) y{t + T),y{t + r)) = \\y{t + r)|
Inequality F6) with the member ip — 0 implies that
F7) ^{t + T)<{l + TC^)E{t)<e''^*E{T) for t>
L(t)-

422 Stability Theory of Difference Schemes
provided condition F4) holds. The energy identity for t = 0 admits the
form
2r (E@) - 0.5rA{0))y{t),yit)) + £{r) = \\ym\\,-^ + 2r (^@), j/,@)) .
With conditions F4) and (p = 0 in view, we deduce from here that
F8) ^{r)<\\ym\l^,y
Collecting F7) and F8) with regard to problem A) with (p = 0 we establish
Such a reasoning discovers actually only one essential difference
between the cases of variable and constant operators.
We now sumimiarize the above results in somie aspects as the analogs
of Theoremis 5 and 7 that furnish the justification for what we wish to do.
Theorem 12 Let operators A = A{t) and B = B{t) be dependent on t
and conditions F3) and F4) be satisfied. Then for scheme A) we have the
estimate
F9) ||l/„+ilU„ <Mi
x{lli/olU„ + ll^olU-. + ll^nlU-+ E HI(^-V),-JUj,
where Mi = exp {^c^tg] and An = A{t^).
If Cg = 0, then the operator A is independent of the variable t and
F9) becomies C7) on account of the equality
Theorem 13 Let operators A = A[t) and B = B{t) be dependent on t
and conditions F3) and F5) be satisfied. Then for scheme A) we have the
estimate
1 "
G0) WVn+iWl^ < M^ {||j/olll„ + ^ E Hk. IP}, Ml = e°'^^3*o.
k = 0
Estimates C7) and D5) are obtained for the case of constant operators
A by merely setting Mi = 1 or Cg = 0.

Classes of stable two-layer schemes 423
12. Example. In order to apply the general stability theory for particular
difference schemes, one needs to perform several manipulations;
• to reduce a two-layer scheme to the canonical form A), that is, to
specify the operators A and B involved;
• to introduce the space Hh of all grid functions and reveal the
desirable properties of the operators A and B such as positiveness,
self-adjointness, etc. as operators acting in the space Hh]
• to verify whether or not the scheme belongs to the primary family
of schemes as well as the fulfilment of sufficient stability conditions
F4) or F5);
• if the preceding conditions are satisfied, the scheme at hand is stable
and a priori estimates hold for it such as, for example, F9) and G0),
The first step within this framework is to reduce a scheme to the
canonical form, but the above sufficient conditions provide a real possibility
of writing stable difference schemes immediately in canonical form.
We cite here only one possible example which helps motivate what is
done. Those ideas are connected with the heat conduction equation
du d'^u
M(a;,0) = Mo(«). M@,t) = M(l,t) = 0,
and associated asymmetric scheme which is given on the grid
'hr
^h X U>^
•^h = {^i = ih, Q<i<N], w^ = {tj = JT, 0 < i < io) ,
and asquires the form
G1) j/i,-+i = —■— {ayi_^J_^_^ + {l-~a)y^^■^J+y^_^_^J-~{2-iu-~a)y^J) ,
UJ -\- CY
where uj = K'^/t and a is a parameter.
We do follow established practice in a step-by-step fashion. 1)
Reduction of the scheme which interests us to the canonical form. Denoting
Vi j = Vi c^nd y- j_^_i = y^ we first rewrite G1) as
G2) {uj + a)yi = ayi_^ -^ A -a)l/i_i + Vi+i -~ {2 - u> - a)yi .

424 Stability Theory of Difference Schemes
After that, taking into account that
l/i_i =-/il/s,j + l/i, Vi+i =hy^i + yi, h y^ ■ - hy^ ^ = h^y^^ ^ ,
substituting these expressions into G2) and omitting the subscript i, we
come to
G3) ojryt = h"^ y^^ -ahry^.^.
Dividing G3) by K^ generates
G4) ^* + X ^** " ^""^ '
0
2) Let -ffft be the space of grid functions fi ^ (see Examples 1 and 2
in Chapter 2, Section 4.1) defined on the grid w^^ = {xj = ih,0 < i < N}
under the inner product structure {y, v) = '}2i=i Vi '"i ^- In conformity with
the results obtained in Section 4.1 of Chapter 2, the operators Ay = —y^^
and Riy = ij/j. involved in the scheme are positive definite: {Riy,y) =:
j;(Ay, y). The operator A is self-adjoint, || A \\ < A/h'^.
3) The operators A and Ri are constant. Because of this, it will be
convenient to write scheme G4) in the form
G5) (^ + ari?i)j/j + Aj/=0,
so that B — E -\- cxtRi-
The condition B > ^tA is satisfied for a > 1 — 2/(r|| A||). Indeed,
for any x £ H
{{B -0.5tA)x,x) = {{E+arRi -0.5tA)x,x)
= {{E+0.5T{a- l)A)x,x) ,
yielding
5-0.5rA> ^+0.5r(«- 1)A> f t-!—+ 0.5 r (a - 1))^ >
0.
4) Since |j A|| < 4//i^, scheme G1) is stable in the space H^ (in the
grid norm of the space 14^2^) for
G6) a > 1- — .
Z T

Classes of stable two-layer schemes 425
Together with scheme G1) one can write down one more asymmetric
scheme which after reducing it to the canonical form becomes
{E + aTR2)yt+Ay = Q, i?21/= - ^ ^/.^. ■
Since {Riy,y) = (-R2J/, J/), this scheme is stable under the same condition
G6). Condition G6) shows that the asymmetric schemes are
unconditionally stable for a > I.
13. The case of a skew-symmetric operator A. The main results of stability
theory for two-layer schemes
Byt + Ay = If
have been obtained under the agreement that A = A* > 0 is a positive
self-adjoint operator, while the operator B > 0 may be, generally speaking,
non-self-adjoint. An exception is a weighted scheme in which the operator
B is of the special type B = E + arA and A* 7^ A.
Assume that A is a non-self-adjoint operator with the approved
decomposition A = Ao -I- Ai, where Aq = \{A + A*), Ax = \{A - A*),
Ag = Ao, A* = --Ai, that is, Ai is a skew-symmetric operator,
{Aiy,v) = -{y,Aiv)
and
(Aij/,j/) = -{y,Aiy) = 0.
To avoid generality, for which we have no real need, we restrict
ourselves here to the case when A = Ai is a skew-symmetric operator involved
in the weighted scheme
j/, + Aj/(^) = 0,
where A* = —A, [Ay,y) = 0, y'^"^ = (ry -\- [\ — a)y, and attempt it in the
form
Byt + Ay = 0, B = E+arA.
Since A and B are non-self-adjoint operators, we cannot use the obtained
results of the general theory, so there is some reason to be concerned about
this. With this aim, we proceed to develop a new approach in such matters.
Observe, first of all, that
{By,y) -{y + (TTAy,y)- {y,y) +aT{Ay,y) = \\y\\~,

426 Stability Theory of Difference Schemes
that is, there exists an inverse operator B"^. By inserting y = B~^x it is
plain to establish the relations H^^^xlp = {B~^x,x) < \\B'~^x\\ ■ \\x\\,
giving, II B'~^x || < || a; || and || 5"^ || < 1- The next step is to rewrite the
scheme in the form
By = By - rAy, y = y- rB^^Ay,
and calculate the inner product
{By,y) = {By, y) - t {Ay, y) - t {By, B^^Ay) + t' {Ay, B~^ Ay).
For later use, it will be sensible to represent the operator B in the form
B = E + a T A :^ {E - a T A) +2a T A = B* + 2(T T A
and make the obvious transformations
{By, B~'Ay) = {B*y, B^^ Ay) + 2<7 r {Ay, B-^ Ay)
= {y,Ay)+2aT{Ay,B-^Ay)
= 2aT{Ay,B~'Ay).
As a final result we obtain
\\y\\^ = \\y\\^-{2a-l)T^B-'Ay,Ay),
yielding
||2/||<||j/|| if<T>0.5,
||j/|P<||j/|p + (l-2<T)r2||A|r'^.||j/|P<(l + c„r)||j/|p
or
l|y||<p||y||, P-e°■'="^ c, = {l^2a)c,,
if cr < i and r II A IP < c^.
We have obtained through such an analysis the following estimates for
the norm of the transition operator:
IIS'll^l for cr>0.5,
IIS'll^p for cr<0.5 and r||A||2<c2.

Classes of stable two-layer schemes 427
This implies the estimate for a solution of the nonhomogeneous equation
BVt + Ay = tf>:
lll/n+ill</'lll/nll + r|kJ|<p"+^||l/oll + Erp"~*||^,||.
k = 0
Let now i7 be a complex space and S be a non-self-adjoint operator.
Then a necessary and sufficient condition for the stability in the space Hjn
with respect to the initial data of the scheme
5 j/j + A J/ = 0
is of the form
5o = Re5 > 0.5rA, A = A* > 0 ,
where Bq — ^{B -\- B*) — B^. In particular, the scheme
i j/j + A J/ = 0 , A = A* > 0
is unstable in the space Hj\, since B = lE and Re 5 = 0. However,
A' = —iA is a skew-symmetric operator (A')* = —A' and, as stated before,
this scheme is conditionally stable in the space H:
||l/„ + ill<p"+'l|l/oll, /' = e=<'^ c,=c, ifr||A|p<c,.
0
In the case of the Schrodinger equation Ay — —y^.^, J/ G fi ft [0, 1], we have
II A II < 4//i^ and the restriction on r takes the form
4
h
- 16 '
which is improper for parabolic equations. However, the weighted scheme
ij/,+Aj/('^) = 0, A = A*>0,
will be stable in both spaces H^ and i7 for cr > |. If so,
Bo = ReB = arA > krA
and ||j/„|L < l|j/oll/i- Oi"^ the other hand, in dealing with the skew-
symmetric operator A' = —iA we get the estimate || ?/„ || < || J/q II fo'" "" ^ i'

428 Stability Theory of Difference Schemes
6.3 CLASSES OF STABLE THREE-LAYER SCHEMES
1. The problem statement. In this section we establish sufficient stability
conditions and a priori estimates for three-layer schemes on the basis of
their canonical form
Bye +t'^ Ryft+Ay = ip{t), 1/@) =% , l/(^) = l/i ,
A)
0 < t =: nr < to ) n = 1,2,... , Uq — 1, 1^ = 71^7.
Here and below, y^ and j/j are arbitrary given vectors of a finite-
dimensional real space H, (p(t) is a given arbitrary abstract -ff-valued
function of the variable t (z UJ^; A, B and R are linear operators in the space
H. The dependence of y{t) ~ yhrit), <p{t) = <PhT{i)> Mi) = ^hr, B, R, y^
and j/j on h and r is not explicitly indicated. We proceed as usual. This
amounts to introducing more compact notations:
y-y{tn) = yn> y = y{tn + t) = Vn+i, y - y{tn - t) = yn-i,
yt = iy-y)/T, yi = {y-y)/T, y. ={y-y)/{2T),
yft = {y-'^y + y)/^-
t
2
allowing a simpler writing of the ensuing formulae.
All the tricks and turns remain unchanged by analogy with Section 2:
first, a solution of problem A) can be arranged as a sum y = y + y, where
j/ is a solution of the homogeneous equation:
Byo +7"^ Rytt + Ay:^0, 1/@) = j/q , yir) = y, ,
(la)
0 <t = TIT < tg ,
and y is a solution of the nonhomogeneous equation with the zero initial
data:
Byo +T^Ryt-t+Ay=^{t), j/@) = i/(r) = 0 ,
(lb)
0 < t = nr < tg ,
Second, an alternative form of writing appears useful:
E + 2 ri?) !/„+!=$„,
B)
$„ = 2 Bi? - A) Ty„ + {B- 2tR) i/„„i + 2r^„

Classes of stable three-layer schemes 429
{A, B and R are, in general, variables, that is, they depend on t„). From
such reasoning it seems clear that problem A) is solvable if an inverse
operator (B + 2tR)~^ exists. In the sequel this condition is supposed to be
satisfied. Moreover, we take for granted that
C) the operator B + 2tR is positive definite.
Further development of the three-layer scheme C) is connected with
the functional known as the compound-norm:
D) II Yn + l 11^ = \ WVn + i/n + lll(i^^ + ll^" + i ^ ^"Il(l2) '
where || ■ jL. , and || ■ ||,. , are suitable norms on the linear system H.
In order to understand the structure of this norm a little better, it is
appropriate to introduce the space H'^ = H ® H being the direct sum of two
copies of H. The space i7^ is defined as the set of all vectors of the form
y = {y(i)-KyB)} ei72_ yC")^//, a =1,2,
where the operations of addition of vectors and multiplication of a vector
by a number are carried out in a coordinate-wise fashion:
y + y = {y(i) + y(i), yB) + yB)}_ ay = {ay(i), ayB)} .
The norm on the space H^ is natural to be defined by
||y||2 = ||y(i)||2 .||yB)||2
II II ll-' 11A,) ^11^ Ii(i2)'
In our case the vector
yn + l ^ \\{yn + l + 'i/n). Vn + l - Vn] ^ H~
passesses the coordinates y„^^\ = |(l/„4.i +!/„) and Y^_^\ = y„_^i - j/„. It is
easy to see that functional D) satisfies all the axioms of the norm, namely
||ay„+i|| = |a|||y„+i||, ||y„+i|| > O for any j/„ G H, y„_^^ e H and
II y„ + i II = 0 only for y„ = y„^, = 0; || y„+i + y„+i || < || y„+i || + || y„ + i ||.
We now in a position to define the notion of stability of scheme A).
The three-layer scheme A) is said to be stable if there exists the norm D)
and for all sufficiently .small t < Tq and \h\ < hg one can point out positive
constants Mi and M2 independent of r, h and disregarding to the choice

430 Stability Theory of Difference Schemes
of J/oi J/i I fit) such that for any j/q, j/j , (p(t) and all t = r, 2r, . . ., [rig — l)r
a solution of problem A) satisfies one of the following estimates:
E) lin^ + r)||(i^<Mi||y(r)||(^„^
+ A^2^maxJ|^(t')||B),
F) lini + r)||(i^<Mi||y(r)||(j„^
where || • II2 is some suitable norm on the space H, \\Y{t + t)\\,,-. and
||y(r)|Los are defined by the formula of the form D), so that
G) \\Y{t + T)\\l^ := \ \\y{t + r) + y{t)\\l^^ + Ut + r) - y{t)\\l^^,
(8) WYir^.o^ = \ \\y, + l/oll(iO) + l|j/i - J/o 11A0^ ,
where || ■ |Los and || • |Lon are suitable norms on the fspace H.
As far as constant operators A and R are concerned, the norms ||y||Qs
and ||y|Los normally coincide. In the general case \\Y{t + T)\\f^-. and
ll'i^@ll(-2') depend on t = nr, so that one should write ||y(i + 7')|L j^ instead
of \\Y{t + r)||(,^ and Mt)\\^^_^^ instead of Mt)\\^^y
As we will see later, the norms || ■ ||,. , and || ■ ||,. , are energy norms
constructed for the operators A and R. For this reason we will assume that
these operators are
(9) self-adjoint: A = A*, R = R*,
A0) positive: A>0, i? > 0,
if i7 is a Hilbert space.
2. The basic energy identity. We will carry out the derivation of the energy
identity for the three-layer scheme A) with variable operators A = A{t),
B = B{t) and R = R{t). This identity is aimed at achieving a priori
estimates expressing the stability of a scheme with respect to the initial
data and right-hand side.
By virtue of the relation.s
1 , , 1 .. . . 1
r
2
y=^{y + y)-^{y-'^y + y)=^i:ii + y)-'^ yn

Classes of stable three-layer schemes 431
we represent scheme A) in the form
B j/o + r^ (i? - i a) j/fj + i A (y + j/) = <^ ,
A1) ^ ^
1/@) = i/o. y{T) = Vi,
where A = A(t„) = A„, 5 = B{t„) = 5„ and R = i?(t„) = i?„. Taldng
the inner product of A1) and 2rj/o = r(j/j + j/f), we find that
A2) 2T{By.,y.) + T''({R-\A){y,-yi),y, + y,)
+ \{A{y + y),y-y) =2T{^,y.).
Let A and i? be self-adjoint operators. Then
R-0.5A^ {R-0.5A)*.
By Lemma i of Section 2 we thus have
A3) ((i?™iA)(j/,-j/f),% + j/,-) = ((i?-iA)(j/,,j/,)
-((i?-iA)(j,f,yf),
A4) (A(y-Fj/),y-j/) ={Ay,y)-{Ay,y).
At the final stage we add and subtract {Ay, y) on the right-hand side of
A4). The outcome of this is
A5) (A {y + y),y-y) = [{Ay, y) + {Ay, y)] - [{Ay, y) + {Ay, y)] .
Lemma 1 Let A = A* be a self-adjoint operator. Then
A6) {Av, v) + {Az, z) = ^ {A{v + z),« + z) + 1 {A{v - z),v - z)
for any vectors v and z of the space H.
Proof Since A = A*, we have {Av, z) — {v, Az) — {Az, v) and
{{A{v + z), V + z)) + {{A{v - z), V - z))
= [{Av, v) + 2{Av, z) + {Az, z)] + [{Av, v) - 2{Av, z) + {Az, z)\
= 2[{Av,v)^{Az,z)\.

432 Stability Theory of Difference Schemes
By inserting in A6) v = y and z = y we transform A5) into
A7) {A{y + y),y-y) =0.5 [{A{y + y), y + y) + {A{y - y),y - y)]
- 0.5 [{A{y + y),y + y) + {A{y ^y),y- y)] .
We now substitute A7) and A3) into A2) and take into account that
{A{y-y),y-y) - T'^{Ay^,y^), y-^ - {y-y)/T = j/j,
{A{y-y),y-y) = T\Ay^,y^) ,
making it possible to establish the basic energy identity for the three-layer
scheme A):
A8) 2T{By.,y.)+[\{A{y + y),y + y)+T'^{R-\A)y„y,)
= ~{A{y + y),y + ij) + T'-'(^{R-^A)ijt,yt)^ +2r(^,j/=).
In giving it we preassumed only property (9) concerning the self-adjointness
of the operators A and R and no more.
3. Stability with respect to the initial data. Recall the definition of stability
with respect to the initial data and the right-hand side. Scheme A) is said
to be stable with respect to the initial data if for problem (la) the a priori
estimate holds:
A9) \\Y{t + r)\\^,-^<M,\\Yir)\\^^oy
Scheme A) is said to be stable with respect to the right-hand side if for
problem (lb) the estimate
B0) 1|y(t + r)||(j^<M2^maxJ|^(OIB)
or the estimate
B1) \\Yit + r)||(^^ < M, ^max^(||^(t')|B) + ll^f(i')lB))
holds.
Making use of the triangle inequality and collecting A9) and B0) or
B1), we obtain estimate E) or F).

Classes of stable three-layer schemes 433
In our basic account A and R are taken to be constant self-adjoint
positive operators and B refers to a non-self-adjoint nonnegative operator:
B2) A = A*>0, R=R*>0, B>0.
These restrictions permit us to specify and extract a primary family of
schemes.
With the regard to problem (la) identity A8) takes the form
B3) 2r Eyo, j/o)+II y(t + r) IP = II y(t)||2, t = nT,
where
B4) II Yit + t) f = \ {A{y{t + r) + y{t)), y{t + r) + y{t))
+ r2((i?-iA)j/,,j/,),
B5) II Y{t) IP = \ {A{v{t) + y{t - r)), y{t) + y{t - r))
+ r'(^{R-\A)y,,y,).
To avoid cumbersome calculations, we will also use index denotations by
setting Y{t -\- t) = Yn + i and
ll^n+llP = H^B/n+l+l/n),l/n+l+?/„) + ^^ ((^- \A)y^„,y^,^^
— 4 II Vn + l + Vn II 4 + ll?/n+l ^ 2/nllfl„ i^l ■
Observe that relation B4) implies that || Y{t-\-T) |p > 0 for any y[t) 7^
0, y[t -|- r) 7^ 0, provided that the operators A and R — A/4 are positive,
A > 0 and i? > A/4.
Theorem 1 Let A = A* > 0 and R= R* > 0 be positive operators. Then
the conditions
B6) B = B{t) > 0 for all t e w^ ,
B7) R>^^A
are sufficient for the stability of scheme A) with respect to the initial data.
Under conditions B6) and B7) for problem (la) the estimate holds:
B8) lin^ + r) II < II y(r) II,

434 Stability Theory of Difference Schemes
where || Y \\ is defined in accordance with rule B4).
Indeed, for B >0 identity B3) implies that
II Y{i + t) f < II Y{t) f, II Y{t + r) II < II Y{t) || < • • • < || Y{t) \\ .
Remark 1 Theorem 1 holds true under the constraints
R> \A, A> 0.
However, in that case ||y|| may fail to be a norm, but it is always a
seminorm.
Remark 2 The three-layer scheme (la) can be reduced to the two-layer
scheme
B9) BYt+AY = 0, YoeH'\
space H'^. This can be done using the vector
where Y = Yn ^ H'^, Yt = {Yn-^i — Yn)/T, A and B are operators in the
Yn = {\{yn +yn~l),yn " Vn-l}
and treating the operators A and B as operator matrices with elements
being operators in the space H:
/ B + Q.brA t{R- \A)
\-r{R-^,A) \{R-\A)
If A = A*, i? = i?*, then the operator A : H"^ ' H"^ is self-adjoint:
A = A*, while the operator B is non-self-adjoint. All this enables us to
conclude that
||y||^ = (y4y,y) = (Ay(i),yfi))+((E-iA)yB),y<2)),
In addition, we have A > 0, provided that A > 0 and R > ~A.
It is easy to verify that the stability condition in the space H^ of the
two-layer scheme B9)
C0) ^-?-^

Classes of stable three-layer schemes 435
is equivalent to the requirements
T '
B6*) B = B{t)>0 for all tew
B7*) R>Ia,
which assure us of the validity of the estimate
B8*) ll^-n+iii^sliyilU,
where
ll^'n+lll^ = ^WVn+l+ynWl + Wyn+l -J/nllLlyl '
In concluding this section, let us stress that condition B6*) is not only
sufficient, but also necessary for the validity of estimate B8*).
4. Stability with respect to the right-hand side. We now consider problem
(lb) under conditions (9), A0) and B7), which are put together for later
use. Since A and R are constant operators, identity A8) for (lb) is of the
form
C1) 2rEj/o,j/o) + ||y(t + r)||2 = ||y(t)||2 + 2r(^,j,o), t^nr.
In the further development of a priori estimates of the form B0) or
B1) a key role is played by the estimation of the functional 2t{<p, j/o ). First
of all, we give below the obvious inequality
C2) 2r(^,j/o)<r£o||j/o|p+-L||^||2_
t t eg
where e^ — const > 0 is independent of r and h both.
Theorem 2 Let A = A* > 0 and R =■ R* > 0 be positive operators. Then
under the conditions B > eE, R > A/4, e = const > 0, a solution of
problem A) satisfies the a priori estimate
C3) \lY{t + r)\\<\\Y{r)\\ +
1
/2e
t
\^
lk(OI
1/2
It suffices to estimate only a solution of problem (lb), since Theorem
1 remains valid for B > eE. By merely setting in C2) e^ — 2e we deduce
from C1) that
C4) \\Yit + T)f <\\Yit)\\' + f^\\^it)\\'.
To finish the proof of the theorem, it remains to sum up this inequality with
respect to the variable t = t, 2r, . . . , nr, exploit the fact that || Y(t) || = 0
and apply Theorem 1.
We cite here without proving the following assertion.

436 Stability Theory of Difference Schemes
Theorem 3 Let the conditions of Theorem 1 be satisfied. Then scheme A)
is stable with respect to the right-hand side and for it the estimate holds
for t > t:
C5) II Y{t + r) II < II Yir) \\ + Wh ^max^ (||^(OIU- + ll^f^OlU-) ,
where M2 = const > 0 depends only on tg.
Theorem 4 Let A = A* > 0 and R =■ R* > 0 be constant nonnegative
operators and B = B[t) be a variable non-self-adjoint positive definite
operator
C6) B > eE, e = const > 0,
where a number e is independent of h and r, and let the condition
C7) ^ > ^^
be satisfied. Then a solution of problem (lb) admits a priori estimate
C8) \\y{t + T)\\<
EHk(OI
i' = T
Look at identity C1). It follows from C6) and C2) for e^ = e that
C9) re\\y,f + \\Y{t + T)f <\\Y{t)r + ~\\^{t)\\\
With the relations || Y[t) || = 0 and || Y{t + r) |p > 0 in view, summing up
over t = t,2t, . . . ,nT yields
D0) eJ2-\\yt{t')f + \\y{t + r)f<\ E^ll^(^')ll'
e
t' = r t' = T
or
t ^ t
D1) E^\\ydi')\\'<^ EHk(i')i
i' = T
Further development of estimate C8) from D1) is based on the
following assertion of auxiliary character.

Classes of stable three-layer schemes 437
Lemma 2 If j/@) = j/(r) = 0, then
t
D2) \\y^t)\\'^ + \\y(^t + r)\\''<At X]Hlj/f(^')ll'-
Indeed,
y{t + t) + y{t) = 2 Y^ Ty,-{t'),
t'-T
t
\\y{t + T) + y{t)f <^tY, r\\yi{t')\
D3) ''^l
In the notation w^ = j/„ — j/„„i we obtain
which implies the inequality
ll«^n'+i II < 2r||j/j-„' || + ||w„'||, n' = 1,2,... ,n.
Summing up the preceding over n' from 1 to n we find that
n
ll«^n+lll<2 Y. H|l/f,n'll
D4)
\\y{t + r)^y{t)\\<2Yr\\y,{t')\\,
t'-T
t
y{t + r)-y{t)f<^tYrU{t')f
Putting inequalities D3) and D4) together with the obvious identity
D5) \\y{t + T) + y{t)f + \\y{t + T)~y{t)f = 2{\\y{t)\(' + \\y{t + T)f)
we arrive at D2). Substituting estimate D2) into D1) yields C8).

438 Stability Theory of Difference Schemes
Theorem 5 Let A = A* > 0, R = R* > 0 and B = B* > 0, A and R be
constant operators and R > |A, Then for scheme A) the estimate holds:
1 "
D6) \\Yn+i\\'<\\nr + -J2r\M\l-^-
k = l
The fact that the operator B is self-adjoint is kept in mind in the
estimation of the expression
D7) 2r(^,j/f) <||j/f|lB ■ MB-^<^^\\yf\\l + ^\Ml-^'
which is involved in identity C1). Putting e = 2 and substituting this
estimate into C1), we obtain the inequality
r
1 I
D8) \\yn+i\\'<\\Yn\\' + ~\\^,X-
implying the desired estimate. Note that the operator B may depend on t:
B = B(t). In that case we must write ||(,£3j.|P _i, where B^^ = B~'^[tf.).
5. Schemes with variable operators. If operators A and R depend on the
variable t, the extra property of the Lipschitz continuity of A and R with
respect to the variable t is needed in this connection:
D9) I {{A{t) - A{t ~ r)) x,x)\<TC^ {A{t - t)x, x)
for all a; G if and t = 2r, . .. , (ng —l)r, where c^ = const > 0 is independent
of h and r; the condition imposed on the operator R is analogous. In this
case the compound norm || Y{t+T) \\ = ||^(i + 7')||(-j-) depends on the variable
E0) \\Y{t + r)\\l^ = I {A{t) {y{t + r) + y{t)), y{t + r) + y{t))
+ T-'({R{t)~\A{t))y,{t),yS)),
E1) \\y{%^r) = i 0^^^ - ^) (^('•) + y^^ - ^))' ^(^) + ^(^ - ^))
+ t' ({R{t -T)^\A{t- r)) y,{t),y,{t))

Classes of stable three-layer schemes
439
We modify the expression in the brackets on the right-hand side of
identity A8) by a simple observation that
{A{y + y),y + y) = {A{y + y),y+y) + T {Af{y + y),y + y) ,
{{R - \ A) j/f, j/f) -((/?-! A) 2/f, %-) + r [{R - \ A)^y,, y,)
A = A(t- t) , At = {A- A)/t .
Also, it will be convenient to introduce more compact notations
E2) J==J(t + r) + ||y(t + r)||J^^, J = J(i) = ||y@||(V,) .
With these, identity A8) is recast as
E3)
E4)
2r {By. , j/o) + J = J + 2r
,<^. 1/;
■F.
F=\ {Af{y + J/), J, + J/) + r^ ((i? - i A) - y,, y,)
showing the new members to be sensible ones. If the operators R — Aj\
and A satisfy condition D9), then
\\P\\<\ (i(l/+ !/),!/+ y) + t'c, ((/?.- li).j;,,yf) = C3 J
and from identity E3) for R > A/4 it follows that
E5) 2r Ej/o , j/o) + J < A + r C3) J + 2r (^, j/o) .
In the general case when each of the operators R(t) and A(t) satisfies
condition D9), we obtain
E6) \\F\\<'^{A{y^y),y^y)^T''c,\[Ry,,y,))^\{Ay,,yi)
under the constraint R > ^^A, where e = const > 0 is independent of h
and r, since
£r
l + e
J > ~ i^yt' Vi) for R > —^ A .

440 Stability Theory of Difference Schemes
Identity E3) yields
E7) 2rEj/o,j/o) +J(t + r)< (l + i^±il^ r) J(t) + 2r (^, j/o) .
Having at our disposal the energy inequality E7), we can derive the a priori
estimates in just the same way as was done for constant operators A and
R. For example, for 5 > 0 inequality E7) for problem (la) implies the
estimate
E8) lini + r)||(,)<Mi||y(r)||(^) if i? > 1±^ A,
where Mi depends only on e, Cj and t^.
We bring together the basic facts in the following assertion.
Theorem 6 Let variable operators A = A{t) — A*{t) > 0 and R = R{t) =
R*(t) > 0 be Lipschitz continuous in t and let
E9) R{t) > -^ A{t) for all 0 < t = n r < to ,
where e = const > 0 is independent of t and h. Then for scheme A) the
following estimates hold:
F0) |in^ + r)||(,)<Mi||y(r)||(^)
+ M2 ^max^ [Mnh'^t') + ll^f(OIU-(,')]
for B{t) >0,0<t-nT<to,
F1) \\Y{t + T)\L.<Mi\\YiT)\l +M, max 11^@11
v/ v^ T<t<.t
for B{t) > eE, where e = const > 0, Mi > 0 and M-j > 0 are independent
of T and h both.
To avoid needless repetitions, we omit here the proof of the theorem.
Remark Some requirements of Theorem 6 can be relaxed. For instance,
the condition 5 > 0 can be replaced by the following one:
F2) B > -c.t'^A,
where c^ = const > 0 is independent of r and h both. Under condition
F2) estimate F0) holds true for r < Tq, Tq = 1/D c^).

Classes of stable three-layer schemes 441
6. Weighted schemes. In practice the reader frequently encountered the
weighted schemes
F3) *
1/@) = Vo, y{T) = Vi,
where y('^i''^2) = (j^ y _|- (l — (j^ __ (j^^y _l_ Q-^y^ The accepted view is that
stability and accuracy of the scheme are governed by selection rules for real
numbers a^ and a^. In Section 1 scheme F3) was written in the canonical
form A), making it possible to recover the operators
F4) B = E + T{a,-~a^)A, R^-"'^"^ A.
Assumming that there exists an inverse operator A"^ and applying it,
on the same grounds, to both sides of A) with operators F4), we obtain
5 j/o + r^ /?, j/j-j + A J/ = (^ , T <t = iiT <to ,
F5)
2/@) = Vo, y{T) - Vi,
where B = A^^+ia^-a^) t E , R = ^^^^-^ e , A = E , (p = A-^^p.
This implies that constant operators A and R are self-adjoint.
On account of Theorem 1 with regard to F5) the operator inequalities
hold:
F6) R-~-A=(^^^^^-~-JE>0 for a^ + a^ > 0.5 ,
F7) 5 = A~^ + (cTj - (T2)r-e;> 0 for a^ > a^ and any A(t) > 0 .
Theorem 7 If A(t) is a variable positive operator and the conditions
F8) ""i > ". ""i + " > 0.5
are fulfilled^ then scheme F3) is stable and the estimate holds:
t
F9) II Y{t + r) II < II Y{t) II + \/2(<T, + <T,) ^^ r \\ ^{t') \\ ,

442 Stability Theory of Difference Schemes
where
G0) \\Y{t + T)r = \\\y{t + T) + y{t)\\'' + l{a, + a,~^^)\\yit + r)^y{t)\\\
Proof In what follows we distinguish two separate tasks.
1) StabiUty with respect to the initial data. Since the conditions
R > A/4 and 5 > 0 of Theorem 1 are fulfilled, we have for a solution of
problem F5) with (p = 0 the estimate
\\Y{t + r)\\<\\Yit')\\, t'<t,
and, in particular,
G1) lin^ + r)||<||y(r)|i,
where || Y(t + t)\\ is specified by formula G0) being a particular case of
formula B4) with A = E and R = |((Tj + (Tj)!? incorporated .
2) Stability with respect to the nght-hand side. Consider problem
F3) for j/@) = j/(r) = 0 and seek its solution in the form
n
G2) yn + l = E T9n + 1,,, l/o = 0.
s=l
where g„,i j as a function of n for fixed s = 1, 2,, .. , n satisfies equation
F3) with (,£3 = 0 for n > s + 1 and the initial data
G3) fif.^i,+ 2(Ti rAfif.^i, = 2<^, , 9^,^-0-
Substituting G2) into F3) and taking into account G8), we conclude
that G2) is just the solution of problem F3). As we stated in G1), on the
strength of stability with respect to the initial data we have for g^ ^
G4) ||G„+i,J|<||G,+i,, II for fixed s=l,2,...,
where || Gn+i,s \\ is expressed through g„ ^ and g„.^i by relation G0). We
find from G3) that (/s+i,s = 2(i?+ 2a^TA)~^if and establish the relations
\\E+ 2(TjrA)~^ II < 1 and \\g,.^i^, \\ < 2||<^, ||. This is due to the fact that
E+2a,TA> E for a,>0
By assumption, g^ ^ = 0. This provides support for the view that
II G, + i,. IP = \ II ^,+1,, II' + \{<r,+a,- i) II 0,^1,, IP
= 2 ('^1 + ^2) II 9s+i,s IP < 2 (<Ti + a,) II ^, IP,

Classes of stable three-layer schemes 443
giving
G5) II Gn+i,s II < II G,+i,. II < y^2{a,+a,) \\ ^, \\ .
Substituting G5) into the right-hand side of the inequality
n
\\Y„+,\\< E H|G„+i,.||
5 = 1
we obtain for a solution of problem F3) with j/@) = j/(r) = 0 the estimate
t
G6) II Y{t + r) II < \/2KT^ E ^ II ^(^') II .
i' = T
which in combination with G1) implies F9).
Theorem 8 If A(t) = A*(t) > 0 is a positive operator and conditions F8)
are satisfied, then for a solution of problem F3) the inequality
\A~\t')
G7) \\Yit + T)\\<\\YiT)\\ + ^\J2rMt')
holds, where \\ Y{t + r) || is given by relation G0).
To prove this assertion the estimates
2r(^,j/o)=2r(/l-V,l/=)<2r||j/o||^_. • ||^||^_.
<2r||j/o||^_,+I|M|^_,,
2r(Sj/o,j/o)>2r(/l-ij/o,yo) = 2r||j/o||2_,
are incorporatred in identity A8) for scheme F5).
Applying TheGrem 3 to scheme F5) with a constant positive operator
A, it is plain to derive under conditions F8) the estimate
G8) II Yit + r) II < II Yir) \\ + +M2 max (|| A" V(i') II + 11^" Vf(i')ll) ■
T<t'<.t
Note that estimate F0) holds true for scheme F3) if A{t) = A*{t) > 0,
A{t) being Lipschitz continuous in t, and
CTj > 0-2
1
r\\A\\

444 Stability Theory of Difference Schemes
while the estimate of the form F1) is valid for
7. Examples. It seems worthwhile giving several schemes of particular
forms.
1) Scheme A) with the operators R = xE and B — E
G9) yt + xT'^ytt+Ay = (p
is stable for xE > A/A, that is, for
(80) X > \\\A\\.
A particular case of scheme G9) is the Du Fort-Frankel scheme known
as the "rhombus" scheme for the heat conduction equation
— = —-, 0<s<l, i>0, u{x,Q)^u„{x),
ot OX''
u{Q,t) = M(l,i) = 0,
emerging from the explicit unstable scheme
y{x + h,t) — 2y{x, t) + y[x — h, t)
yt + Ay = 0, Ay = -y^^
1 iJ+^ , J-~^^
upon replacing y{x,t) = yj hy the half-sum ^(y^ + y^ ) = i (y^. + y-).
The outcome of this is
.„,, Vi- iii Vi+i - ik + y») + Vi-i
^^^) .^^ = h^^ ■
Also, it will be sensible to write (81) in canonical form. Since
y + ij = 2y+ r'^y^f ,
the right-hand side of (81) equals j/j^. — ji^ytf Therefore,
^^ An,, M . M 4 2 tt/i 4
yt + -f^yu+Ay = Q, Ay= -y^^ , || yl || = -p- cos — < — .

Classes of stable three-layer schemes 445
Comparison of this equation with G9) gives x = l/h'^ > ||yl||/4, that
is, the Du Fort-Frankel scheme is stable for any r and h. This is a way
of producing an analog of tliis sclieme for the case when L is any elliptic
operator with x still subject to condition (80).
2) The asymmetric three-layer scheme
3 1 . ,^ ^y — Ay + y
l^Vt- l^Vt + ^y = f or — + Ay = (p
suits us perfectly for solving the heat conduction equation. Using the
formulae
2
2/i = j/= + 2 Vit' yt = y° - 2 ytt' y = J/ + T y= + Y vu.
it is plain to reduce it to the canonical form
(E+tA) yo +t'' (^-+-Aj y^t +Ay = ^,
giving B = E + tA and R = E/t + ]^A. These assure us of the validity of
the conditions
l + £
R > —— A, 0<£<1, B > E
for any A>0.l{A = A*>0, the scheme concerned is stable in the norm
\\yii + r)\\i^ = HMt + ^WA + \\ymi) + rM\i.
8. Three-layer schemes with non-self-adjoint operators. The three-layer
explicit scheme with a self-adjoint operator A
yo+Ay=:0, A = A* > 0^
is absolutely unstable; the necessary stability condition R > ^A is violated
for this case, since R = 0. This scheme is unstable in any norm || • ||„ and
refers to a generalization of the well-known Richardson sclieme for the Iieat
conduction equation
yj - yj ^ yj-i -'^yj + yj+i
The implicit scheme Byo -\- Ay = 0 witli any operator £? = £?*> 0 is also
absolutely unstable.

446 Stability Theory of Difference Schemes
We now consider the scheme with a skew-symmetric operator A:
(82) yo+Ay = 0, A=-A*.
We are going to show that this scheme is stable for t||^|| < 1 and for it
the energy identity holds true:
t'?-i,+i
£,
where £„+i = || y„+i |P + 2 r( 2/„_^i, yly„) + || j/„ IP > 0.
All the tricks and turns remain unchanged: first, write the difference
equation in the form
y+TAy-y-T Ay
and compute the squared norms on the left and right-hand sides
\\yf + 2r{y,Ay) + r'\\Ayf = \\yf -2r{Ay,y) + T'\\Ay\\\
then add \\y\\'^ to both sides and take into account that A is a, skew-
symmetric operator, that is, {Ay^ y) = —(y. Ay). As a final result we obtain
^n+i — ^n ~ ■ ■ ■ = £i- What is more, we claim that £„+! > 0- Indeed,
4+i>||?y„+il|=^-2r||y„+i|| . \\Ay^\\ + \\y„\\'
>\\yn\\'-r'\\Ay,,r>0
under the restriction r||yl|| > 1. Here H is a real space. In the case of a
complex space H we have the quantity
4+1 = II Vn+i IP + 2 r Re (j/^+j, Ay J + \\ j/„ f ,
with respect to which both the above reasoning and results are still valid.
Example Of special interest is the Schrodinger equation
du d'^u
* ~7r = TT^ ~ 9 " . 9 = const > 0 , 0 < x' < 1 ,
at ox^
M@,i) = M(l,i) = 0, u{x,{]) = Ua{x).
We introduce, as usual, on the segment 0 < x < la uniform grid
Lo^ = {xi = ih, i = 0,1,... ,Af; hN=l}.

Classes of stable three-layer schemes 447
Let H = Q,h be the space of all complex functions defined on the grid
Cu^ and vanishing at the points s = 0 and x — I. Also, it will be sensible
to introduce
N-l
(y, v) = J2 Vk Vk h ,
J; = l
where Vf. is the complex conjugate function of Vf.. The next step is to write
the initial equation in the form
du . d'^u
dt dx
i -^^ + iqu
and introduce in the space H the operator
Ay = i Vsx - *' 9 « ■
The operator A is skew-symmetric: {Ay,v) = —(y^Av), since
N-l N-l
k=l k=l
N-l N-l
Yl yk (*' '^si-.J h-J^Vk (-«■ q v),, h = ~{y, Av)
k=l k=l
The norm of the operator A admits the estimate
Mil < ^ + 9-
The explicit scheme (82) for the Schrodinger equation takes the form
0
.2/° +«'fe.x- - «9y = 0, y e Ah ■
It is stable for r|| yl || < 1, that is, under the constraints
/ 4 X ^ h^
7^ + 9 r < 1 or r < — .
As the second possible example we look at the scheme
j/o + 2/0 = 0

448 Stability Theory of Difference Schemes
associated with the transfer equation
du du
ot ox
which is stable for r/h < 1.
Observe that the explicit three-layer scheme
ytt+Ay = 0
with a skew-symmetric operator A = —A* is absolutely unstable, while
the scheme with a self-adjoint operator A = A* > 0, as stated before, is
conditionally stable for
r2 2
E > — A or r < ,
(for Ay = —y^x we obtain r < h).
9. Other a priori estimates. Together with scheme A) the reader may be
confronted with schemes written in the form
(83) {E+T^R)y^t + Byo+Ay = ^, y{Q) = y^ , y{r) = y, .
Such schemes formally can be obtained from A) by replacing Rhy R = R +
E/t'^. With this substitution in mind, one can easily conclude that scheme
(83) is stable for R > A/A and write down the appropriate estimates.
Compound norms || y || arise naturally in connection with the energy
balance equation. Their structure seems to be rather complicated. It is
desirable to possess a priori estimates for solutions of problems A) and
(83) in the usual energy norms of the spaces Ha and Hr. We proceed to
the derivation of such estimates. This amounts to setting any three-layer
scheme in the form
D yjt + Byo +Ay = ip{t), 0 < i (E w^ ,
(84)
y(o) = j/o. y((o) = j/o,
where D = D(t), A = A(t) and B = B{t) are linear operators. In
particular, D = t'^R for scheme A) and D = E + t'^R for scheme (83).
Together with (84) we consider the problems
(84a) Dytt + Byo+Ay = 0, j/@) = j/^ , y0) = y, ,
(84b) Dyj^ + Byo+Ay = ^{t), j/@) = j/,@) = 0 ,

Classes of stable three-layer schemes 449
assuming that
(85) A{t) = A* {t) > 0 , D{t) = D* {t) > 0 , S(i) > 0 ,
(86) A(t) and D{t) are Lipschitz continuous in t with constant c.^ .
Theorem 6 implies that scheme (84) under conditions (85)-(86) and the
extra restriction
(87) D > '-^r^A,
where a number £ > 0 is independent of r and h, is stable with respect to
the initial data and a solution of problem (84a) admits the estimate
(88) ||y„+i||(„^<Mi||yi||(j^,
where Mj = Mi{c^,e,tg) > 0 is independent of r, h, n and
(89) ll>'n + ^ll(„)=Jl|j/n+yn + llll(,,,)
2
(90) \\yA\li, = \\\y. + yA\%^^,
+ [[D{r)-^^A{r)) 2/,@), 2/,@)) .
The constant Mi equals 1 if the operators A and D are independent of i.
In order to turn from (88) to the estimates in the spaces Ha and Hd,
we shall need yet some bilateral estimates of the functional || Yn+i ||.
Lemma 3 Under conditions (85) and (87) one has
(91) ll>'n + l||(„)<||y„IU(,„) + ||j/t(iJ||^(,,^),
(92) ll>'n + l||(„)>y^||j/„ + llU(,„),
(93) ll>'n + l||(„)> ^/^ (l|j/n + llUo„) + ll2/.(^.
'-DOn)

450 Stability Theory of Difference Schemes
Proof To make our exposition more transparent, it is convenient to deal
with
J = \\\y + y\\\ + {{D-"^A)y,,y,).
To prove inequality (91), we concentrate mainly on
J =]^{\\yf^ + 2{Ay,y) + \\y\\\)
-l{\\yf^-2{Ay,y) + \\y\Q+{Dy„y,)
= {Ay,y)+ M\l
with respect to a new variable y = y+ ry^. With this relation established,
we arrive at
J^\\y\\l+riAy,y,) + M\l<\\y\\'i + T\\y\\^ ■ \\ yA\ ^ + HWl ■
Condition (87) yields the estimate
ll2/.IL<;7^lkll..
so that
J<\\y\\l + ^^\\y\\^ -llj/JI^+lkll' <(l|ylL + ll%ll^)
This implies the first inequality of the lemma.
A simple observation that J — {^Ay, J/)+l|j/(lln justifies the forthcoming
substitution y = y — ry.^, leaving us with
J^iAy,y)-T{Ay,yt) + \\y^\\l.
Making use of the generalized Cauchy-BunyakovskiT inequality
(Ay,yt)<\\y\\^ ■ WvAIa
and taking into account (87), we obtain
J>\\y\\l-r\\y\\^ ■WvAlA+Mln
l-r\\yh ■\\yt\\A+\\yt\\l
>\\y\\l-~^^J\y\\A ■11%11^ + IIjaII^
^/TT^

Classes of stable three-layer schemes 451
Applying now the inequality ab < Sa^ + 6^/D5) yields
(94) ^>(l-^)ll^lli+(l-^)lk
S{l+e)J'"""D
Assuming the second coefficient to vanish, we deduce that 6 = 1/A + e)
and
The second estimate of the lemma is proved.
In order to obtain the third estimate, we require the equality of the
coefficients at the members ||2/|R and HyJI?, in (94); this yields
1 -S
x/TT^-i
Since \/l + e < 1 + £ for any £ > 0, it follows from the foregoing that
1 ^ ^ > 2(iT7) ^"^^
J>^^^i\\y\\'i + \\yi\\l)>X^^i\\y\\A + \\yi\\nY-
Thus, the lemma is completely proved.
Upon substituting (91)-(93) into (88) we obtain estimates for problem
(84a)
(95) ll2/n+ilL(,„) < ^^ \[^ (llj/olU(.) + UmUir)
(96) ll2/n+ilL(,„) + Il2/.ll^(,„) < 2Mi yi±^ (llyJI^^^) + \\ym\n(^r
In order to prove the stability of scheme (84) with respect to the right-
hand side, let us employ the superposition principle and seek a solution of
problem (84b) as a sum
(97) 2/„=Er5„^,, n=l,2,..., y„ := 0, s = l,2,...,
5 = 1
where g^ ^ as a function of n for any fixed s satisfies equation (84a) and
the initial conditions

452 Stability Theory of Difference Schemes
Since D > 0 and the space H is finite-dimensional, D > 6E and the
inverse operator D~^ exists (S > 0). As far as £? > 0 and D = D* >^ SE,
we have for a solution of the equation @.5 tB + D) lu = (p the estimate
\\M\d ^ ll^llc-'' *^o that
\\(9t)sjDit,) < WVsWo-'it,)-
By virtue of estimate (95) we are led to
Making use of (97) and the triangle inequality, we derive the estimate for
a solution of problem (84b)
(98) ik+iiu„)<M,yi±i x:-ii^ji^-(.j-
5 = 1
We summarize all the results obtained in the following assertion.
Theorem 9 Let conditions (85)-(87) be satisfied. Then scheme (84) is
stable with respect to the initial data and right-hand side and a solution of
problem (84) satisfies the a priori estimate
(99) Wvr.
+ ilU((„)
< Ml yi±^ ( bmUr) + \\ym\n(r) + E - W^^Wn-^iu))
its)] ■
s = l
Corollary If D = E + t^R> E and D'^ = E, then ||^J|^-i < ||^|| and
for a solution of problem (84b) the following estimate is valid:
s = l
Finer estimates that are similar to estimates established for the string
vibration equation (for more detail see Chapter 5) hold true in a more
narrower class of schemes
A01) Dy,, + Ay = p, 0 < t = nr < t, , y@) = Vo , 1^@)=^%.

Classes of stable three-layer schemes 453
In order to understand the nature of this a little better, we take for
granted that
A and D are constant operators,
A and D are self-adjoint positive operators.
Then under condition (87) for any scheme of the form A01) estimate (99)
is vahd with Mj = 1. By merely setting x = D'^l'^y and C = D'^I'^AD'^I'^
we reduce A01) to the form
A03) Xt, + Cx = (p, .■c@) = xo, .■c,@) == x-o .
Applying the inverse operator C~^ to scheme A03) yields
A04) C~^x^^ + x = C"V , ^^(O) = x^ , ^((O) = x„ .
Comparison with scheme (84) reveals the correspondences
With this in mind, condition (87) acquires the form
C-'>^-^t''E or E>^-^T^C.
~ 4 - e
Involving estimate (99) with Mi = 1 and taking into account that C
is a constant operator, we arrive at
n
A05) \K+i\\ < \l^-^ ( ||x@)|| + WxMWc-^ + 1^ r \\C-'^l
5 = 1
with X = D^l'^y and (p = D~^''^f incorporated. What has been done is to
derive the desirable expressions
||x,@)||^_, = (C-^x-,@),x,@))
= {D"''A-'D^I''D"''y,{id),D'l\j^{Q))
l|c-i^|p^=(c-v,^)
= (A-V,^) = IHI^.,
so there is some reason to be concerned about estimate A05) in the original
variables:
A06) Wvn+iWn < V^ (\\ym\n + II^^WIL- + E ^II^JL-) ■
^ 5 = 1 ''^
Thus, we have proved the useful assertion.

454 Stability Theory of Difference Schemes
Theorem 10 If conditions (87) and A02) are satisfied, then for scheme
A01) a priori estimate A06) takes place. In particular, for scheme A01)
with D = E and y^ = y^ = Q we have
(see Chapter 5, Section 6.2).
As an example consider the weighted scheme
y^^+A{ay + {l-2a)y + ay) =ip, A*=A>6E, 6>0.
Substituting y + crr^j/fj for ay -\- {I — 2GJ/ + <^y we obtain
A07) {E + aT^A)y^^ + Ay^^,
yielding D = E + ar'-^A. The stability condition D > ^t'-^A or E >
(A + £)/4 — a)T'^A is ensured by
^ l+£ 1
a > —-
r'\\A\\
For the explicit scheme with G = 0 this implies that
2
r <
y(i+s)iMir
Having stipulated this condition, the explicit scheme (y£f = y^^) for
the string vibration equation is stable for r/h < 1/y'A + e) (see Chapter
5, Section 6).
10. On regularization of difference schemes. Stability theory of difference
schemes outlined in this chapter may be useful for the statement of a general
principle (the regularization principle) providing with schemes of a desired
quality, that is, stable, generating an approximation and satisfying the extra
economy requirement of minimizing the arithmetic operations necessary in
computer implementations of resulting difference equations.
The economy requirement in the case of nonstationary problems in
mathematical physics generally means that the number of arithmetic
operations needed in connection with solving difference equations in passing
from one layer to another is proportional to the total number of grid nodes.

Classes of stable three—layer schemes 455
In studying the canonical form of two-layer and three-layer schemes
A08) {E + TR)y^+Ay = ^, 2/@) =2/0, B = E + t R,
A09) By.+T^Rytt + Ay=^, 2/@) = 2/0, 2/(r) = 2/1,
it has been disclosed that the operator R (regularizator) is responsible
for stability.
Sufficient stability conditions have now simplified forms
A10)
R> Gn A , G„ = ,, , ,, for two-layer schemes,
- "  t\\A\\
R > - A or R> A for three-layer schemes.
4 - 4
Stability or instability of a scheme from the primary family depends
only on selection rules for the operator R. From the point of view of stability
theory the arbitrariness in the choice of the operator R is restricted by the
following requirements:
• a scheme should belong to the primary family, that is, B = E + tR
for A08) and R= R* >0 for A09);
• conditions A10) should be valid.
To obtain a stable scheme of a desired quality, one is to construct
a scheme generating an approximation of the attainable order and being
economical, that is, it is required to solve the equations (E + TR)y = F for
A08) or {B + '2TR)y = F for A09) in a minimal number of operations (in
a certain sense).
First of all observe that if scheme A08) or A09) with an operator R
is stable, then so is a scheme with an operator R > R. Common practice
in designing difference schemes involves the development of a scheme which
generates an approximation of the attainable order and is economical. With
the indicated properties, its stability will be given special investigation.
The main idea behind regularization of difference schemes is that the
schemes of a desired quality should be sought in the class of stable schemes
starting from an original scheme and replacing it, by changing the operator
R, by another scheme of a desired quality belonging to the class of stable
schemes.
Many modes of constructing scliemes of a particular form can be
treated as simplest regularization modes. The canonical form of a scheme
is convenient not only for practical tests of stability, but also for proper

456 Stability Theory of Difference Schemes
evaluation of the order of approximation. The factor at the member R in
A08) is r, while in A09) is r^. Therefore, if in the case of two-layer schemes
the condition || Ru^. \\ = 0A) with such a variety of R continues to hold (u
is a solution of the initial differential equation), then the approximation
error changes by a quantity 0{t) when R changes. In the case of three-layer
schemes the condition || Ru£f \\ = 0A) ensures that the regularization leads
to schemes with the approximation error differing by a quantity 0{t'^). For
this reason three-layer schemes are good enough for the purposes of the
present chapter in designing stable schemes of second-order approximation
in r.
The main problem here is connected with selecting the regularizator
R. Since regularity conditions became operator inequalities, it seems
reasonable to choose as R operators of the most simplest structures which are
energetically equivalent to the operator A. Let, for instance, A and Aq be
energetically equivalent operators with constants j^ and j^, so that
A11) yiAo<A<y^Ao, 7i>0, 72 > 0 ■
Keeping then R = aAo we have at our disposal stable schemes: for a > cr^j^
(or a > ^72) in the case A08) and for a > 7,74 in the case A09).
A simplest form of R is the operator R — aE (Aq = E). Stability
conditions are satisfied if d > (To|| A || for scheme A08) and a > ||| ^ || for
scheme A09).
Example 1 The explicit three-layer Du Fort-Frankel scheme for the heat
conduction equation from Section 3.7 belongs to the family
A12) yo+aT'-'y£, + Ay=0, cr>|||^||, A ^ A* >0.
Indeed,
M . M 4 ^nh 4 1
Ayrr-Ay, ky = yg^, |m| = — cos — < -^ , o-= -^ ,
that is, the condition a > t||^|| is satisfied. This scheme provides a
conditional approximation of 0(h'^) for r = O(h^).
It is not difficult to write down an explicit stable scheme for the heat
conduction equation with variable coefficients
A13) i > 0, 0 < s < 1 ,
w@,i) = u{l,t) = 0, u{x,Q) = Ua{x).

Classes of stable three-layer schemes 457
In this case we agree to consider in A12) a = c.^/fr^, Ay = —Ay and
For the multiple heat conduction equation
A14) | = I:£(m^>0|^), . = (......xgcG,
w|p = 0 , u(x,0) = Ug(x) ,
G being a parallelepiped @ < s^^ < /„, a = 1, 2, . . . , p), 0 < Cj < A;„ < Cj,
other ideas are connected with
p , p
a=l « a=l
where /i„ is the step of a grid uif^ = js = (x■^ , . .. ,x ) E Gj along the
direction x^.
Let A(j — Ai + A2, where Ai and A2 are adjoint or "triangular" (with
a triangular matrix) operators, so that
{Aoy,y) = 2{A,y,y) =2(A2y,y).
Setting R = aAi or R = aA2 we arrive at scheme A08), which is stable for
a > 2j^2ag {j^ is a constant involved in A11)).
Example 2 The asymmetric scheme for the heat conduction equation
du d'^u
belongs to the family of "triangle" schemes having the form
(J T
A15) yt + — yg^= Ay, Ay = y^^. .
Here
Ay = -Ay , Ry=-y-, Aq = A , 7i = 7^ = 1 .
Scheme A15) is stable for d > 1 — |;^ and conditionally approximates the
heat conduction equation to 0{h) as r = 0(h'^).

458 Stability Theory of Difference Schemes
For equation A13) we accept
72 = C2 , Ay= -{aijg) , Ry = -y^
and take
With regard to problem A14) we thus have
p I
G > Cj fl —
or Ry =
2tAJ '
P I
p I
A-C.E;?
a=l «
It is important to note that while constructing the Du Fort-Frankel
scheme as an original scheme the explicit unstable scheme j/o = Ay
generating an approximation of 0(r^-|-/i^) has been taken with further modification
corresponding to the regularizator of the simplest type (R = -rjE, a = A-
in A12)).
Afterwards when the sweep formulae became customary, one began
to analyze in full details two-layer implicit schemes (weighted schemes) for
which R = a A. These schemes obviously represent a particular case of the
scheme with R = aAo.
We point out one more selection rule for R. Let Aq — Ai + A2,
A2 = A\ > 0. Choose R in such a way that the two-layer scheme possesses
the factorized operator
B^ {E+aTAi){E + aTA-2)^E+T{aAo + a''TAiA2),
so that
R= (JAo + f'^TA1A2 .
Since (Ai A2y,y) = (A22/, ^22/) = 11^22/11" > 0, this scheme is stable if
<^ > 72(^0 •
Various schemes with the factorized operator
B = {E + ioRi)iE + ioR2), R2 = R
are in common usage as iteration schemes for solving equations of the form
Ay - (p.

Homogeneous Difference Schemes
for Time-Dependent Equations
of Mathematical Physics
with Variable Coefficients
7.1 HOMOGENEOUS DIFFERENCE SCHEMES FOR THE HEAT
CONDUCTION EQUATION WITH VARIABLE COEFFICIENTS
In the present chapter the objects of investigation are various homogeneous
difference schemes for the heat conduction equation and a second-order
equation of hyperbolic type with variable coefficients in several settings:
on nonequidistant grids, with the boundary conditions of the third kind,
etc. The results obtained in the preceding chapters find a wide range of
applications in designing homogeneous difference schemes and establishing
their stability. Especial attention here is being paid to one-dimensional
problems.
1. The original problem. We begin by placing the first boundary-value
problem for for the heat conduction equation in which it is recjuired to find
a continuous in the rectangle Dt = {0 < x' < 1, 0 < i < T} solution to the
equation
A) ^ = Lu + fi.,t), Lu=-{ki.,t)-),
satisfying the initial condition
B) u{x,0) = Ua{x), 0<x-<l,
459

460 Homogeneous Difference Schemes for Time-Dependent Equations
and the boundary conditions
C) M@,i) = Wi@. "(!.<) = @. 0<i<T,
under the assumptions that the coefficient k{x,t) is bounded from below
and from above:
D) 0 < c^<k(x,t)<c,, {x,t)eDT,
where Cj and c, are constants.
Also, we take for granted that problem (l)-C) possesses a unique
solution with all necessary derivatives.
2. Homogeneous difference schemes with weights. In a common setting it
seems natural to expect that a difference scheme capable of describing this
or that nonstationary process would be suitable for the relevant stationary
process, that is, for du/dt = 0 we should have at our disposal a difference
scheme from a family of homogeneous conservative schemes, whose use
permits us to solve the equation Lm + / = 0.
One way of covering this for the heat conduction equation is to
construct a homogeneous conservative scheme by means of the integro-interpo-
lation method. To make our exposition more transparent, we may assume
that the coefficient of heat conductivity k = k{x) is independent of i. The
general case k = k{x,t) will appear on this basis in Section 8 without any
difficulties.
We proceed as usual. This amounts to introducing the following grids:
an equidistant grid on the segment 0 < x- < 1 with step h
Lo,^ = {x^ = ih, i=0,l,...,N, h=l/N};
a grid on the segment 0 < i < T with step r
Q^ = {tj =JT, i = 0, 1,.. . , VVo , T = T/No};
a grid in the rectangle Dt
^ftr =^ft X ^r = {(»i>^j). ^i&'^h, ^j&'^t]
and to forming
ui,^^ = ui,^ X ui^ = {(x',-,<j) , X-,; = ih , 0 < i < N , t^ = jr , 0 < i < Nq} .

Homogeneous difference schemes for the heat conduction
461
The starting point is the balance equation written in the rectangle
{^i-i/2 £ ^ ^ ^i+1/2}' ^j < ^ < ^j + i with regard to the governing equation
dt
Lu + f{x,t), Lu=Uk{x)^£)
The outcome of this is
'j+i
E) / [u(x,tjj^y)~u{x,tj)\dx= / [w(x-,;_^i/2,0-«^(^"i-i/2H]
dt
+ dt / f(.r,i)dx, w(xj:)=k
du
where the integrals and derivatives are yet to be replaced by
/ u{x,t) dx '^ h u{x^,t) ,
-1/2 "^ "i "«,i
'j + i
/ H^i-l/2,t}dt - Crw^+j/2 + A - cr) W^_j/2
dt / f{xj)dx '^ hripj. ,
where ss designates approximation, a is a numerical parameter and the
coefficients a,; are expressed through the values of k{x) for x'^_j < x < X;
by means of pattern functionals yl[A;(s)], — 1 < s < 1, so that
a(xj-) = a^ = A [fe(x', + sh)] or — = A
1
A;(x; + s/i)
Here yl[A;(s)] is linear nondecreasing functional, for which the conditions
A[l] = 1 and A[s] = -0.5 hold.

462 Homogeneous Difference Schemes for Time-Dependent Equations
From such reasoning it seems clear that the operator
Lu- -W-[k-K~)
ox \ ox J
is approximated to .second order by the difference operator Ku — [aug)^,
meaning
ku~- Lu^Oih^) if k{x)^C^^\
In so doing 0 < Cj < a[x) < c^.
Upon substituting the resulting expressions into E) and replacing u
by y we obtain the difference scheme for the grid function y{x^,tA:
F) yL^A = A{ayj + '+{l^a)yi)+^l,
1=1,2,... ,N^1, j>0,
where Ay = [ayg),..
The simpilest formulas
suit us perfectly for determination of cpl and «,,
When the difference equation F) is put together with the
supplementary conditions 2/° = Ug{x,i), j/^ = j«i(^j)i i/n — f'-ii^j)) there arises naturally
the difference boundary-value problem
G) y, = A{ay + {l^a)y)+^, xeto,,, t = JT>0,
y{x,0) = Uo{x), xeco,^,
y{0,t} = ^h(t), y(i,t) = ^i,(t), teu>,,
Ay = {ayg)^ , 0 < Ci < rt, < C2 ,
within the usual notations
y^yf = ij{xi,tj) , y = yi + ^ = y{xi,tj_^_^) .

Homogeneous difference schemes for the heat conduction 463
The next step is to reduce the resulting scheme to more convenient
forms in trying to avoid cumbersome calculations. The case G = 0, relating
to the explicit scheme, is simple to follow:
yj' + i = ?4 +r(Aj/| +ipl)
permitting us to find a solution on every new layer.
The case G^0, relating to the implicit scheme, is connected with the
equation related to the unknown y ~ y-''^^'
arAy — y=—F, F — y + T{l~~a)Ay + T(p,
which can be written in the augmented form
(8) Aiyi_,^Qyi+A,+,yi^^=^F, i = 1,2,... , N - I,
Ai = arajh" , d = Ai + Ai+y + J ,
f
+ /^(^"'^) («<2/':-i +ai+i2/i+i) + ^ f^
with the supplementary boundary conditions for i = 0 and i = N
yo= ^^i{ij + i)> 2//V =-«2('^j+l)•
The difference boundary-value problem associated with the difference
equation G) of second order can be solved by the standard elimination method,
whose computational algorithm is stable, since the conditions Ai ^ 0,
\Ci\ > \Ai\ + |A-+i| are certainly true for a > 0.
The recurrence formula
a a
is aimed at finding the right-hand side Fi = F^ of equation (8) with a
smaller volume of computations.

464 Homogeneous Difference Schemes for Time-Dependent Equations
The following implicit schemes are frequently encountered in the
theory and practice:
a) the symmetric scheme {a = 0.5)
Vt = 0.5 A {y + ij) + if ;
b) the forward scheme or purely implicit scheme (a = 1)
y^ = A J/ -b ¥>.
3. Stability and convergence. The general stability theory for two-layer
schemes applies equally well to the stability analysis of the weighted scheme
G). With this aim, the appropriate difference scheme with the
homogeneous boundary conditions comes first:
(9) yt = A{ay + {l^a)y) +ip, x&uj,^, t>0,
y{x, 0) = Ug{x), y — 0 for x- = 0 , s = 1 ,
All the tricks and turns remain unchanged: we, first, introduce the
0
space fi of all grid functions given on the grid lo/^ and vanishing on the
0
boundary for i- = 0 and .v — 1 and then define in that space -ff = fi an
inner product
and associated norm \\y\\ ~ \/(j/7y)- By means of a linear operator A
acting in accordance with the rule
Ay = -A y = -(ay^)^^ for y e H
the preceding scheme (9) can be rewritten as
A0) y,+A{ay + {l~-a)y)=ip, t=JT>0, y{0) = u, .
Observe the operator A so defined is self-adjoint and positive definite:
A*=A>0, 6E <A<AE, 6>0,
where
S= min A,.(yl), A = |M||= max XJA)
l<k<N-l '"^ ' " " 1<J;<A'-1 *^ '

Homogeneous difference schemes for the heat conduction 465
and Aj.(^4) is the A;th eigenvalue of the operator A. It follows from Green's
formula
iAy,y) =-((aj/j.).^,2/) = {a,{ygf]
that
0 0
Ci A < A < C2 A ,
0
where Ay = —y^x fo^" V ^ ^, with further reference to the relation
[Ay,y)^{{y,)\l].
0
The smallest and greatest eigenvalues of the operator A are given by the
fornuilas
0 4 .> TT h 0 4 .-, TT h
5 = ^siir~, A=^cos-~,
0 0
thereby justifying the estimates 6 > c^ S and A > c^ A ■
We know from the general stability theory that scheme A0) is stable
in the space Ha with respect to the initial data, that is,
l|j/^'IU<l|j/°IL if v' =0
under the constraint
1 1
G > Gj , G, = z~ 1 0 < £ < 1
In the case where
1 1
2 " 7a
a solution of problem A0) satisfies the estimate
A1) ll?/ + ^IU<l|j/°IL + 4^(Er||^"|p) .
j'-o /
For the explicit scheme (G = 0) we might have
A2) yi + ' = A ~ -^ (a, + ai+i))y^ + ~ {a-yl_^ + a,^,y{) + r^^ ,
where the coefficient at the member yj is non-negative, provided the
condition
, 2r
l-^c,>0

466 Homogeneous Difference Schemes for Time-Dependent Equations
holds. Under this condition the relation
l|j/^' + 'llc<IIJ/''llc + llv^llc
takes place. Summing up the preceding other j = 0,1, 2,.,. , we obtain the
inequality
A3) llj/^'+'llc<l|y°llc + EHl^^'llc.
i'=o
which expresses the stability of scheme A0) in the space C if
This condition is only sufficient for the indicated property. A necessary and
sufficient condition for the stability of the explicit scheme with respect to
the initial data in the space Ha is
2 4c
A5) T<—, where A < —^ .
But it may happen that the coefficient k{x) varies very fastly. In that
case the estimate A < '^c^/h^ is too rough and condition A4) gives a severe
restriction. In mastering the difficulties involved, we are forced to apply
the maximum principle to the weighted scheme (9) written in the canonical
form for any a:
A6) (l + ^ {a, + a,^,)L+^ = ^{aM^l + a.+Mtl) + Fl ■
A — (T) r / ■ ■ ^ ,:
+ —p— («^j^-i + «»+iJ^+i) + ^n-
The boundary conditions j/@) = y(l) = 0 together with Theorem 3 in
Chapter 4, Section 2 for equation A6) give us the estimate
lhy + ^llr;<ll^'llc-

Homogeneous difference schemes for the heat conduction 467
The coefficient at the member yj becomes non-negative for
which assures us of the validity of the relations
ll^^^llc<ll?/llc + Hl^^llr
and
A8) y+'\\c < \w\\c + ^ w^'Wc < \\y°\\c + E ^ ii^'"iic ■
Summarizing, the weighted scheme (9) is stable in the space C, provided
condition A7) holds. For the purely implicit scheme with a — 1 estimate
A8) is valid for any value of r.
The accurate account of the accuracy is stipulated by more a detailed
exploration of the residual
ip — A (a u + {1 — a) u) + If — u^
on the solution u of the original problem (l)-C). Upon substituting the
expansions
U + U T „ T -r ^ Os
Uf - u + 0(r^)
with the members
i+i/2 ■ <9w
dt
it is plain to calculate the residual
. u + u
^ - A —~ \- (a - 0.5) r Atij + ip -
= (Am + /-u)
+ (Au - Lu) + (ip- f) + (a- 0.5) T All + ©(r^)

468 Homogeneous Difference Schemes for Time-Dependent Equations
Taking into account the relations Lu + / — « = 0 and
A9) Au^ Lu + 0{h^), (fi = f + 0{T^+h^),
we arrive at
B0) i>^ (a-0.5)TLii + O{T'^+ h'^).
This supports the view that the order of appiroximation for a given value of
a coincides with the order of approximation established before for the unit
constant coefficient A;(x') = 1:
i> ^ 0{t'-^ + h'-^) for G = 0.5,
tP^Oir + h-^) for aj^O.b.
Along these lines, the error of approximation z^ — y^ —u^, where y^ is
the solution of problem G) and u — u[x,t) is the solution of the governing
problem (l)-C), can be most readily evaluated with the aid of the equation
Zj = A {az + A - (t) z) + V", X EiOh , t - jr >0 ,
z(a;,0) = 0, z@,i) = 2A,0 = 0.
Some progress can be achieved by having recourse to relation A1), leaving
us with the estimate
B1) ||.^' + '|U<-^f^r||^^'| "''
2^ V=o
Using the results obtained in Chapter 2, Section 3, namely the relations
ikii,^< 1A, (.,J]'/' = 1^A77)
and the estimate Cj \\z\\c < ||-iK; w<? derive the inequality
A
B2) ik;-iic<2^(EHi0Mp'^'^
thereby justifying by virtue of representation B0) that scheme G) converges
uniformly with the rate 0(/i^ + r'"'), where
2 for G = 0.5,
1 for G 7^ 0.5 ,
provided that the conditions hold under which the describing scheme is of
accuracy 0(/i^ + r'""') and G > G^ for 0 < £ < 1. A similar conclusion was
drawn in Chapter 5 for the relevant equation with a constant coefficient
k{x).

Homogeneous difference schemes for the heat conduction 469
4. The equation with a discontinuous coefficient of heat conductivity. In
the general setting the convergence of the weighted homogeneous scheme
G) is of our initial concern, A case in point is that the coefficient k(x) has
a discontinuity of the first kind on the straight line s = ^ in the plane {x,t).
Some consensus of opinion is that the usual condition of conjugation
B3) M = 0, [A:^]=0 for x = ^, t>0,
L ox J
is fulfilled on every discontinuity line. In the physical language, this is a
way of saying that the temperature u{x,t) and the heat flow {—ku') are
continuous.
By relating the functions k[x), f(x,t) and a solution u[x,t) to be
sufficiently smooth everywhere except this discontinuous line we proceed to
evaluate the residual (the error of approximation)
-ip = K[a u -\- {I ~~ a)u) -\- (f ~~ u^
,(-)
a u + (I ~~ a)u
Let now S, = x^ -\- Oh, x^ — nh, Q < 6 < \, n >^ 1. As far as a three-point
operator L is concerned, we might have
B4) Vi = 0(h^ + r"') for all i 7^ n , i^n + l.
On this basis it remains to calculate V'i for i = n and i = -n + 1. Also, it
will be sensible to introduce
B5) h V-n = w[% - w\^'^ +h(p„~~h Ut„ , Wi = a Wj ,-.
Since
for k e C'^^^[2;,>_.,, x;,] and u G C^'^^[xj_i,x^], we obtain
B6) w„^(ku')„_,/, + 0(h')
= {ktiX,,-{e + o/o)h{kti')l,^ + o{h'),
where v^^J■f- = v{^ — 0,t). By the same token,
B7) h V-n+i = w[^^2 - «^i+i + h ^„+i - h Mj „_^i ,
B8) r«„ + 2 - (« «,x^)n + 2 = {k «')nght + (^-^ - 0) ^ {k ti')' + O(h') ,

470 Homogeneous Difference Schemes for Time-Dependent Equations
where t;^^!,^ = v{^ + 0,t).
Furthermore, using the expansions behind
u„ = u
@-Shiil,^ + Q.be'h'ti'(^^^ + O{h')
«„+! = «@ + {^-e)h <g,, + 0.5 A - 9f h' <g,^ + 0{h^).,
we find that
w.
a„,, u.
n + i — "n + 1 "'.f.n + l — "n + 1 V' "left
a„
•u
U + (i-'^)<.ht)+o(/0
As a corollary to the condition of conjugation [ku'] = 0, we might have
(^ «')left = (^ «')i-ight = "^(?. 0 . «U
ii.t)
left
W{^,t)
right ^
ght
SO that
B9)
+ ^^\iv[^,t) + 0{h).
\ ^left ^i-ight J
Upon substituting B6), B8) and B9) into B5) and B7) we finally get
e i-e\
«n+i 7r- + i— I -1
'>{Li):
'•left '^right /
h {i>n + i'n + l) = ■4 + 2 - 'wi''' + h ((fn + ^„ + i) - h (Mt_„ + i + Mt^„)
= A.5 - (?) ft ((fc u');.ight)^''^ + @.5 + ^) M(fc uOieft)'"^
+ h((fi^ +¥>«+!) -/^("t,„+l +'Ut^n) + 0{h^),
showing the new members to be sensible ones. The limiting values emerging
from equation A) are taken to be
(^"Oieft = («-/)..=?. (^"'):,ght = ("-/)-?.
since [it] = [/] = 0 for x = ^. The next step in this direction is to insert
the assigned values in the preceding expansions with further reference to
lb + w
fr
= /„ + 0{h^ + t') , .&(-) = '^^^ + {a- 0.5) r m

Homogeneous difference schemes for the heat conduction 471
making it possible to find that
= 0{(a -0.b)Th + T'h + h^)
and deduce for any weighted scheme of the form G) that
C0) hij, = 0{l),
C1) h {ij^ + ij^^,) = (cr - 0.5) 0(r h) + 0{t' h + h').
Under the special choice
(=^> »« = «<"=(/iK^TII)) '
it is straightforward to verify by analogy with the available procedures (for
more detail see Chapter 3, Section 3) that
Q,, = 0{h), hi,^ = 0(h).
From such reasoning it seems clear that the following estimates are valid:
C3) /z,^,-„ = 0(i),
C4) h @f,„ + V^f.„+i) = (cr - 0.5) 0{r h) + 0{t'' h + h'),
thereby justifying the representations
C5)
r = 0{h'^ + r , V-; = 0 for i^n, i^ n + 1
where S^ „ is, as usual, Kronecker's delta.
The intervention of a new grid function
2 — 1 0
Vi= H hi>,., ??i = 0, f = 2,3,.,. ,Af,
k-l

472 Homogeneous Difference Schemes for Time-Dependent Equations
arising from the representation
0
C6) V-" = Vie + 'V* . that is, ip = rj,^ ,
gives rise to a perfect model for a more simpler estimation:
,,^^. ' = / ° '^°^" * < " + 1 ' ^n + l = '^ 'I'n .
I h{^ip„ +'(/'„+i) for i > n+ i .
The order of accuracy of scheme G) can be most readily evaluated
with the aid of the representation for the error z — y — u as
Z = V + z" ,
where v and z* are solutions of the related problems:
0
v^ = k [a V + {I -■ a) v) + ip , Vg = vj^f ::^ 0 , v{x,0) =0 ,
z; = A {a z" + {I - a) z") + 4'\ z*^ = z*^ = 0 , z*(,t;,0) = 0.
In the accurate account of v and z* we apply the results of the general
stability theory (Theorems 9 and 11 from Section2, Chapter 6):
C8) \\v' + '\\<\\A-'r\\ + \\A-'r\\
-A
+ 22 ^11-4"'-^'f II for a>a,,
3
C9) lk*^ + Ml< E^ll'^*'"ll f°i' ""^""o, cr>0.
j'=o
0 , 0
Since A"^ rj) = C is a solution to the equation AC^ = ip = rj^., we have
occasion to use the relation
D0) \\A-'i>\\<^ih\v\],
the right-hand side of which can be modified on account of C7) into
N
(l,h|]= Yl /^l%l = /^'l'^nl + /j|'^n+'^„+ll(l-^„ + l)-
i=n+l

Homogeneous difference schemes for the heat conduction
473
By substituting estimates C0) and C1) into inequality D0) and, in turn,
0
estimates C3) and C4) into a similar inequality for ||A~'i/' || and then
applying estimates C5), C8), C9) and the inequality || z || < || z* \\ + || v ||
to the resulting expressions we establish the convergence of scheme G) in
the class of discontinuous coefficients,
Let k[x) possess a discontinuity of the Brst kind for
X ^ ^ and conditions B3), B4), C0) and C1) hold
simultaneously. Then for a > a^, a > 0, scheme G)
converges in the norm of the gnd space L-j with the rate
0{h'^ + r"'+''), while the best scheme with coefficient
C2) does the same with the rate 0{h^+7"^+"); m„ = 2
for a — 0.5 and ni„ = 1 for a ^ 0.5,
A priori estimates obtained in Chapter 6, Section 2 provides the
sufficient background for a> uniform estimate of accuracy in the norm of the
grid space C such as
D1)
ll^^+ML<lh0°IL- + lh0^'IL-
D2)
+ EHr^i'lU- for a>
y'=i
Y^rwr^'
1/2
for a > a^
V'=o
where \\^\\\ = D;^,-;) = (a, (-j)"] and tlie inequalities (see Chapter 2,
Section 3) are taken into account:
\VA\c < 1A. i^,n"^ \\A\\ = («, (%)'] > q A, (%.)'] .
In conformity with Chapter 2, Section 4 we deduce that
1/2
I'^l
1^.11.-^ < 4
M^"
Vi

Tn + 2
474 Homogeneous Difference Schemes for Time-Dependent Equations
which in combination with C7) leads to
= h (hxP,,)'' + A - X-„ + i) (/l (V-n + i>n + l)Y,
thereby justifying that
IIt^I] = 0(V7r+r"') for any scheme G) ,
IIt^I] = 0(/i3/2 _^ T-"^.^ for the best scheme with coefficient C2) ,
Having stipulated the same conditions B3), B4), C0) and C1) as
before, scheme G) converges uuiformly with the rate 0(\//i + t'"'):
\\z'\\a^\W-u^l. = 0{s/h + r^-^).
This estimate can be improved for the forward difference scheme with
G = 1 by means of the maximum principle and the method of extraction of
"stationary nonhomogeneities", what amounts to setting
-J _ yj _j_ ^y] ^
where w^ is a solution to the equation
D3) Aw=°iP=7]^,
so that
M\c<^ih\v\], \H-\\c<^i^Avi\]-
Here v^ needs to be recovered from the relations
D4) Vf + Av = xp , ijj ='4>* - Wf , ■yB;, 0) =-w(;i-,0) .
In this regard, the maximum principle with regard to equation D4) yields
ll'^^'+Nlc;<IK'°llc + E^(llV^*'"llc + ll"^fllr)-
j' = i
By inserting here the estimates for ||'t«||,-,, ||'ff(|L- and ||'i/'*||r. we arrive at
||t;^'||^ = 0{t + h) and, hence, \\z^\\(j = 0(t + h). For the scheme with
the coefficient specified by C2), it retains the order of accuracy in the
class of discontinuous coefficients as occurred before in the stationary case:

Homogeneous difference schemes for the heat conduction
475
5. Homogeneous schemes on nonequidistant grids. There is no doubt that
the users come across uonequidistant grids in both x and t variables in
practical implementations of some or other problems. All the preceding
results and estimates remain valid for the two-layer scheme being used on
a nonequidistant in t grid. That is to say, the nonuniformity in t has no
considerable impact in such matters and it should be taken into account
only in the selection rule for the step r = r^ depending now on the subscript
j. The order of approximation in t remains unchanged, but the symbol
0(r™) will stand either for 0(r™) or for 0(( max r,-)™), where there is
^ ^ i<J<Jo ^
uo danger of confusion. The situation in which the grid is nonequidistant
ill X needs investigation by exactly the same reasoning as before.
Let
= {.
0,1,
N, Xg = 0, Sjv = 1} be an arbitrary
grid on the segment 0 < x' < 1 with steps h^ = x^ — x^^-^, i = 1,2,... , N.
In line with established priorities from Chapter 3, Section 4 the operator
Lu =^(k —
dx V dx
IS approximated by the difference operator
A 2/ = (a rjs
-liy+i-Vi) ai{yi-yi_^)
h.
i+i
k
with the same coefficients a^ as was done on equidistant grids. This is
acceptable if we take, for example, under the constraint —1 < s < 0 (see
Section 2)
D5)
ai-= A[k{xi + shi)] or
1
A
1
k (x^ + shi
The right-han-d side is calculated by the simple formula
D6) ^^^^■=/(x,,i^.+o.5) = /(x-,,i),
assuming f{x,t) to be a continuous function of the argument x. When
some such function may have discontinuities of the first kind at the nodal
points, the obstacles involved can be avoided by setting either
D7)
V;
2h,

476 Homogeneous Difference Schemes for Time-Dependent Equations
where /,±o = /(a-'iio.O. ov
1^1 + 0,5
D8)
Vi =^i
h,
f{x,t)dx .
••i^i-O.S
In order to clarify the main idea behind this approach, we turn to the
scheme with weights
D9) yt=A{ay + {l-a}y)+ip, xe^;,, Q<t=JT<T,
y{x,0) = y„(x) , xeui,^, y{0,t) = u^(t), y{l,t) ■= u^{t) ,
Ay = {a{x) yg)^, 0 < q < a < c, ,
whose augmented form is suitable for subsequent calculations. The
elimination method unveils its potential once again and permits us to find 2/^ + -'
on every new layer t = iy_|_j in terms of a known value y^ on the current
layer t = tj'-
Ai jji^-, - a y, + Bi yi^, =-Fi, i = 1,2,... , N - 1
A,. =
h.: hi
Bi^
-'i+i
Fi
A - G)r (tti
a = A, -F Bi -F 1
i+i
hi h
i+\.
A-Cr)
flijt-i , a-i+i Vi+i
hi
h.
+ 1
Under such a choice of the computational algorithm the accuracy of
scheme D9) will be given special investigation. We are going to show that
it converges uniformly with the rate 0{h^ + r'"") in the case of smooth
functions A;(;c) and f{x,t).
For this, we proceed as usual. This amounts to evaluating the error
zj = y'- — u\, where j/] is a solution of the original problem (l)-C) and

Homogeneous difference schemes for the heat conduction 477
u{x,t) is a solution of the preceding difference problem. Upon substituting
y = z + u into D9) the complete posing of the problem for the error z is
zix,0) = 0, xELo,^, z@,i) = z(l,i) = 0, 0<t = JT<T,
E0)
2(a;,0) = 0, x-ew;, , z{0,t) = z{l,t) = 0, 0<t=JT<T,
where
E1) ipix^t) = A[au +{1 - a)u)+(fi - u
is the error of approximation of problem (l)-C) by scheme D9). In such a
setting the balance equation on the segment x^_q 5 < ^ < ^i+o 5 with the
ends Xj_Q 5 = Xj — 0.5 /),- and Xj_^_q g =: x.^ + 0.5 /ij_,_j gives
■^'j + tl.S ■''i + OJi
E2) / -^(x,i)dx = w(Xi^^y5,i)-w(x^_or^,i)+ / f(x,t)dx.
Here
w{x,i) = k{x) —{xj.).
Some modification of the residual ^ is possible with minor changes.
Dividing both sides of identity E2) by fij = 0.5 (/?.,• + Aj_|_i) and then substracting
the resulting expressions from E1) reveal
where v = v(xi_^ ^,i) and
•^■j + 0.5
i>z = i^i-V'i + ^ / -g^(x,t)dx-Uti.
Using an expansion arising from Chapter 3, Section 4 such as
and allowing an alternative form of writing
E3) 0 = 7,^ + V* ,
E4) V{=a,4']'-iku'tljl+fii^-f')
i~-l/2

478 Homogeneous Difference Schemes for Time-Dependent Equations
with ii' — d'^u/dtdx and /' = djjdx incorporated, we arrive at
0; = 0(fi2 + r^) .
It seems clear that representation E3) for the residual ?/> is an
immediate implication of the balance equation E2) by observing that
{k «'),^^/^ = {k u')^ + 0.5 ft,+i {k u')[ + I hl^ {k u')': + 0(ftf^ J ,
{ku')^^^^^^{ku')^-Q.^h,{ku')\ + \h]{ku')'l + 0{h^)
and, hence,
{ku')' =Uku') 1 --}-[h\(ku')"-hHku')"]+0{h').
By virtue of the relations
{ku'y = u-f, {ku')" = u'-f'
the desired result will be substantiated if we succeed in showing that
i [h^^, Vi - e V,-) = (ftf t;,_.i/2),,,, + o{hi).
An alternative form of the residual
4, = Au'-"'^ + ^-Ut - {{k u')' + f - ii) ^^^
is best suited for our purposes in the applications of formula E4), thereby
justifying the final results
7] = 0.5 a (m^ + Ug) -Wu' + T{a- 0.5) a u^^ + 0{h^) = 0(h'^ + r'"') .
The next question we have raised above is the accuracy of scheme D9),
the accurate account of which can be done using a priori estimates of the
problem E0) solution with further reference to the special structure E0) of
the right-hand side of ih. Following established practice, we introduce the

Homogeneous difference schemes for the heat conduction 479
0
space fl = H of all grid functions given on the grid w^ and vanishing at
the points x" = 0 and x = I and then define several inner products by
N-l N-l N
i — y i~l i~\
In this direction we refer to the operator A : H i-^ H with the values
Ay ■= —A y = —(a y^),,-. for any y £ H .
We know that the operator A so defined is self-adjoint and positive definite:
A = A*>{),
since [Ay,v) = (ay^^v^ = {y,Av). By the same tol<;en,
{Ay, y) ={ay^, t/j,] > c^ (y^., ty^] ,
1 1
\\y\\c = max \y^\ < 7. B/s,2/s]'^'' < ;^^= {Ay^yfl^ .
0<'i<N Z Z w^Cj
By obvious rearranging of the problem E0) solution as
Z = V + W ,
where t; is a solution of the same problem with another right-hand side
ip = % and w is a solution of problem E0) with the right-hand side ip = ip*.
As can readily be observed, scheme E0) is stable under the constraints
where hr, = min hi.
" l<i<N
other ideas are connected with a priori estimate D1) for v and
estimate D2) for IV. When providing such manipulations and establishing the
relation
M M i ■■ ^.

480 Homogeneous Difference Schemes for Time-Dependent Equations
where ||??]| = ^/(l, if] and H^H < \\v\\ + \\iu\\, we get for a > a^, a^
0.5 - A - £)/{t\\ a II), the estimate
E5) lk^' + '|lc<^ Ih
2y^^
Vj'=o
In what follows it is supposed that A;(x'), f[x,t), Ug[x), u^{t) and u.y[t)
are smooth functions and the conditions under which
E6) ,, = 0(|/i|2 + r'"'), ,,^ = 0(|ft|2 + ^m.)^ ^* ^0(|;,|2^^2^
hold. Then scheme D9) converges uniformly on any sequence of nonuniform
grids {loj^} with the rate 0(r'"' + |/ip), where |ft| = max /i,, if d > G^,
l<j<iV
This fact follows immediately from the combination of estimates E5) and
E6).
We touch upon briefly the convergence of scheme D9) in the class of
discontinuous coefficients and will pursue some analogy with the stationary
case which has been considered on the same footing in Chapter 3, Section
4 under the following assumptions:
(a) the functions k{x) and f{x,i) may have only a finite number of
discontinuities of the first kind on straight lines parallel to the
coordinate axis Ot;
(b) the grid Qj^ = iD;,(/\) is chosen in such a way that all of the
discontinuity lines of the functions k{x) and f{x,t) will pass through the
nodal points;
(c) the functions k{x), f(x,t) and u[x,t) have all necessary derivatives
in the regions lying between the discontinuity lines so that formulas
E3)-E4) and estimate E6) are still valid at all the nodes of the
grid W;,(A').
We note in passing that f^ is determined by formula D7), due to
which scheme D9) converges uniformly with the rate 0(r™' -|- |ftp) under
the aforementioned conditions on sequences of special grids.
Remark 1 Convergence in the norm of the grid space L2 occurs with the
same rate if condition c) is relaxed:
E7) ^ = 0(|ft|2 + ^™.), ^* ^ o{\hf + T^).

Homogeneous difference schemes for the heat conduction 481
This fact follows immediately from a priori estimate D2):
1 / ^ V
\j'=l /
Remark 2 Uniform convergence with the rate 0(|ftp + r) of the forward
difference scheme with G = 1 can be established by means of the maximum
principle and the reader is invited to carry out the necessary manipulations
on his/her own.
6. An one-point heat source. Of special interest is the nonstationary heat
conduction problem in the situation when a heat source is located only at
a single point s = ^ under the agreement that at this point the solution of
problem (l)-C) satisfies the condition of conjugation
E8) [u] = 0 .
k ^\ - -Q for X = £^
ox i
where Q = Q{t) is a power of the heat source.
The discontinuity condition of the heat flow
oxi
is to be understood as the discontinuity property of the first derivatives
dii
k 7^. That is to say, the solution u = u(x,t) has a week discontinuity on
ox ^
the straight line x = ^ by relating at the same time the coefficient k(x) and
the function f{x,t) to be smooth enough.
By means of the integro-interpolation method it is possible to
construct a homogeneous difference scheme, whose design reproduces the
availability of the heat source Q of this sort at the point s = ^. This can be
done using an equidistant grid ujj^ and accepting £. = x^-^ + 9h, 0 < 0 < 0.5.
Under such an approach the difference equation takes the standard form
at all the nodes x^ ^ x"„ (f 7^ n). In this line we write down the balance
equation on the sagment a;,j_w2 < ^ < ^'n+1/2 f°'' fixed t = t = ^j+o,5-
With the aid of the relations
»'n + 0.5 4" -i^n + O.S
/ {ku')'dx= j {ku')'dx+ j iku'Ydx
^'n-0.5 ■■'-'n-U 5 C
= ku'iy-"-"^ + [ku'] = W„+i,2 - 1«„_W2 - Q ,
■^n + 0.5 ' '

y^= K y^"') + (p + -^ for x = x^ ,
482 Homogeneous Difference Schemes for Time-Dependent Equations
vve arrive at
^n + l/2
E9) / —{x,t)dx = w{x„_^_^/■-,,t)~w{x„^y2,i)~Q{i) + h'P„.
■'■n~-l/2
The usual transition to a difference equation leads to
q([)
h
where Ay = (ay^)^.
Summarizing, the difference scheme for the problem described by A)-
C), E8) is of the form
F0) y^=Ayi-) + ^+j^Q{tNi_^, 0<x = x, <1, t = t^>0,
2/o = )«i . 2/iv = )«2. y{x,0) =Ug{x) ,
where S^ „ is, as usual, Kronecker's delta.
For the error z = y — u we have equation with the right-hand side
F1) ij = Au(^"^+^+j^Q{t)S,^,,-u,
and the boundary conditions
F2) ^0 = 0, 2iv = 0, z(s,0) = 0.
On account of the balance equation E9) the residual is representable by
F3) V = '?:r + '0* > V = a u^:"^ - K'
F4) ri=Vi-k-Ut.^"J^ J -^{^,i)dx\.
Here we adopt v^ = v(Xj_q 5,f) as was done before. For the sake of
.simplicity we take 9 < 0.5, that is, a;,, < ^ < x„_^_^,2, permitting us to deduce
that
,;. = 0(ft2 + r"') for all i^n+l,
4'* = 0{h-^ + t'-^) for all i ^ ?z.

Homogeneous difference schemes for the heat conduction 483
By analogy with the case of discontinuous coefficients we find that
%,n + l = 0 «U + i^-0) «nght + 0-5 /^ [A - Of <ght - 0' <J + Oih') .
Since a„_^i = k{^) + @.5 - 0)hk'(^} + 0{h'^), we might have
a„+i Wj,„+i = dwy^jt + (!-(?) w,,ght + 0(/i).
With the aid of the relations
'"^nght - 'f«left = -Q . (^ "')„ + l/2 = «^nght + ^ @.5 - 6*) (fc U )',,^y^^ + 0(/)^)
we establish
F5) ,?„+, -(?Q + 0(ft)-0(l)
by observing further that
F6) < = 0(ft).
The accurate account of the error z can be done as in Section 4,
0
leading to the same rate of convergence. No progress is achieved for a = a
in line with approved rules, because the choice of the coefficient should not
cause the emergence of a higher-order accuracy. From the formula t?„_|_i =
9Q+0(h) it is easily seen that i]n+i = Oih) and, hence, || z || = 0(/z^-|-r'"')
[{0 — 0, meaning that the heat source is located at one of the nodal points.
This guides a proper choice of the uonequidistant grid uij^{Q) so that
the heat source will appear at one of the nodal points. When this is the
ca,se, scheme F0) converges uniformly, on the same grounds as before, with
the rate 0(|Ap + r'"'). But a special choice of the coefficients a, given
by the formulas of the truncated scheme with second-order accuracy (see
Chapter 3, Section 7) improves our chances of constructing the difference
scheme of accuracy^ 0(|/ip + r"*') for any 6 (E [0,1], that is, disregarding to
the possible locations of the heat source.
7. A concentrated heat capacity. We now consider the boundary-value
problem for the heat conduction equation with some unusual condition
placing the concentrated heat capacity Co on the boundary, say at a single
point X' = 0. The traditional way of covering this is to impose at the point
X = 0 an imusual boundary condition such as
F7) Co -^ = k^, .T = 0 , Co = const > 0 ,
ot ax

484 Homogeneous Difference Schemes for Time-Dependent Equations
thereby completing the statement of the problem under consideration:
rjli
F8) — = Lu + f{x,t), 0<x<l, t>0,
Co-^ = A; — for x = 0, u{l,t) = 0, u{x,0) = Ug{x) .
The design of a homogeneous difference scheme necessitates
approximating the boundary condition at the point s = 0.
The first step during the course of the integro-interpolation method is
to rely on the balance equation, say in the rectangle {0 < x < x"w2 = 0.5/),,
tj '£i < ^7+1 }i leading to
•'^■1/2
/ [u{x,t,'ij^^) — u{x,t,j)\dx
0
«j + l «i + l •'^1/2
= / [w{x-^i2,t) - w{Q,t)]dt + / / f{x,t)dxdt,
where iv(x,t) = k ——, The next step is to substitute here
ox
wiO,t)=(k^) =Co^@,i)
v oxJ i:=o at
dxJx=o dt
and take into account that
Co -^@,t) dt = Co (m@,«, + i) ~ u@,tj)) = Cu r «; 0
Then upon replacing the integrals in x by the simplest expressions O.bhUg
and O.bh fg and the integrals in t by the expressions tw]"! and rfg , the
difference boundary condition is taken to be
F9) C2/,.o = a,2/i.';^ + 0.5ft4''), C=Co + 0.5/i.

Homogeneous difference schemes for the heat conduction 485
In this way, the difference scheme
G0) ijt = A y'-"^ + ifi, 0< x = ih<l , t =JT>0,
C'2/(,o = ai2/i-'^o+0.5/j4''\ i>0, t/w = )«2 > y{x,0) = u„{x) ,
is responsible for problem F8). It is plain to recover from the condition at
the point x = 0 that
,_^, . „ aa. T
G1 y, = x^y, + iy^, "^i = ^. -. STrT '
V, = h [0.5ftr4'') + ai A ~ cr) r2/,._o + Cy]/{Ch + a.ar) ,
yielding 0 < >fj < 1 for d > 0. In turn, the boundary condition of the first
kind is imposed for i = N:
G2) yj^, = /.jj .
With these, for determination of y^ = yj we obtain a second-order
difference equation supplied by the boundary conditions G1)-G2) that can
be solved by the standard elimination method.
The intuition sugge,sts that in such a setting the governing difference
equation and the boundary condition at the point s = 0 have one and the
.same order of approximation 0(r™' + h'^). To make sure of it, it suffices
only to evaluate the residual
^.-C«,,o-«i«S-0.5 ft/('').
Substituting here the expressions
Ci Wa-,0 = (a «s)i = {k u')i/2 + 0{h^)
= (ku'\ + 0.bhiku%+Oih^),
(A:u')o = C'o'Wo . (^w')o = «o -./o .
yields
ij,=C («,,o - ^i"^) + 0{h') = 0(r"' + h')

486 Homogeneous Difference Schemes for Time-Dependent Equations
as required.
For the error z = y — u we may set up the related problem
G3) Zt = Ay^'^'^+iij , ip = Au^"^ + ^- Ut, 0 < x- < 1 ,
C «( 0 = «! 4"o + '^u , ^N = 0 , z(x-, 0) = 0 .
One obvious way of proceeding is to introduce the .space H of all grid
functions defined on cof^ and vanishing for i = N. Under the inner product
structure
b. "^j - E ViVih + O-bhyaVg
8=1
we refer to operators A and D acting in accordance with the rules
iAy% = ^iAy), for 0 < f < A^, (Ay), = ^^^ ,
iDy)i = y, for 0 < i < Af, (DyX = ^Cy,.
With the detailed forms in mind, problem G3) can be recast as
G4) Dzt+Az'-''^ = 4>, t = JT>0, 2@) = 0,
showing the new members to be sensible ones. All this enables us too write
B = D + arA.
From the general operator theory outlined in Chapter 2, Section 4 it
seems clear that the operator D is ,self~adjoint and positive definite:
D > c^, E , where c^ = min A,2 C'/h) .
However, this is certainly so with the operator A:
A = A* >0.
What is more, the operators A and D are commuting: AD = DA. Due to
the,se properties the stability condition for scheme G4) is expressed by
B--0.bTA = D+ +(G-0.5)ryl> 0,
which is valid only for
More a detailed proof of convergence of this scheme is concerned with
the form G4) and a priori estimates obtained in Chapter 6, Section 2 and
so it is omitted here. As a final result we deduce that scheme G0) converges
uniformly with the rate 0(r™' + h'^).

Homogeneous difference schemes for the heat conduction 487
8. The case when the coefficient of conductivity k depends ont, k = k[x, t).
So far we have preassumed for the sake of clarity that the coefficient of
conductivity k depends only on one variable ;c.
In a common setting the governing equation A) of the general form
dt dx
(k{x,t)^^+f{x,t), 0<c,<k{x,t)<c,
is put together with the boundary and initial conditions B)-C). In
preparation for this, the intention is to use instead of F) another difference
scheme
G5) Vt = A{i) y^^^ + <p ,
where k{t)v = (a(
•^]0%) 1 ^ " ^j+0,5 ^'^'-1 ^h^ coefficient aB;,i) for fixed t
appears by exactly the same reasoning as before (see Sections 2 or 4 of the
present chapter). The error of approximation ^ of the scheme concerned is
of ©(r™' + ft2) if k{x,t^) G C'C)[0, 1] for every fixed t = i, and this is also
consistent with the results obtained.
In mastering the difficulties involved, we refer to a variable operator
0
A(t) in the space Q = H of all grid functions with the values
Ay=-~A{i)y, yGfi.
In connection with its dependence on t the usual practice is to impose, in
addition, the Lipshitz condition with respect to the variable t:
I i(A{t) - A{t - r)) y, y)\ < r c, {A{t ^ r) y, y) ,
making it possible to apply the general stability theory. As can readily be
observed, the Lipshitz condition is ensured if
\k(^x^t) — k[x,t — t)\ < T C;^ k[x,t — r) ,
thereby confirming the validity of the preceding results obtained in Sections
2-4 for scheme G5).
Further development of this trend of research is devoted to a more
complex problem in which the governing equation acquires the form
du d f, , , du
, .OIL Of,, N OU\ „, ,

488 Homogeneous Difference Schemes for Time-Dependent Equations
with c[x,t) > Cj > 0 and 0 < Cj < k(x,t) < c^. In view of this, the
corresponding homogeneous difference scheme becomes
G6) p{x,i)yt = A{f}y<^"^+^{x,f},
where p and if are calculated by means of the same pattern functional
p{x^J) = c{x^J) or p{Xf,t) = \ (c(.i;j — 0,i)+ 0B;,-|-0,i")) in the case where
c(a;,i) is discontinuous at the node x = x^.
With the detailed forms of these functionals, scheme G6) is stable
under the constraints
->-o@, -0=^"^^, ll^ll = l|A||.
Because of this fact, it is unconditionally stable for a > 0.5.
Remark So far we have considered only equidistant grids in t. But it i.s not
difficult to show that the preceding estimates for two-layer schemes remain
valid on nonequidistant grids with step t^ = t- —ij_i being a function of the
subscript j. It is obvious that the grid in t is rather flexible in comparison
with the grid ujf^ as a result of refining the step r- in the regions of the
widely varying right-hand side f{x,t), boundary values f-i^{t), IJ-ii^) and
the coefficient k = k{x,t) in t. With knowledge of the behaviour of the
problem solution on a sparse grid, successive grid refinement will be caused
by the necessity of diminishing the step r- in some intervals during wliich
the solution varies very fastly in t. Any changes in the composition of the
grid ujf^ with changes of i • in the process of calculations are connected with
indeterminate values of the function yj at new (additional) nodes of the
grid cof, in view.
For example, a greater gain in accuracy in r will be achieved once
we perform parallel calculations on several grids uj^ and w^^ by the rules
approved in Chapter 3, Section 4.
The forthcoming procedures serve to motivate what is done on an
equidistant grid loj^^, on which the representation takes place:
G7) y', = d + a,^ ft™' + % r"' + 0{h^- + r"=),
m^ > mj > 0 , ^2 > '^1 > 0 1
where a^ • and /?j ■ are independent of h and r both.

Homogeneous difference schemes for the heat conduction 489
Let 2/^j^ (x^,t,j) and y-j^^ (^x-,t;) be solutions of the same difference
problem with different steps t-^ = r and r^ = t. Via the linear transform
yftr(a;.^) = Ci J/ft^^(K,i) + C2 2//,^3(.-C,<). t = JT , i = 0,l,...,
with expression G7) standing for j/;,^^ and j/^^^,^ both, it is necessary to
reduce to zero the coefficient at the member r"'. All this enables us to find
with reasonable accuracy
y = M + 0(ft'"^ + r"^)
the values
A similar procedure works on two different grids ujj^ and ujq g^^ for fixed r,
thus causing a grid solution y such that
~y = u + 0{h"'' + t"'').
When providing current manipulations, the solution u = u[x,t) and the
available data of the original problem are preassumed to be .smooth enough
and sufficient for the existence of the asymptotic expansion
2/„^ =u + CY h""' + 13 r"' + 0(ft™= + r"=) .
9. The third boundary-value problem. For the moment, the statement of
the problem is
Lu + fix,t), Lu^ —(k(x,t)^] , 0 < .r < 1
ox \ ox J
-~k{l,t) '^^y;^) =f3,_{t)u{l,t)-n,_{t), /?, >0.
In Chapter 3, Section 5 we have formed a difference equation of the
third kind for the stationary equation Lu + f = 0. The formal passage
from the stationary equation to the nonstationary one is provided by the
replacement of / by / — du/dt. This trick has been already encountered in

490 Homogeneous Difference Schemes for Time-Dependent Equations
specifying the difference conditions approximating the boundary conditions
of the third kind. The outcome of this is
yt = A{i) [ay + {l-~a)y)+^, 0 < x, = ill < 1 , tj = jr > 0 ,
ai@ {o-%^i + A " 0-) ;Vs,i) = A@ {o-Vo + A " 0-) J/o) + fh{i)
+ 0.5hyto-~aj^(i) {atjgj^ + (l-~ a)y^^^)
= P2{t) {'^ Vn + A " ct) Vn) + h{t) + 0.5 h 2/( yv •
Here /i, = pi.^{i)-~0.b h f{0,i) and fi^ = pi.^{i)-~0.5 h f{l,i) for f = i^+0.5 r.
The resulting scheme is of accuracy 0(r^ -|- h'^) for a = 0.5 and it is of
accuracy 0(r -|- h'^) for a > 0.5.
The best possible choice for later use of the elimination method is due
to
y
M:Oy = "F, yo = ^iVi + t^i, vn = ^2yN-i + '^2^
a T
aS)
^ ^ ajv@
' a,y{i) + h6,^(f) + h:^/BaT) '
__ A - c^)(ai@ ys,i " Piji) Vo) + 0-5 hyjT - fi^
"' " a{a,(t)/h + p,(i) + h/{2aT))
_ A "^cr)(-ajv(f) t/g jv - P^{i) y^) + 0.5 h y^^/r - p,^
a{a^{i)/h + l3S) + h/{2aT))
F={{l-~a)i\{i)y + y/T + ^(t))a-' .
Having completed the elimination, we observe that the computational
procedure is stable if d > 0, because 0 < >fj < 1 and Q < x^ < \.
10. Monotone schemes for parabolic equations of general form. It is
required to find a solution of the following problem foor a parabolic equation

Homogeneous difference schemes for the heat conduction 491
of the general form in the rectangle Dt = {0<k<1, 0 <t < T}:
G8) c{x,t)-^ = Lu + f{x,t),
u{0,t) = u^[t) , u{l^t) — u^it) , u{x,0) = Ug{x) ,
T d (,, , du\ , . du , ,
Lu-= 7— A;B;,i) -- +r(x-,i) q{x,t)u,
ox V ox/ Ox
0 < c^ < k{x,t) Kc^, c{x,t)>c.^>0, q>0.
A similar problem has been solved in Chapter 3, Section 5 for the stationary
equation Lu + / = 0 through the use of monotone schemes of second-order
accuracy attainable for any step h and the function r(x').
In order to construct a monotone scheme for problem G8) for which
the maximum principle would be valid for any h and r, we involve in
subsequent considerations the equation of the same type, but with the perturbed
operator L:
Ot ox \ ox J Ox
x = (l + R)-\ i?=0.5/i|?-|/A:.
As usual, the operator L is approximated for fixed t = i = tj + u-j by the
difference operator
'\y = x{aij^)^ + b+a^+^^y^ + h~ay^. ~dy,
where (for more detail see Chapter 3, Section 5)
a = A[k{x + sh,t)\, d^ F[q{x + sh,i)\, b^ z^ F[f^{x ^ sh,!)] ,
?^=r^/k, r+= 0,5(r-b |r|) > 0, r" = 0,5 (r - |r|) < 0 .
Here the same pattern functionals A and F are adopted in achieving much
progress as was done in Chapter 3, Section 2, making it possible to generate
an approximation of order 2.

492 Homogeneous Difference Schemes for Time-Dependent Equations
To equation G9) there corresponds the purely implicit (four-point)
homogeneous scheme
(80) p{x,i)y^ = ~k{t)y + ^,
y{x,{)) = Uo{x) , y{0,t) = u^{t) , y{l,t) = u^{t) ,
where the coefficients p and (p are calculated by the same formulas as before
for d and b . A special structure of the operator A built into this scheme
results in the error of approximation ip = 0(r + ft^) and may be of help in
achieving this aim.
The maximum principle applies equally well to the estimation of the
problem (80) solution with zero boundary conditions y^ = jy^y = 0 in
tackling the governing equation in the canonical form
{Pi/r + a^ + Pi + di)yi = a, y^_^ + Pi y^^^ + F, ,
Fi = PiVi/'T + ft, o:i = a^(xi-hh-)/h'^, Pi=a^^^{xi + hb+)/h^.
The conditions of Theorem 3 in Chapter 4, Section 2 are easily verified for
this equation, due to which we might have
||2/||^ = ^max^|2/,|<
Dt = pjT+di.
Substituting here the appropriate expressions for Fi and Di yields
W'-'\\c<\\y'\\c + '
p
<l|j/^'llc + rll^'ll
which assures us of the validity of the estimate
[y''-'\\c<\\y°\\r + 7-E^\\^'\\c
-1 j' = 0
This provides enough reason to conclude that scheme (80) converges
uniformly with the rate 0{t + K^).

Homogeneous difference schemes for the heat conduction 493
11. Cylindrically symmetric and spherically symmetric heat conduction
problems. In explorations of many physical processes such as diffusion
or heat conduction it may happen that the shape of available bodies is
cylindrical. In this view, it seems reasonable to introduce a cylindrical
system of coordinates (?•, 99, z) and write down the heat conduction equation
with respect to these variables (here x = ?•):
In the physical language, this is a way of saying that the temperature
is independent of (p and z.
In the case of a spherical symmetry the heat conduction equation
acquires the form
Homogeneous difference schemes for stationary equations in spherical and
cylindrical coordinate systems have been designed in Chapter 3,
Still using its framework, the starting point correcting that situation
is to impose at the point x = 1 the usual condition of the first or third kind
(82) u(l,i)-/^2@
and then require the boundedness of a solution at the point x =^ 0:
nil nti
lim fc s — = 0 for (81), lim A: x-^ — = 0 for (81').
s-*o dx ,i;-*o dx
The cylindrically symmetric heat conduction problem is reproduced by
(83) | = L. + /(.,o, ^-i|:(^%.0£
i > 0, 0 < s < 1,
u[x, 0) = Uq[x) , 0 < s < 1 ,
du
xk ^-
ox
= 0, u(i,i) = M,(i;), f. >o.
x- = 0
In line with the usual practice we introduce on the segment 0 < s < 1 an
equidistant grid
Lu^^ = {x^^ih, i = 0,1,.. . , Af, hN = 1}

494 Homogeneous Difference Schemes for Time-Dependent Equations
and a time grid w^ = {<■ = jr , j = 0, 1, , , , } on the segment 0 < i < T to
expound exploratory devices for obtaining difference schemes by analogy
with Chapter 3,
The operator L is approximated by the difference operator
A@«j = — (•^■<:-i/2«j%,i),. . ~-^«. where a- =aB:.,i),
thereby establishing a correspondence between equation (83) and one of the
weighted schemes
y, = A(i)?/^' + ^^, '^= fixj).
The complete posing of this includes the difference boundary condition
at the point x = Q. The methodology of Chapter 3 furnishes the
justification of the forthcoming substitutions into the stationary equation: we first
replace y^ by y[!''^ and /(O) by [j - —j _ and then — by u^ and u by
y, The outcome of this is
"il'-J Vxfi — 4 Vtfi . Jo '
which admits an alternative form of writing
Vt.o = l"-i (^") 2/i^o + ^0 . fo- fi"^ ■
In this regard, we observe that it is possible to insert /(O, i) in place oi f^".
When the conditions for x = 1 and / = 0 are put together with the
preceding equation, their collection constitutes what is called the difference
boundary-value problem
yt = A{i) y^"'^ + ^ , 0<x = ih<l, t=zJT>0,
where
4
A(<)y = j^a^{t)yx for X = 0 ,
A(i) y = — {x a(x,t) y^,) _ for 0 < s = ift < 1 , x — x^ — 0.5 h .

Homogeneous difference schemes for the heat conduction 495
If G ^ 0, then the difference equation related to y- can be solved by
the standard elimination method,
The statement of the problem for the error z = jy — u amounts to
Zt = A{t)'/"^ + ij , 0 <x = ih <1 , tj>0, Zo=0, z{x,0) = 0,
where tp = ^{i) w*-" ~^ f ~ '^f ^V exactly the same reasoning as in Chapter
3, Section 5 the residual ip is representable by
4>=z - (x- ri)., + i* + ijj** , i] = a «(,'') - (A; «')i-=s-, t=i ,
with the members
?; = 0(/r' + r"''), ^*=0(/i,Vs), i>** = 0(h^ + t") .
Having no opportunity to touch upon this topic, we refer the readers to
the aforementioned chapters of the manograph "The Theory oof
Difference Schemes", in which the method of extraction of "stationary nonho-
mogeneities" was employed with further reference to a priori estimates of
z. The forward difference scheme with G=1 converges uniformly with the
rate 0{h^ + r) due to the maximum principle.
Being concerned with the heat conduction problem in the case of
a spherical symmetry, we are now in a position to produce on the same
grounds the difference scheme associated with problem (81')-(82):
Vt = A{i) y'-"^ + (f for 0 < x = ih < 1, <y > 0 ,
i/n = ^^i^ i > 0, y{x^, 0) = Uo(x-J, X- e oj,,,
Hi) Vi
{x^a(Xi
'M<)i/o =
or (fii :
'i)yx,i),.. for i>0.
6
=/r^ foi' 0 < f < vv
This is also consistent with the results expounded m Chapter 3,
Section 5. The standard elimination method applies equally well to such a
setting. We omit here more a detailed exploration of the residual and the
accurate account of accuracy of the describing scheme. The reader is invited
to do this on his/her own in line with established priorities for difference
schemes on "flowing" grids to,,.

496 Homogeneous Difference Schemes for Time-Dependent Equations
12. A periodic problem. We are now interested in the problem of the heat
distribution over a uniform thin circlic ring 0 < (p < 2ir of radius r^:
du ci d 11
577 = 725^. 0<^<2^, i'>0, «(^,0) = «oH.
A unique determination of a solution u(ip,t') necessitates imposing
the condition of periodicity
u{(p + 2-!r,t') = u{(fi,t') for any ^G[0,2 7r],
which, in turn, can be replaced by the condition of conjugation at the point
^ = 0:
du
du
dip ^==0+0 dip
t/? = 2 7r —0
u@ + 0,i') = MB 7r-0,i').
By interchanging the variables
the segment 0 < <^ < 27r is carried into the segment 0 < a; < 1. In view of
this, the governing equation is modified into
M@ + 0,i) = w(l -0,i),
du{0 + 0,t) du{l-0^t)
dx dx
which is not surprising. On the grid
Cuf^ = {x^ = ih, I = 0, 1,... , Af, ft = l/N}
we have occasion to use the simplest implicit scheme
t/( = yj^, , i) < X = ill <1, t = JT > 0 , y{x, 0) = Ug{x)
which is supplemented by the condition
2/0 = Vn

Homogeneous difference schemes for the heat conduction 497
as a corollary of the condition of conjugation m@ + 0,i) = ■«(! — 0,i). By
analogy with Chapter 3, Section 5 the second condition of conjugation is
approximated by the equation j/q ^ = Vxx o- By identifying the endpoints
.c = 0 and s = 1 it is supposed that
Vn+i === J/i ■
In accordance with what has been said above, the difference scheme
in question is constructed at all the nodes f = 1, 2,.. . , A^ of the grid W/j
under the periodicity condition t/yv+i = Vn imposed at the node i = N.
The same procedures are workable in constructing the appropriate
difference scheme associated with the equation with variable periodic
coefficients
m(s, 0) = Ug[x) , 0 < s < 1 ,
and the boundary conditions of periodicity that are known to us as the
conjugation ones:
u@ + 0,i) = «(l-0,i), k -:^
ox
du
x=o+o dx
,i;=l-0
Here all the functions k{x,t), f(x,t) and Mo(^) ^^'g periodic of period
1 so that
Uo(x-+ 1) = «oB;), f{x + l,t) = f{x,t) , k{x + l,t) = k{x,t) .
Let us stress here that the available coefficients k{0+0,t) and k{l—0,t)
may be different: A;@ + 0,i) j^ k[l — 0,t). When this is the case, the
derivatives du/dx involved happen to be discontinuous:
du{0 + 0,t) (9«(l-0,i)
dx dx
By identifying the endpoints x = 0 and x' = 1 on the same grounds as
before, the condition of periodicity is to be understood as the condition of
conjugation at a discontinuity point of the coefficient k(x^t). From such
reasoning it seems clear that, having stipulated the condition j/yv+i = 2/i i
the design of the scheme in question includes all the nodes f = 1, 2,, ,, , A^

498 Homogeneous Difference Schemes for Time-Dependent Equations
of the grid at hand. As a final result we get the homogeneous difference
scheme with weights
yt=A{t)t/"^+^{x,i), x = 2h, 2= 1,2,... ,7V, i = (j + 0.5)r,
y{x, 0) = Ua{x), 2/yv+i = 2/i , Vo = Vn ,
where Ay = [a{x\i) y^'j and the coefficients a and (p are given by the usual
formulas. For example, it is fairly common to deal with
The solution can uniquely be found from the above conditions. This scheme
has the approximation order 0{{a — 0.5)r + r^ + /i^).
In these concerns, there arises the problem for determination of y =
AiUi^i " BiVi + A + iy, + i ="F,, i = 1, 2,.. . , Af,
Vn+i = 2/i > 2/o = 2/yv , A, = a t ajl? , Q = yi^ + yi, + i + 1 ,
which can be solved by the cyclic elimination method established in Chapter
1, Section 2,
In an attempt to cover all the issues, we should raise the questions
of stability and accuracy for the approximation just established. With
this aim, we introduce the space H of all grid functions y{Xj) given for
j = 1, 2,, .. , A, A+ 1 and satisfying the condition of periodicity y-^^y = y^,
yyy = y^. An inner product and associated norm in that space are defined
by (y, w) = ^^^ v.iUih and || t^ || = \/{u,v).
The operator A is specified by the relation
Ay = -Ay for y e H .
It seems clear that Green's formulas are certainly true in the case when
the operator A is defined in such a way. Moreover, A =z A* > 0. All
this provides the sufficient background for the possible applications of the
general stability theory outlined in Chapter 6, within the framework of
which the scheme concerned is unconditionally stable for a > 0.5,
For G=1 the maximum principle is in full force for any r and h,
due to which the resulting scheme is uniformly stable with respect to the
initial data and the right-hand side. What is more, the uniform convergence
occurs with the rate 0(r^ + /i^).

Homogeneous difference schemes for hyperbolic equations
499
7.2 HOMOGENEOUS DIFFERENCE SCHEMES
FOR HYPERBOLIC EQUATIONS
1. The original problem. In a common setting it is required to find in the
rectangle
L»T = [0 < x- < 1] X [0 < i < T]
a solution of the first boundary-value problem for a second-order equation
of hyperbolic type
A)
B)
C)
d'^u
d
di
du
u(x, 0) = Wo(a
dt
{x,Q) = u,{x),
t<@,f) = Mj(i), T<(l,i) = t<2(<)>
0 < Cj < fc(X-, <) < Cg ,
where Dt = @<x<1)x@<<<T], under the following assumptions;
the problem is uniquely solvable, its solution is continuous in the closed
domain Dt and possesses all necessary derivatives which do arise in the
further development, the coefficient k(x,t) and the right-hand f{x^t) may
have discontinuities of the first kind on a finite number of straight lines
parallel to the axis Ot ("immovable discontinuities"), on every discontinuity
line X = ^j.
1,2,
J, the conditions of conjugation relating to the
continuity of the functions u and k du/dx for x = ^j
hold:
1,2,
•Sq , must
D)
[«] = u{i, + 0, t) - u(^, - 0, i) = 0 , [k du/dx] = 0 .
2. Homogeneous difference schemes. In preparation for designing a
homogeneous weighted scheme associated with problem (l)-C), let
{Xi, 1 = 0,1,
vv.
0, X
N
1}
be a nonequidistant grid on the segment 0 < ;c < 1, w^ = {i • = ir,
j = 0,1, 2,... , Jo} be an equidistant grid on the segment 0 < t < T and
let a suitable grid in the rectangle Dt be made up by uij^^ = uij^ x ui^. A
homogeneous difference scheme for solving problem (l)-C) can be obtained

500 Homogeneous Difference Schemes for Time-Dependent Equations
for fixed t G <^^ through the approximation Au + f = (^a{x,t) u^), -\- ^p io
Lu + f. The forthcoming replacements (9^'u/(9i^|j__j. ~ Mj-^ and Lu + f ^
i\{t-)u^'^^-'^''"' + ip, where
A(ij)u= (a(a;,/^-)'%).-
u = u' , ii — u-'"', u = u^'^\
complement subsequent constructions. As a final result we obtain the
homogeneous weighted three-layer scheme
E) y,-, =A(i^.J/(^-^=) + ^.
The coefficient a can be taken on the middle layer t = i,-.
Substituting y — y + ryo + 0.5r^j/j-j and y = y — rt/o -|- 0.5r^j/j-j, where
y» = (y - y)/Br) and y^-^ = (?/ - 2?/ -^ ^)/r2, into E) yields
y(a. .a.) = y + (^^ _ ^J ryo+ O.bia, + a.^y^, ,
which admits an alternative form
F) {E - 0.5 {a, + a^) t'K) rj^^ - {a, ^ a,} t Ayo = Ay + ^,
where E is the identity operator. For cr^ — a^ = a\i becomes the symmetric
scheme
G) {E-(rT''A)yti=Ay+^{x,t), 0<t=JT,
which will be given special investigation in the sequel.
Before going further, we must append to G) the initial and boundary
conditions. These depend on the range of variables; thus, the boundary
conditions and the first initial condition are specified exactly;
(8) y{0,t) = u,{t) , yil,t)=u,{t), y{x,0) = u^^x).
There are two ways of approximating the second initial condition
du/dt\i^Q = Ug{x), one of which generates an approximation of order 2
in r:
(9) ytix,0) = t\{x) , where Ugix) = Uo{x) + 0.5 t{LUo + f}t=o ■

Homogeneous difference schemes for hyperbolic equations 501
The second one is connected with the difference equation for
determination of J/(t):
(9') {E^a T^A{0)) y,{x, 0) = u,{x) + 0.5 r (A u, + f{x, 0)) .
Summarizing, the homogeneous difference scheme G)-(9) or G), (8),
(9') is put in correspondence with the original problem (l)-C) under
consideration.
The computational procedures for three-layer schemes were
established before. For their successful realization we need to know the values y^
and j/^"'-' on two preceding layers for searching y = y-''^^ by the elimination
method being used on every new layer t = t-_^_^ in solving the boundary-
value problem with respect to i) = y^~^^:
A0) (E-(XT'^A)y = F , 0<x = ih<l, % = u, , Vn = ^^ ,
F{t) = 2y - y - T^A{{2(r - l)y - ay) + T^^ , t>T,
F@) = -Uo + r2@.5 - a) A@) u^ + tu^(x) + 0.5 r'^fix, 0).
3. The error of approximation. Let 'u{x,t) be a solution of the original
problem (l)-C) and y{xf,tA — yj be a solution of the difference problem
G)-(9). The next step is to set up the difference problem for the error
zj = j/^ —m] , where uj = u{x^, i ■), by inserting in G)-(9) the sum y = z + u:
A1) {E-(rT^A)ztf = Az + i>{x,t), 0 < ;c < 1 , f>0,
z{x,0) = 0, z@,i) = z(l,i) = 0, zt(x,0) = i^(x),
where
A2) ip(x,t) = A(t)u-Uff +ar'^^Auff ,
v(x) - 0.5t{Lu„ + f)t^o +t\{x)-Ut{x,0)
are the errors of approximation on the solution of problem (l)-C)
associated with equation A) and the second initial condition B), respectively.
If the coefficient k{x,t) and the right-hand side f{x,t) possess only a
finite number of immovable discontinuities, the grid tD^ ~ ujf^{K) will be so
chosen that all discontinuity lines will pass through the nodal points (the

502 Homogeneous Difference Schemes for Time-Dependent Equations
main idea behind this approach was explained earlier in Chapter 3, Section
4 and Section 1 of the present chapter).
For convenience in analysis, the residual -ip is representable by
A3) i; = Vi + r.
where
V OxJi-il2 V> \0X Ot'^ ox / i-1/2
A5) V* = 0{t^ + h-).
Apparently, the current situation needs certain clarification. Having
integrated equation A) with respect to x at a fixed moment t = i- from x-_-^,.2
to •'''i + l/2
16 [k tt) - (k -^)
^ ox J ^-^x ^ + 0.5 V OX/x:^,;,-0.5
d'^u{x,tj)
'■i + ll2 ^i + 1/2
+ / f{x,tj)dx- I "^2^^' dx=:0,
'^i-1/2 ^i-1/2
we then divide this identity by H^, subtract the resulting expressions from
the right-hand side of representation A2) for the residual ip and, finally,
get
\ ^ OxJi^ii2j
i+l/2
1 [ ( d^u{x,ti)\
A7)
hi .J V ' dt
■■^i-l/2
where the coefficients a^ and 7?^ are given for fixed t = t. by the same
formulas as stated in Chapter 3, Section 4.
Let Xj be a discontinuity point of both functions k and /. To avoid
cumbersome calculations, the usual practice involves the simplest formulas
for finding a^ and ipi:
A8) a. = ^..1/2, y.= '' ^^'^^ ' ■ f^=f{x,±0).

Homogeneous difference schemes for hyperbolic equations 503
Following the procedures of Chapter 3, Section 4 and taking into
account that the second partial derivative d'^u/dt" is continuous on the line of
discontinuity x — ^ oi the functions k(x,t) and f(x,t), we deduce through
such an analysis that
f , , 'd'u{x,i,)\
Because of this, formulas A3)-A5) are an immediate implication of formula
A7) with the preceding expansion involved. From representation A4) it
seems clear that
A9) ,,, = 0(/l2 + ^2) , ,j^^^ = 0(h^ + T').
4. Stability and convergence. No restrictions are made regarding the
smoothness of the coefficients and the solution in the further estimation
of the accuracy of scheme G)-(9). This can be done using various a priori
estimates for the operator-difference three-layer scheme
B0) D Ztt + A z = ^ip(t), t = iT>0,
ziO) = 0, z,@) = /y.
Here D and A stand for linear operators in a Hilbert space H, z(t) and ^(i)
refer to abstract functions of the argument t £ ui^ with the values in the
space H and u is an element of the space H (for more detail see Chapter 6).
0
In preparation for this, H = Q is the space of all grid functions given on
the grid d;^ and vanishing on the boundary at the points x =0 and x = I.
The usual inner products are defined to be
N-l N-l N
i~L i — 1 i—L
In the general setting three types of suitable norms are in common
usage:
|z|| = max |zB;)|, ||z|| = VB, z)^ , \\z\\^ = \/{Az,z)

504 Homogeneous Difference Schemes for Time-Dependent Equations
A comparison of G) and B0) provides enough reason to conclude that these
operators A and D are identical:
A = -A, D = E-o-T^-A = E + ar'^A.
The operator A is self-adjoint, positive definite and satisfies the estimate
(see Chapter 2, Section 4)
II yi II < 4c,//r ■ , /i„,;„ = min h.; .
II II — ^' mm ' mm )<./</V"
The general stability theory outlined in Chapter 6 asserts that scheme B0)
is stable under the condition
D > ^ T^ A or {Dy, y) > ^ t' (Ay, y),
where £ > 0 is an arbitrary number independent of h. We will pursue the
further stability analysis of this with
D^'-±^r^A = E+(a-'-±^)r^A
4 V 4 /
1 / 1
> Tr-77T+ (T ^]t' ] A>0
which is certainly true for
^2
c_ ^ )
a > a^
mm
4 4 r2 C2
If you wish to explore this more deeply, you might find it helpful to
refer to Chapter 6, Section 3 of the monograph "The Theory of Difference
Schemes", in which the following estimates were derived for problem B0):
i—1— / ^
1 +£
B1) lk^+^IU(.^,) < A^ V ^ |^ll^*@)II^W + E - W^'Wn-^it,)^
+ max(||/'|U_.(,^^ + ||V.,^'|U-.(Jj.

Homogeneous difference schemes for hyperbolic equations 505
For the weighted scheme B0) these estimates are ensured by cr > (T^ and
\ai\ < Cjfl.
We shall need yet, among other things, some modification of the well-
known estimates on an equidistant grid (see Chapter 2, Section 4), taking
on an arbitrary nonequidistant grid the form
2
B3) IIV^IU-=lh.flU-<^IWI
VI
2
10,11,1-1 < ^^ ll'/i]l for ■'P = Vf
Since D is a self-adjoint operator and
D= E+(rT'^A = E + {(r-o-^)T^A + (r^T^ A
>E + 0.5t'^A- \r^A>eE,
II ^ II
we obtain
D^'<-E and |t^||^., < -1 || ^ || .
£ \/£
As usual, we may attempt the solution z of problem A1)-A3) as a
sum z = V + lu with the members v and w satisfying the conditions
i,-^=At'(''J + 7/.f, v(x,0]^vtix,0)=0, v„ = v^, = 0,
B4) lu^^ = A lyt'') + ^tp* , w(x, 0) = 0 ,
'Wf{x, 0) = u{x), Wg — Wjq = 0 .
Putting these together with B1)-B3) we deduce for v and w that
B1') ||.^ + -^||,^,<-^ m,a^x,(|b?^-]| + ||ry^-]
M
fe 0<k<3
M ' ^
B2') IK+^llc<;;^(llHI^ + E^II^*'llj
where ||z^||2 = {{E + a t"^ A)v,v) = \\vf + aT'^{a,vl].

506 Homogeneous Difference Schemes for Time-Dependent Equations
Theorem Let the functions k(x,t) and f{x,t) hsive discontinuities of the
first kind on a finite number of the strsiight lines x = ^g, s = 1,2,. .. , s^
parsillel to tlie axis Ot and in the regions of the special configursitions
^s={L<^<L + l, 0<t<to), s = 0,l,...,s„ ^0 = 0, e.„ + i = l,
the coefficients k{x,t) &nd f{x,t) &nd the solution u(x,t) are smooth enough
so thsit both conditions A9) and A5) hold true. Then under condition B2)
scheme G)-(9) conyerges uniformly with the rsite 0{t'^ + h?) on special
sequences of nonequidistant grids ^'^^(/l) and the solution of problem (II)
satisfies the estimate
B5) \\z'\\^. = y-u^\^,<M(T'^ + hl), where h^ = max h^ .
To prove tliis assertion, it suffices to bring together a priori estimates
BI)-B2) with relations A5) and A9).
Remark The theorem is still valid upon replacing G) by the scheme
B6) (E-ar'' A)yi,=Ay+^,
0
where the constant operator Ay = J/j£ is adopted as a regularizer. In that
0
case A = —A, R = —cr A and D = E + t'^R. The sufficient stability
condition B1) is ensured if we agree to consider
B7) (r=:{l+e)cj4.
In evaluating the error of approximation all the tricks and turns remain
unchanged except for formula A4) for rj, in which the member crT'^^aUfig
should be replaced by crr'^Uifjr., where a constant a is specified by B7).

Difference Methods for Solving
Nonlinear Equations
of Mathematical Physics
In this chapter the new difference schemes are constructed for the quasilin-
ear heat conduction equation and equations of gas dynamics with placing a
special emphasis on iterative methods available for solving nonlinear
difference equations. Among other things, the convergence of Newton's method
is established for implicit schemes of gas dynamics.
8.1 DIFFERENCE METHODS FOR SOLVING
THE QUASILINEAR HEAT CONDUCTION EQUATION
1. The stationary problem. To avoid misunderstanding, we concentrate
primarily on the simplest problem, the statement of which is related to the
stationary heat conduction problem with nonlinear sources:
A) u" = -f(u), 0<2;<1, w@) = 0, u{l)^0.
An excellent start in this direction is to introduce on the segment
0 < a,- < 1 an equidistant grid ulf^ = {x^ =ih,i = 0,l,...,N, hN = 1} and
proceed to design the difference scheme
B) y,^. = -f{y), x=th, i=l,2,...,iV-l,
507

508 Difference Methods for Solving Nonlinear Equations
making it possible to set up the difference problem for the error z = y — u:
C) Zj;^ + f {y) z =-ip , x-ih, i = 1, 2,.. . , A^ - 1 ,
i'o = i'/v = 0 ,
where y = u-\- 9z, Q < 9 < \, and i/' = u,g,. + f(u) is the residual.
ft seems clear that scheme B) generates an approximation of order 2:
ip = 0{h').
ff f'(y) < 0, then a solution of the difference problem C) satisfies the
estimate
D) \\4c<\mc'
which serves to motivate the uniform convergence of scheme B) with the
rate ©(/i^):
\\z\\^. = \\y-u\\^,=0{h').
What is more, a solution of the difference problem B) is bounded, so that
E) Il2/Ilc.< 1/@I = Co
under the constraint /'(j/) < 0. fndeed, simple algebra gives
fiy) = /(O) + (/B/) - /(O)) =: m+f'{y)y,
wliere y — 9y,Q<9< 1, yielding
fe + /"(y)y = -/@), (/o = 2/,v = o.
Whence estimate E) follows on account of D).
fn this regard, Newton's method suits us perfectly in connection with
solving the nonlinear difference equation B). ft is worth recalling here its
algorithm:
'y\. + ny)Cy' -y) = -f{y),
where k is the iteration number, fc = 0, f, 2, . . ., leaving us with the tliree-
k-\-i
point linear difference equation related to y :
F) ys, + f{y)y =-{f{y)-f {y)y), % = yN=^-

The quasilinear heat conduction equation 509
This can be solved by the standard elimination method, whose
computational procedure is stable under the condition f'{y) < 0.
The convergence rate of such iterations needs investigation with regard
to the error
fc+i k+l
V = y -y,
where y is the exact solution of problem B). Upon substituting two
subsequent iterations y = y -\- y and y = y + y into equation F) we may
set up the problem for the error v :
(8) F =f{y)-fiy} + iy^y)r{y}-
Taking into account the well-established decomposition
f{y) = f{y) + ny)iy-y) + U"ihiy-yf,
?here y = y + d{y - y), 0 < d < I, yve find that
k 1 .,,k
1 fiU~\ *-'2
F=-y"iy)
V
Thus, it is required to evaluate a solution of the problem
(y) Vx:i- + j(y) V =2-' yy>'" > ^'o = "w =0'
If f{y) is a concave function, that is, f"{y) > 0, then due to the
maximum principle we might have
k + l k + l k + l
' V = y -y<0, y <y,
thereby clarifying that the iterations approach the exact solution of problem
B) from below. It is plain to show that for a solution of problem (9) either
of the following estimates
A0) ll'^'llc = T6ll/"(^)llc-|l^llc
\'t'\\c.<q\\'v\\l

510 Difference Methods for Solving Nonlinear Equations
is valid under the condition ||/"(:y)||c. = 16(/.
Indeed, in conformity with the maximum principle (for more detail
see Chapter 4, Section 2) problem (9) has for f'{y) < 0 the majorant
V{x) = Kx{l-x), \\V{x)\\^.<\K,
where/v -i||/"(y)||^ • ||'||2,, so that
ll'J'llc<ll^llc<T6ll/"B/)llc •il^ll^<'/ll'lPc'
With the chain of the relations \\q v ||^. < ||gf ||1 < ■■■ < Ikf ||^-. in
view, we deduce that
uk + l II _^ 1 II 0 ||2'-- + i
II V \\c< ~-\\qv\\^ ,
thereby confirming the quadratic law of the convergence of iterations with
the initial approximation y subject to the condition
'illy ~?/llc < 1 o^' 'ilhllc < 1 ■
When f'{y) > —Cj, c^ > 0, an alternative estimate in the grid norm of the
space L instead of A0) is such that
nn i|fc+i|| ^ ll/"(y)llc i|fc|i2 gi 11^112
A-1) V < 7-77- r v\ = — V ,
2F + cj 6 + Cj
vvlier
e
4 . , tt/z , I
= T^ sm^ —— an
d q, = 7^\\ny)\
/i2 2 " " 2 ''■' ^^^^'c ■
Other iterative methods apply equally well to problem B). Among
them the method with the recurrence relation
'yL = 4.-(i-^)/(y)
will be appreciated. Here the parameter 9 is given by the formula
A r*
A = — , c* = max \f'iy}\[
A + 1 ' 26
in so doing the iterations converge with the rate of a geometric progression
with denominator ^2 = d, so that
II *'+'- II ^ II ^ II ^ fc+i II 0 II
II V II < 92 II i" II < Q2 II ■'^ II •
It is worth noting here that the iterations converge no matter how the initial
approximation y is chosen, because ^ < I.

The quasilinear heat conduction equation 511
2. The quasilinear heat conduction equation. So far we have considered
merely the linear heat conduction equation in spite of the fact that in plenty
of real physical processes the coefficient of heat conductivity is, generally
speaking, a nonlinear function of temperature (and density). In some
problems it gives, in addition, a function of the temperature gradient. High-
temperature processes in plasma physics are in line with these statements.
Being the right-hand sides of the heat conduction equation, heat sources
may depend on the temperature when, for example, the heat transfer is
caused by a chemical reaction. Such processes are described by the
nonlinear heat conduction equation
, s deix.t.u) dw
where the heat flow
du
w = wi x,t,u, ——
ox
is a nonlinear function of temperature u and its derivative. If the heat flow
is linearly dependent on the derivative du/dx and it is governed by the
Fourier law
w = —k(x, t, u) -^r— ,
ox
we obtain as a final result the quasilinear heat conduction equation
u
A3 c x,i,ti — = —{kix,t,u)—) +f{x,t,
at ax \ ax I
c{x, i, u) > 0 , k(x, t,u) > 0 .
When this is the case, the heat capacity c, the coefficient of conductivity k
and the right-hand--side / depend on the temperature u(x,t). In inhomoge-
neous media k, c and / may have discontinuities of various kinds and this
dependence upon the temperature u may be different and depends on the
range of situations to be considered.
In this view, it seems reasonable in a typical situation when the
functions k = k(u), c = c(u) and / = /(u) depend only on the temperature u
cind give rise to the govering equation
A4) ^(^')^ = 5^Nd + ^(^')'

512 Difference Methods for Solving Nonlinear Equations
to introduce a new variable v — L k(^) d(, for later use in equation A4)
with further simplification of the ensuing formulas. The outcome of this is
^ = £ + /», where ^(v) = J dQ d^ ,
0
By merely setting v — /^ c(i^) d^ we are led to an alternative form of
equation A4):
so that
>f(t)) dv = / k{u) du
When c(w) and k(u) can be expressed through the power functions of
temperature u, that is,
c{u) — Cgu" , k{u) = kgU^ ,
it makes sense to introduce one more variable
V = / c(Od£. - Co
a+ 1
0
and take into account that
dxi _ k dv _ kg p_^ dv _ kg /a + 1\ ^+f ^qr dv
dx C.Q u" dx Cq dx Cg \ Cg J dx
permitting us to recast equation A4) as
dv d { „ dv\ ~ ^
a + 1 ' " Co V c„
/3-&
P — a kg (CY -\- \\^
3. Some analytical solutions to the quasilinear heat conduction equation.
Nonlinearity of the coefficient of heat conductivity results in the new
physical effects, the main of which is a final velocity of heat conducting. In what

The quasilineai' heat conduction equation 513
follows we expound some exploratory devices for obtaining the simplest
particular solutions to the equation
A5) a^=5^h«^3^j- -o>0, .>0, .>0.
Here the subsidiary information is the temperature at the point x = 0:
A6) u = Uaf\
With these, it is required to find a solution to equation A5) in the domain
{x > 0, i > 0} with the zero initial temperature
A7) m(;c,0) = 0.
We may attempt a solution of this problem in the form of a "travelling"
wave
u(x, t) — U{Dt — x) , D = const ,
where f/(^) is the unknown function which is sought. Inserting this
expression in A5) and taking into account that
du __ dU du dlJ
'dt " ll^ ' 'dx ~ ~'d^ '
we derive the ordinary differential equation for the function U{(,):
DU' ^ {x^WU'y ,
yielding
H^U" U' = D[/ +const .
In the case const = 0, we obtain
x,.U''U' = DV or """y^' =1,
which upon integrating once again becomes
5J;^" = <+'»■
Since t/ = 0 for i = a; = 0 (i^ = 0), we find that c^ = 0 and
«-'«) = (!^«)""^(^)""''"('-lf7)""'

514 Difference Methods for Solving Nonlinear Equations
whence it follows that at the point x = 0
\ H„ J
A comparison with A6) gives
1 D^a „
n — ~ , = u
a
0
0
Thus, we have proved by having recourse to a "travelling" wave that
problem A5)-A7) is solvable and its solution admits the form
\iDt~xy/\ 0<x<Dt,
A8) u{x,t)= <; """ V" DtJ ~ D^/
X > Dt,
provided the condition 71 = 1/cr holds. Any solution of the form A8) is
called a "temperature wave" with a finite velocity. What is more, it
depends on three parameters x^, a and u^ in accordance with the law
At the next stage we focus our attention on the heat flow
In view of this, on the front of the temperature wave x ~ Dt the
temperature and heat flow vanish for cr > 0 and the partial derivative
du Un 1
dx ffD^I" {Dt- xy-'^l'^
tends to oo for a > 1, it is finite for cr = 1 and becomes zero for 0 < cr < 1.
Therefore, it is meaningful to speak for cr > 1 only about a generalized
solution to the heat conduction equation A5).
A case in point is a nonlinear dependence of the coefficient of heat
conductivity upon the temperature. From the formula for D it is easily
seen that we formally have D = oo for the linear heat conductivity when
cr = 0; meaning that the velocity of heat conducting turns out to be infinite.

The quasilinear heat conduction equation 515
It may happen that the temperature front is held fixed, that is, D = 0.
Such a solution always exists in a special boundary regime such as
A9) u@,0=u,
where t^ is an arbitrary constant, under the agreement that the initial
condition was imposed for t — —oc:
B0) u(a;,-oo) = 0.
Still using the framework of the method of separation of variables, a solution
to equation A5) is sought in the form
u{x,t) = v(x) T(t) .
Upon substituting this product in A5) and separating the variables we
obtain
Id/ „dv\ 1 dT ,
XnV ^ I = „ . . ~ = A ,
wliere A is a separation parameter. Whence it follows that
(.2) ^ ="'"'■
Along these lines, we may attempt a solution to equation B1) in the
form
v'^ = a (xj — xf ,
where the numbers a and C are free to be chosen and x^ is an arbitrary
number. Substituting v"^ into B1) yields
^^^1 + 1/- ^ f^ + ;3 _ l) (^^ _ ^)/'/-+/'-2 _ ^ „l/.(^^ _ ^)/J/. = 0
(X \cr I
and reveals
,« = 2.
2xoB + (r)
Having integrated the equation related to T:
T{t}= [a\{c,-t)]~"\

516 Difference Methods for Solving Nonlinear Equations
where Cq = const, we find the function in question
but minor changes are needed in complying it with the special boundary
regime A9):
m=l/^, c,=t,, u, = [-^) =i2^^(^ + 2)J
and, therefore,
Thus, equation A5) with the special boundary regime A9) possesses
the solution
, 0 < X < X,,
B3) uix,t)='> "V^^T^
[ 0 , X > x^,
where x^ is the width of the region of the heat distribution.
As a matter of fact, the front of the temperature wave becomes
immovable, since x^ = const is independent of t and depends only on the
parameters x^, a, u^ of the problem concerned. Moreover, at the front the
heat flow and temperature vanish for any cr > 0, while the partial
derivative becomes du/dx = oo for cr > 2 (at the front of the "travelling" wave
du/dx = oo for o" > 1).
A solution known as a "staying wave" exists during the interval of
time t < tg. This is stipulated by the special boundary regime A9) relating
to the regimes with "breaking down".
For the heat conduction equation with a heat source depending on the
temperature in accordance with the law
, , du d / „ du \ f.
dt dx V dx
there exist both types of the aforementioned solutions which fall within the
category of travelling waves for j3 < a -\-l and the category of staying waves
for /? = 0--M.
Numerical solutions of such problems cause some difficulties during
the course of many methods in connection with nonlinearity and tendency
to infinity of the partial derivatives at the front of the temperature way. It
is hoped that the exact solutions obtained in such a way help motivate what
is done and could serve in practical implementations as "goodness-of-fit"
tests.

The quasilinear heat conduction equation 517
4. A difference scheme. Newton's method. We now proceed to constructing
difference schemes for the quasilinear heat conduction equation.
For this, it seems unreasonable to employ explicit schemes with fastly
varying ingredients k{u), c{u) and f{u). The power functions of
temperature reflect in full measure the difficulties involved in such a case. For any
implicit Scheme one possible stability condition
r 1 min c(u)
/z^ 2 max k(u)
is connected with successive step refinement in i to a considerable extent.
Quite often, it depends on the values of k and c in a smaller number of nodal
points. This supports the conclusion that explicit schemes are useless for
our purposes, fn an attempt to fill that gap, a considerable amount of effort
has been expended in designing unconditionally stable implicit schemes,
fn order to understand some things a little better, the governing
equation is put together with the boundary conditions
B5) ^ = S. ^<-<^'
u{x,0) = Ug{x), u@,i) =/Ui(i) , u{i,t) — fi^{t) .
A nonlinear difference scheme with respect to t/^ + '-
B6) ^U^-')-^(,') ^ ^., ,
X = Xi =ih, 0 < i < N , hN = 1,
may be employed in such a setting under the conditions (fi'{y) > Cj > 0 and
lv"B/)l £ ^2, providing its stability and convergence in the space C with
the rate 0{t+ K^). The proof of these assertions is somewhat lengthy and
cumbersome and so it is omitted here.
The nonlinear equation
is aimed at determining ij^^ on every new layer by making several iterations
of Newton's method
B7) ^{y) + ^'[y)CV ~y)-ryif-^iyn-

518 Difference Methods for Solving Nonlinear Equations
Under the boundary conditions
B8) 'I' =,_^^[tj^,), X=/'2«, + i)
the elimination method is quite applicable in giving y . There is no doubt
that its stability is ensured by the condition
^'(y) >0.
Indeed, this fact is an immediate implication of an alternative form of
writing the governing equation
T k+i / fc 2r\ fc+1 T k+i k
i = 1,2,... ,Af- 1,
where f. - tp{y) - ^{y) - tp'{y )y and y - y^.
The rate of convergence of iterations can be evaluated by means of
the difference
k + i k + i . .. j + i
This can be done by inserting y. = y,; + J. and y. = y^ + 'y^ in B7)
,,*: s k + i k + i , ,k I k i
fKy) V -TV ^.^. = f{y) - 'fi{:y ) + if {y) V + T y^;,.
k k k
= fill) -'fi{y) + 'fi'{y){y -y)
and taking into account the basic relation
^{y) = ^C^) + ^'(y ){y-y) + k ^"iy)iy -y?
and its corollary
f>{y) - f>{y) + f'iy){y -y) = ^^"{y)v'^,
where y ~ y + dv , 0 < d < 1. The outcome of this is the equation
B9) ^'iy)''v' - T^'t\,, = i /'(t/) I - =F , x = ih , 0 < i < N ,

The quasilinear heat conduction equation 519
with the homogeneous boundary conditions
C0) 't,' = 0, 'J^ = 0,
In this regard, the maximum principle states that
C1) fVl,<OSoy{y)/^'{^)\\^. ■\\'v\\'<qtvt,
where q = 0.5 ||^"(?/)/^'(y )||^. < 0.5 |j^"(y)|i^-,/q < 0.5 c,,/cj = q„. since
f'iy) > Cj. > 0 and \f"{y)\ < c^.
Using this estimate behind, it is not difficult to establish that for the
convergence of iterations in accordance with a quadratic law, it suffices to
choose the initial approximation so as to satisfy the condition
C2) \\y -y\\c<2cjc,.
The meaning of this is that we should have for the choice y z= y = y^
T||2/illc < 2ci/c2,
which is always valid for sufficiently small r.
In practical implementations Newton's method converges with any
prescribed accuracy £ only if
^iy) = y", a<l, ^'{y) = atr\
^"{y) = -a{l-a)y"-\
If y > 0, then Cj = oo and unfortunately the preceding estimates are
meaningless.
However, due to the maximum principle a solution of the boundary-
value problem B9)-C0) is non-negative;
k + l k + 1 k + l „
V = y -y <0, y <y,
provided the condition ip"{y) < 0 holds. The preceding justifies that the
iterations approach the solution from below. Because of this tendency, the
first iteration is such that y < y if the initial approximation was taken so
that y < y. But it may happen that y < 0, thus terminating subsequent
computations.

520
Difference Methods for Solving Nonlinear Equations
5. Various implicit schemes for the quasilinear heat conduction equation.
Other ideas are connected with two types of purely implicit difference
Schemes (the forward ones with cr = 1) available for the simplest quasi-
linear heat conduction equation
C3) | = |:(K«)|^)+/W. 0<.<1, 0<i<T,
u(x,0) = Ug(x) , u@, i) = Wj(i) , u(l,t) = U2{t) ,
where k(u) > 0.
The structures of both schemes are well-characterized by
C4)
Vi - Vi
a,+i(y) —^T ^^{.y) -,
/B/i
for the scheme a) and by
C5)
%-2/i
L(y)
\)i\\
h
a-i{y)
yi ~ yi-
h
m
i+i
for the scheme b), where y^ = yl ' ', Hi
example, we might agree to consider
C6)
C7)
C8)
a(vi
hc'-
H
2kiv,
i+Vi\
2 )'
~i)K'"i}
k{vi
k{i
For
A greater gain in accuracy in connection with the temperature wave
depends significantly on how well we calculate the coefficients a^{v). In
the case where k = k^u" is a power function of temperature, numerical
experiments showed that formula C8) is useless and formida C6) is much
more flexible than C7), so there is some reason to be concerned about this.
Further comparison of schemes C4) and C5) should cause some difficulties.
Both schemes are absolutely stable and have the same error of
approximation 0{t -\- /z^). The scheme a) is linear with respect to the value of the
function y^'^^ on the layer tjA.i and so the value y^'^^ on every new layer

The quasilinear heat conduction equation
521
i _,_i can be found, for example, by the elimination method in terms of the
values of the function y^ on the current layer i ■. Because the scheme is
absolute stable, the well-founded choice of the step r is stipulated by accuracy
reasoning only. Unlike the preceding scheme, the scheme b) is nonlinear
with respect to the value of the function y-''^^, so there is a real need for
employing the iterative method. Still using its framework, the iteration
process is governed by the rule
C9)
y i
Vi
■'i-l-l
, (« + l)
iv —S"
y i
A')
^iiy
y !
y ,
■fiy.
As a final result of such operations a revised scheme becomes linear
s-l-i . . .
with respect to the value of the function y and only the initial
approximation y remains a.s yet unknown. One way of avoiding this obstacle is
to accept y = yJ. We note in passing that most of the iterative methods-
converge in practical implementation.? for rather broader classes of
coefficients k and / after two-three iterations performed. Even if the process
in view is divergent, two iterations can result in improved accuracy of the
describing scheme. In trying to adapt the iteration scheme C5), C9) the
usual practice is connected with specifying the condition
|(s+i)
max y j
@,
<e,
where either the total number of iterations is known in advance or a
desirable accuracy e is beforehand prescribed.
Let us stress here that the iteration scheme C5), C9) requires the
double storage in comparison with the scheme a). This is caused by the
5-1-1
necessity of calculating and saving the values of the function y in terms
of the values of two functions y and y.
One more difference lies in the fact that the transition from the value
i i-l-i . .
y to the value j/ is possible only after several iterations made in scheme
C5), C9), while the value y immediately follows in the algorithm of the
scheme a).
But it would be erroneous to think in light of the same properties of
the indicated schemes such as their absolute stability and the same order of

522
Difference Methods for Solving Nonlinear Equations
approximation that the scheme a) offers in such matters more advantages
than the scheme b).
Practical implementations showed that the computational procedures
of the scheme b) can work with a larger step in time, thus reducing
essentially the total volume of computations and the time complexity despite
the extra iterations recjuired in this connection.
However, a preference relation between such schemes is some
consensus of opinion. The reader can encounter in the theory and practice various
schemes generating approximations of order 2 in time and space:
Vt
^\[{<y)hh + i<y)y,).] + f{
y + y
As can readily be observed, they are not monotone, thus causing some
"ripple". This obstacle can be avoided by refining some suitable grids in
time. When solving equations of the form A3) with a weak quasilinearity
for the coefficients k = k(x,t), f = f('u) and c = c{x,t), common practice
involves predictor-corrector schemes of accuracy 0{t" + /r). Such a
scheme for the choice c = k = 1, f = f{u) is available now:
D0)
y-y
0,5 r
y.«-+/(y). y = y{tj+ii2)
y-y 1
whose composition is depicted in Figure 1,
,fB/).
y —
■-i-i
Figure 1.
'i+i
*i + i/2
We omit here theoretical investigations of the preceding schemes
relying on cumbersome calculations and leading to unsatisfactory and rough
estimates that can result in wrong reasoning. Such difficulties are,
generally speaking, typical for nonlinear problems in many branches of science.

The quasilinear heat conduction equation 523
engineering and technology. In tackling nonlinear problems some
preliminary tests may be of help in verifying the quality of numerical methods.
The traditional way of covering this is to compare numerical solutions of
a simple specimen problem with known analytical solutions of the same
problem.
It is worth noting here that Newton's method is quite applicable for
solving problem C5) in addition to the well-established method of
iterations.
6. Calculations of the temperature waves. Of special interest is the case
where the coefficient k(u) is a function of temperature such that
' dt dx V dx
As we have mentioned above, the process of heat conducting emerged in
that case with a finite velocity and the derivative du/dx tends to oo behind
the front for cr > 1.
The temperature waves can be found through the use of the scheme b)
relating to "continuous through execution" ones. No fixation of the front
applies here. Under the guidelines of the preceding section with regard
to problem A5)-A7) having the exact solution given by formula A8), the
calculations permit us to discover that almost everywhere except several
near-front nodes the approximate solution deviates very slightly from the
exact one, not exceeding 0.002 for x^ = 0.5, cr = 2, D = 5 and the total
number of nodal points A'^ = 50. In so doing the number of the necessary
iterations is no greater than 3 and t < 0.2. When the temperature wave is
moving from the left to the right along the zero temperature background,
more and more grid intervals are captured in a step-by-step fashion in the
process of numerical solutions in a number of different ways in connection
with po,9Sible computation.s of the coefficient a,(i/).
Apparently, the main idea behind this approa.ch needs certain
clarification. For example, formula C8) necessitates imposing a nonzero
background temperature prior to the front. In spite of this fact, there are some
delays in introducing new intervals, thus causing large deviations of a
solution in a vicinity of the front. Formula C7) is useless for very large values of
the index o" (cr > 20). Formula C6) has the best accuracy and reproduces
rather accurately without concern of the background temperature.
7. The Stephan problem (problem of the phase transition). Subsequent
considerations include two phases with the coefficients of heat
conductivity ^i(m), k^^u) and of heat capacity Cj{u), C2(w), in either of which it is

524 Difference Methods for Solving Nonlinear Equations
supposed that the temperature satisfies the equation
D2) ^^(^)aF = 5^Md' '-'''■
At the same time, on the boundary of these phases the temperature is
constant and coincides with the temperature of the phase transition: u(x,t) —
u*. The velocity of the boundary i^ of the phase transition is subject to the
equation
du
du
x-=c+o ox
x=^~o dt
if 11 < u* ill the first phase and u > u* in the second one.
With the boundary condition for the phase transition in view, we
rewrite equation D2) by means of the i5-function as
Cj (m) , U < U* , ( fcj (u) , U < U* ,
c{u) = <; k{u) = I
c^iu) , 11 > u* , [ k2{u) , u > u* .
The method of smoothing is available for solving the Stephan problem.
As a matter of experience, this amounts to replacing the E-function by a
nonzero E-type function 6(u — u*, A), not equal to zero only on the interval
(m* — A,u* -|- A) and must satisfy the normalization condition
u' + A
6{u - u*,A)du = 1 .
The quasilinear equation
du d /; , du\
arises in the process of smoothing the functions ^i(m), k^iu), Cj[u) and c^lu)
on the interval {u* — A, u* -|-A). All the schemes we have mentioned above
find a wide range of applications for its numerical solution. However, there
are other numerical methods for solving the Stephan problem concerned.

Conservative difference schemes of nonstationary gas dynamics 525
8.2 CONSERVATIVE DIFFERENCE SCHEMES
OF NONSTATIONARY GAS DYNAMICS
1. One-dimensional equations of nonstationary gas dynamics in Lagrangian
variables. Plenty of problems arising in mechanics and physics such as
the harnessing of nuclear energy, the creation of nuclear reactors, aircrafr
(planes and space-vehicles) design, the dynamics of space flights, plasma
physics (governed thermo-nuclear synthesis) led to equations of gas
dynamics that are, generally speaking, nonlinear and can be solved by the
universal difference methods.
In spite of the fact that problems in gas dynamics began to spread
to more and more branches of science, engineering and technology as they
gradually took on an important place in real-life situations, for the time
being there are no rigorous mathematical results regarding convergence of
schemes even in the simplest and typical situations. The desirable
quality of schemes are verified with the aid of linear models in the acoustic
approximation by applying numerical tests for specimen problems, whose
analytical solutions are known to the users in explicit form.
As a rule, equations of gas dynamics are discontinuous. From a
physical point of view it is fairly common to distinguish weak discontinuities
relating to "cutting waves" and strong discontinuities relating to "shock
waves". For these reasons successive grid refinement can be made with
caution when the accurate account of accuracy of numerical methods is
performed.
In this section we initiate the design of difference methods for
numerical solutions of the simplest problems in gas dynamics. Of our initial
concern is the problem about one-dimensional nonstationary gas flow in a
plane with the following ingredients: velocity v, density p, temperature T,
pressure p, internal energy e.
In preparation for this, the equations of gas dynamics will reproduce
the conservation laws of impuls, mass and energy that can be written in
a number of different ways with respect to Eulerian {x,t) or Lagrangian
(s,t) variables, where x is the coordinate of a particle and s is the initial
coordinate of a particle or the quantity
p{LO)dC
0
that is, the value of mass in the volume 0 < i^ < a;. The usual practice

526 Difference Methods for Solving Nonlinear Equations
involves the system of gas dynamics equations in Lagrangian variables (s, t):
dv dp
A) ~a~ ~ "~15~ (^'^'^ ^^^ of the impuls conservation),
1 dx
C) - = -— (the law of the mass conservation),
p OS
r\ 2 O O
D) — fe + — j = - — {pv) - ■—- (the law of the energy conservation),
E) p = p{p, T) , s = s{p, T) (the state equation),
where lu is the heat flow.
A combination of the second and third equations we have mentioned
above gives
d [l\ dv
^'^ Ft[-p) = Ts
by observing that equation B) can be excluded from the governing system
through the possible separate integration.
The expression for the heat flow
G) iu=-h(p,T)p^,
OS
where x = h{p, T) is a coefficient of heat conductivity, is intended to
complete the above system of equations. It should be noted here that h can
usually be expresse,d through the power function of T and p.
The functions p{p,T), s{p,T), H[p,T) for this system of equations
must be given.
For example, the state equations of the ideal gas are of the form p =
RpT and e = e{T). The readers can encounter £ = CgT, where R and c^
are constants, R/Cq = 7 — 1, 7 is constant, so that
(8) £ = p/(G-l)p).
In this context, two limiting cases of interest are as follows:

Conservative difference schemes of nonstationary gas dynamics 527
a) The adiabatic flow of the ideal gas when w = 0, that is, x = 0.
Because of this, equations of gas dynamics A), F), D) for the
adiabatic flow of the ideal gas can be represented by
dv _ dp d /1\ _ dv
which will be put together with equation (8)
A1) £=p^--.
G- 1)P
Thus, the resultant system of equations comprises -1 equations with respect
to four unknowns v, p, p and e.
We will use below the volume i^ = 1/p instead of density p. In such a
setting the preceding equations can be represented by
A2) ^-^
A3) pJT=(j-l)£.
Equation A0) capable of describing the tootal energy can be replaced by
one of the newly formed equations
, 1/N de dv
at OS
A5 -- — -P -;— .
^ ' dt ' dt
Indeed, taking into account the first equation (9) and A2), we obtain
(9 / v^\ d , ^ de f dv dp\ dv
de dv de dij
b) The isotermic flow of the ideal gas when the temperature of gas
T = const and the equation of energy is missing. The condition T = const

528 Difference Methods for Solving Nonlinear Equations
corresponds to the case when x —> oo. The system of equations of gas
dynamics for the isotermic flow of the ideal gas comprises the equations
dv dp d (\\ dv 2
where c = const > 0 is the velocity of sound, or
dv dp drj dv ,
which are consistent with the adiabatic case indicated above. In the sequel
our exposition is mostly based on more a detailed exploration of three
equations (9), A0) and A1) capable of describing the gas dynamics.
The complete posing of this necessitates specifying the boundary and
initial conditions in addition to equations (9)-A0). Knowing
A8) v{x,0), p{x,0), p{x,0),
we may attempt, for example, the boundary conditions in the form
A9) p@,t) = Po(/) for s = 0 , p{M,t) = p^{t) for s = M ,
or
A9') v{0,t) = v„{t) for s = 0, p{M,t)=p^{t) for s = M.
Summarizing, it is required to construct difference schemes for
equations of gas dynamics (9)-A0) in the complex closed domain {0 < s < M,
t>0}.
2. Equations with psevdoviscosity. In practical implementations there is
a real need for forming homogeneous and conservative difference equations
of gas dynamics. The meaning of homogeneity here is that difference
equations are written at- all the nodes of the grid in just the same way regardless
of the possible continuity or discontinuity of a solution so that subsequent
calculations should be carried out by the same ensuing formulas.
Homogeneous schemes or "through execution" schemes of gas dynamics contain the
extra members with psevdoviscosity, a key role of which is to spread the
front of shock waves over several intervals of the grid. From a formal point
of view, the viscosity ui arises as the additional member to the pressure p,
so that equations (9)-A0) contain for now instead of p the sum
g =p + ui,

Conservative difference schemes of nonstationary gas dynamics 529
where the "viscous" pressure to = uj{p, v'^, h) depends on p, v' and the step
h. The reader can encounter two types of viscosity:
a) a linear viscosity
B0) a; = -—
dv
b) a quadratic viscosity or Neuman's viscosity
/.TIN "o ,2 9v dv
dv
where i^^ is the coeificient of viscosity. It follows from the foregoing that
the function ui = 0 for dv/ds > 0 and u) ^ Q for dv/ds < 0, that is, only
within the zone of the shock wave.
Thus, the psevdoviscosity may emerge only within the zone of the
shock wave. The accepted view is to use
V dv
rj ds
or
v (dv\2
Tj \OS
assuming that the coefficient of viscosity depends on the sign of the partial
derivative dv/ds, so that // = 0 for dv/ds > 0.
3. Conservative homogeneous schemes. The presence of psevdoviscosity
makes it possible to design homogeneous difference schemes or "through
execution" ones, permitting us to reveal the gas distribution caused by shock
waves. Since the equations of gas dynamics express the conservation laws
of impulse, mass and energy, the scientists wish to have at their disposal
conservative difference schemes for which the difference equations involved
reproduce the analogs of these conservation laws on the grid.
Equations of gas dynamics in integral form are aimed at designing
conservative difference schemes by means of the integro-interpolation method:
B2) (h {vds-pdt) = 0,
B3) i {7jds + vdt) = 0,
B4) j {{e + 0.bv'^)ds~pv dt) = 0,

530 Difference Methods for Solving Nonlinear Equations
where integration is accomplished along a closed curve in the plane (s,<).
To make our exposition more transparent, we introduce the grids
c;j^ = {s. = ih, i = 0,1,... ,yV, hN = M} ,
^T - {t] = J'T, i = 0,1,...,io, joT = t„},
retaining the notations u, r], p, e with respect to difference equations and
regarding the function v to integer points s = s,,- on the grid 10,^ and p, i], e
- to half-integer points s = Si_|_i/2 on the same grid.
Other ideas are connected with setting equation B2) in the rectangle
{■-S:-i/2 < s < ■Sj+i/2i tj <i < *j+i}-
*'!-l/2 *j
and writing equations B3)-B4) in another rectangle {s^ < s < s, , j^, t, <
*i + l 'i+1
H+i 'i + i
The integrals built into these identities are replaced by the newly formed
expressions
tj + i *i+i *i + i
pdt^p(-^'^T, [ vdt^v^^'-''>T, [ {pv)idt = p[\'K[''*\,
where
j{oc,) _ a^f^^^ + A ~ (^a)!^ ! ^a is arbitrary parameter ,
Ph = 0.5(pi_i/2 +Pi+i/2), a = 1,2,3,4,

Conservative difference schemes of nonstationary gas dynamics 531
^■ + 1/2 «i + l
/ vdsK,v^h, 1 rj ds f^ rj^.ii^h, etc.
*i-l/2 *J
s a final result we get the difference scheme
Vl+' -v{ ^ /Pi+i/2-P.-l/2V-'^ ^
'I h J -^'
Vl+i/2 - Vl+l/2 _ /^i + 1 - Vi\(^2)
1
T
\'^^ + l/2+ ^ j \^^ + l/■2 1 4 !
B5)
B6)
B7)
h
which falls within the category of conservative schemes for any admissible
values of parameters cTj , o",, (T3, cr^. In particular, for cr^ = 0, cr^ = 1, cr^ = 1,
(T^ = 1 we constitute a system of difference equations, whose solution is
found successively by the explicit formulas: at the first stage - vf , at the
second stage - V'ili/2 ^^'^ ^^ ^he third stage ^Pi_|_i/2) i = 0, 1,. .. , A*'— 1 by
the elimination method from the energy equation and the state equation
pi] = G — l)e being supplemented with the suitably chosen boundary
conditions for i = 0 and i = N — 1. For example, we might agree to impose
conditions A9).
In thi.s context, it should be noted that conservative difference schemes
may be good enough for the equation of the total energy, but approximate
poorly the equation of the internal energy A4)
de _ dv
This can result in improper choices of computational procedures in
giving the temperature. The lack of energetic balances cannot be avoided
by refining the grid in a spatial variable s. A presence of energetic
disbalances in a scheme can be interpreted as a presence of energetic sources of a
purely difference nature connected with some "lack of agreement" between
separate difference equations of a scheme being inconsistent each to other.

532 Difference Methods for Solving Nonlinear Equations
The emerging disbalances are characterized by a typical solution with a
wide range of values. For example, they are sufficiently small on smooth
functions, but could grow on solutions varying fastly in time and space
prior to the total energy of the system concerned.
4. Fully conservative schemes. Other ideas are connected with succe,ssive
use of conservation laws and more detailed balances of the kinetic and
internal energy.
All the schemes with these properties are called fully conservative
schemes. As a matter of fact, the requirement of the full conservatism
is equivalent to being approximated of both equations A4) and A5) in
addition to the usual requirements of approximation:
de _ dv "^e _ d-q
Before going further, it will be sensible to introduce more compact
notations
Pz = Pi + L/'2 , Vz = %+i/2 > £i = £i + l/2 . P = P'i , V^V\
r lPi+i/2 ~ Vi-ifi) - Ps. , - "i- ■
etc,
Where there is no danger of confusion, we will omit the symbol "bar" over
p, -rj, e. Within these notations, equations B5)-B6) can be reduced to
B8) Vt = -Pf'\ Vt = yi"'^-
Instead of B7), let us consider the scheme generating an
approximation to equation A4) capable of describing the law of the internal energy
B9) et = -p^"''^v^f'K
In this connection .there arises a four-parameter family of schemes, from
which a fully conservative scheme needs to be selected through the
approximations to equations A5) and A0) by appeal to scheme B8)-B9).
We will use the obvious relationship
C0) /'''^ = /"^ + r E-«)/,,
where a and (] are arbitrary numbers and /'•"' = a/ -|- A — a)f- The
quantity of interest e^ can be discovered from

Conservative difference schemes of nonstationary gas dynamics 533
which, in turn, can be obtained from the equations 77^ = f(''2) and t)(''^) =
v''^^' — T{a^ — (T2) Wsj = i]i + T{a^ — a^)vg^. From here it seems clear that
C1) i.s consistent with equation A5) only for cr^ = a.^-
At the next stage scheme B8)-B9) is obliged to be conservative. This
can always be done by multiplying the equation v^ = —p/ by v^^'^'^ =
0.5(f + i)), adding the resulting equation
C2) i(,2)^^_,@.5)^,(^O
to equation B9):
C3) (£ + 0.5 v^)t = -p'^"'^ v[^'^ - ^(;(°'^)p^/')
and rearranging the right-hand side of C3) by means of formula C0) as
C4) p(<'3)^(<''')-K^@-5)p(<'i)= (p(^^) + r{(r^-(r,)pt)
X (^,@ 5) + ^(^^_0.5),^^)+,;@-5)p(/l)
where
P(-n — Pi-l — Pi-l/2 '
S,E=T [a, - (r,)v[°''^pt + T{a,- 0.5) p'"'' v^^
+ ^^@-3-0-1) @-4 -0-5)Pi^.i-
The outcome of rearranging equation C3) is
C5) (e + 0.5v\ = -{p[:p'^°''))^-6,E.
In the preceding the quantity 6^E means the disbalance of the internal
energy. By equating 6^ E to 0 for any p and v we find that a^ =■ cTj and
(T4 = 0.5 and, hence, cr^ = 0.5.
Thus, under such an approach a one-parameter family of fully
conservative schemes is given by
C6) ^',--p^^ V,^v(f'\ £^ = _p(-0,(o,5)^

534 Difference Methods for Solving Nonlinear Equations
within which the third equation can be replaced by one of the following
equations:
C7) £^=._p(-0,^^,
C8) (£ + 0.5^2), = -(pjl\))^(°'S))^.
Observe that instead of the latter equation there seem to be at least two
alternatives
C9) (e + 0.bvl,^}, = -{p(^^K('>'^))^,
D0) (e + (,2+,2^^P/4)^^_(p(..),@,5))^^
Equation C9) can be derived from the foregoing by involving equation C2)
at the (i + l)th node
D1) O.Hvl;),=.-v^f^p^P or 0.5(vl^^),=.-v[ll]p(;^\
Combination of C8) and C9) gives immediately D0).
Further comparison of D0) with B7) shows that a new family of fully
conservative schemes is contained in family B5)-B7) of describing
conservative schemes with four parameters as a result of employing the integro-
interpolation method.
Another conclusion can be drawn from the preceding that for any cTj
scheme C6) generates an approximation of 0{t + K^) and for cTj = 0.5 it
provides an approximation of 0{t''^ + /i^), that is, only one scheme of the
form C6) can guarantee a second-order approximation in r:
D2) •., = -pf^), '?. = ^i:°^), e, = -pf"'^)tf^),
With the psevdoviscosity in view, we replace everywhere m formulas
C6) the pressure p by the approved rule g =. p -\- lo, leaving us with
D3) v^=-g(p\ Vt = y?''\ e, = -g(^^^K(°-'\ g = P + u^ ■
For the ideal gas and the linear viscosity, this amounts to
D4) p^ri= (y -l)e, lo = --v^.

Conservative difference schemes of nonstationary gas dynamics 535
Putting these together with the supplementary boundary conditions for
i — 0 and i = N, say with the values p^, p^ of pressure, we must write
down the equation of the motion for vj not only at the inner nodes, but
also on the boundary for i = 0 and i :^ N \
0.5ft / ' T \ QSoh
giving f^+-^ and v-'j^ . The remaining quantities rj, e, p are sought only at
the inner half-integer points s^i2, s^j^,---, Sn-i/2-
It seems worthwhile giving one possible example of nonconservative
schemes. A "cross" scheme was very popular and much applicable in recent
decades. Any scheme of this structure can be written on a "chess" grid by
regarding e, p, i] to half-integer nodes (si_|_i/2, ^,-_l_i/2) and v, x to integer
nodes (s,;,l) of the grid at hand.
Within the notations pJ,i/2 = P\ = P ^md v\+ii2 — Vi'^ — Vt etc.,
the "cross" scheme
J+3/2_ ,-1/2 +i +i
'/; + l/2 'li + l/2 _ f'i + i - Vi
i+3/2 ^i-l-1/2 ,_|_i ,'_|_i
= -pi
^i + l/2 ^i + 1/2 _ i-1-3/2 '"i + 1 - •";■
'+1/2 h
admit.s an alternative form
D5) Vf=.-pg, fif = v,, e.t = -pv,,
showing the new members to be sensible ones. In this line the values on
every new layer is found by the explicit formulas. No wishing to load the
book down with full details on this point, we cite only a final result after
multiplying fj = —pj by t)(°-^' and repeating the preceding manipulations
in such a setting:
(e + 0.5.^)^ = -(p,-i,.'"^'),-^^.
where SE = TPfV^°'^'> + O.brpv^f + O.br'^PfVgf, that is, scheme D5) is not
conservative, thus causing some limitations in practical implementations.

536 Difference Methods for Solving Nonlinear Equations
5. Numerical solution of difference equations by Newton's method. As
can readily be observed, the system of nonlinear equations capable of
specifying the values f^+-^, g^'^^ and 77^+-^ on every new layer will be solved
by making several iterations of Newton's method. This can be done by
reducing equations D3)-D4) to the following ones:
■V + o-iTf/j = v-([- ai)Tg-^ ,
i] — 0.5 Tv^ = 77 + 0.5 r fj ,
^ ?} — £ G — 1) + // ■y^, = 0 ,
After that, applying Newton's method yields
D6)
k+l , .k+1 *
AY-0.5rAT=/^
. k + l k I „ , . k + i ,k . . k + i *■
D7) A^r+f/'-^-'A )) +a,{:iT -^])i\g = /;
k + l , * A*^+'- I '' A *'' + '- 1 A*^ + i
k
D8) -A'^I'+agAri +ar] A g + avAV/ = /,
^ = 0,1,2
where
a =1/G-1),
I =1=0 for ^ > 0 ,
° 0 ., , 0
/j =v~v-{l~(Xi)Tgg- a,T g,j
0 0 0 s
fo_ = '/- '? +0-^t{v, + V,),
f,=-l+e-'g(^^H^-,),

Conservative difference schemes of nonstationary gas dynamics 537
k k k k k
/4 =s-agri-avv^ ,
. k+i k+i k
lA V = V — V ,
k+1 k+l k
A i) = T] — r] , etc.
During the course of the elimination of A £ , A ry and A v from
the foregoing we obtain the three-point difference equation for determina-
,. ,*■ + ! . k+i
tion ot y — A (J
where JP is expressed in terms of /^ , /^ , f^ and f^ . The elimination
method can be employed for the last equation, permitting one to find,
first, A V , A rj , A e with knowledge of A gr = y and, second,
k + l . k + L k k + L . k + L , k ,
g = A g + g , v = A v + v , etc.
6. Convergence of Newton's method. We are now in a position to find out
the conditions under which Newton's method converges. With this aim,
the differences
k + l k + l ^ k + l k + l , k + l k + l
0 9 = 9 ~ g, 0 1] = 1] - 7], b V - V - V,
where g, 'fj and v are the exact solutions to equations D3), will be given
special investigation. For this, we write down a typical equation related
to such a difference. By the linearity of equations D6) the homogeneous
equations
D9) 6^V =-(t,T6^j^ , 6^tj' =0.bT6X , fc = 0,l,2,,,.,
follow immediately from the foregoing, Putting equations D7)-D8)
together with the newly formed differences
.k + l ffc + i ft .k+l k + l k
A e = & e - oe , A i] = 6 r] -dr] ,
.k + l ,-A- + l ,-A,- .k + l k + l k
At) = d V - 0 V , A g ~ o g - o g

538 Difference Methods for Solving Nonlinear Equations
one succeeds in showing that equation D7) becomes
= o-i('? -V)^9 -(^iiv-r])Sg ~ [(£-£)+gr(''0(^_ n)]
- a^bri 8g ,
since e — £ + g'^"^'{fi — ?^) = 0 on account of D3). Because of this, we thus
have
E0) 6 e +g^""b rj +a^{Ti-i])b y =(r^bi]6g.
An alternative form of equation D8) is
E1) —0 e +agb i] -\-arj b g -\-avb v =abrjbg.
Indeed, from D8) it follows that
0=-(E e -be)+ay{b rj - brj) + arj [b g - b g)
/ rk + l fix /f *: , - *' * k \
+ av{b v^ -Ov^)-(oe+e-agri-avv^)
r k + l , k fc + l fc fc + l .k + l-l ^
= [-<)£ +ag b rj +ari b g + aub v, \ ~ F4 ,
where
5^ i , i , t , fc c * , ■ '-■■''■ k
F^ ~ ag b rj + arj b g + av b V. + s — arj g — a u v. ■
Substituting here e = agfj + avii^ yields Fa, ^ abr] bg .
Having completed the elimination oi b e from E0) and E1) both,
we arrive at
E2) {al + ^-0) 5*'+i + ((« + ^j,^; „ ^^^) s 't/ + avb't^
k k
= (a + a^) b 7] bg .

Conservative difference schemes of nonstationary gas dynamics 539
-n • i' ck + 1 J- k + 1 , f;k+l ck + l , ,
By inserting now b v^ ~ —cr^TO g^^ and b i] — O.oro v^ we deduce
with regard to
afc = a,T [av + 0.bT{ag + g^^'^)]/[(a + (r,)i] ~ a,!]] > 0
that
E3) b g ~ a^b g^^ = qk& 9 , Qk = -^ 1 .
k
provided the condition (a + (Xj) r] — cr^rj > 0 holds, The meaning of this is
that
E4) ^>—^^?? for all ^ = 0,1,2,,..
a + o-j.
was supposed before proceeding to further derivations,
When the pressure is prescribed for i = 0 and i = A'^, the boundary
k-\-1
conditions for b g are certainly homogeneous;
E5) 5^=0, 6%=0,
After scrutinising the canonical form of equation E3) with respect to
S g (s,)
A(P)y{P)= J2 B{P,Q)y{Q) + F{P)
Qein'(P)
we can be pretty sure that
^(P)>0, B{P,Q)>0, D{P) = A{P)~ Y. B{P,Q)=\.
QdllViP)
This serves to motivate the validity of the maximum principle with regard
to equation E8) supplied by the homogeneous boundary conditions E5).
By utilizing this fact it is plain to show that the estimate holds;
E6) PT|lc<k.lll^^llc.
whence it follows that the iterations converge under either of the following
conditions:
r Vlk\ < « < 1 for all fe = 0,1,2,... ,
E7) .k
<9, b
*■ I ' ' a -\- a,
^ 1] — 0 1] '1

540 Difference Methods for Solving Nonlinear Equations
The preceding is equivalent to the inequalities
fi + b a 7] k i) ~ h qri
E8 \ , ^ ' <^ <-. -, fl>bqv,
1 + 9 1 -?
thus causing, in fact, some restrictions on the step in time in connection
with the dependence upon the variations of volume i] (or density p = l/i]).
0
By setting ^ = 0 and choosing r] = rj we obtain
or, what amounts to the same,
E9) A-9 (!-&))??<'?< A + 9A-&))'?.
When the isotermic flow of the ideal gas A7) is considered, scheme D3)
can be written in simplified form, since the energy equation was disappeared
because T = const . A proper iteration process is governed by the same
rule as in the adiabatic case, the convergence of which can be established in
a similar way without difficulties. In the isotermic case with the assigned
values Y = 1, a = i30, 6 = 0 we deduce instead of E8) that
<r] < -—!— and ~ < q .
I + q 1-9 1]
Under the first or second condition E7) the following relations occur:
thereby justifying the convergence rate as a geometric progression for the
iterations just established.
Numerical calculations for j — 5/3 (« = 1.5) showed that the itera.-
tions within the framework of Newton's method converge even if the steps
r are so large that the shock wave runs over two-three intervals of the
grid LOf^ in one step r, Of course, such a large step is impossible from a
computational point of view in connection with accuracy losses. Thus, the
restrictions imposed on the step r are stipulated by the desired accuracy
rather than by convergence of iterations,

Conservative difference schemes of nonstationary gas dynamics 541
7. Equations of gas dynamics with heat conductivity. We are now
interested in a complex problem in which the gas flow is moving under the heat
conduction condition, In conformity with (I)-G), the system of differential
equations for the ideal gas in Lagrangian variables acquires the form
dv
PV =
dg
ds '
= RT,
In
c —
dv
= c„r,
dt
LO =
dv
OS
( dv
dw
^)'
dT ,
where w = —xip, T)—- i,s the heat flow and g = p + ui.
ds ,
It is plain to create for this system of equations a fully conservative
scheme such as
^t
~gi"\ %=vr^
F0)
w = -kTs, g=p + ui, ui =ui{ri,v^,i^) ,
P^RT/v, e=c,T,
where k = k^ = ><@.5 (??,-„o,5 +'?i+r).5). 0-5 (r,.„o 5 + 7',;_^„ rj). Here a > 0
and f] > 0 are arbitrary numbers.
Generally speaking, Newton's method may be employed for nonlinear
difference equations on every new layer, but the algorithm of the matrix
elimination for a system of two three-point equations (see Chapter 10,
Section I) suits us perfectly for this exceptional case. We will say more about
this later.
The main idea behind this approach is to accelerate and simplify the
algorithms by means of the method of separate or successive eliminations.
To that end, the difference equations F0) are divided into the following
groups;
I: "dynamical group"
vt=-9i"\ % = ^1°'^^ g = p + uj,
uj =.uj{-q,v^,v), p=p{ri,T).

542 Difference Methods for Solving Nonlinear Equations
II: "'heat group"
e = e(rj, T), k^x @.5 {p + P(_i)), 0,5 {T + T^.i^)) .
After that, Newton's method of iterations applies equally well to either of
these groups independently. By analogy with the isotermic case the fir,st
group of equations is to be solved with a prescribed temperature, while
the second one needs the assigned values of rj and v. The essence of the
matter in the last case is that the origin of the heat conduction equation is
stipulated by the available sources of a dynamical nature.
The iterations in the first and second groups can be found successively
by the elimination method. Having completed the ^th iteration, for which
the condition of termination HA^II^-. = Hw — v \\q < ^oW^Wr is fulfilled,
where e^ > 0 is a prescribed accuracy, there is no doubt that the values v
h
and rj are known.
With knowledge of v and 7] the mth iteration T is recovered from the
equations of the second group by Newton's method.
The process of the exterior iterations continues to develop prior to the
occurrence of the convergence conditions.
Another way of going further is connected with re-ordering of these
groups in reverse direction or inserting k = I and m := I in all of the
iterations. The separate elimination method may be of assistance in
minimizing the total volume of the available information in the storage of high-
performance computers.

Economical Difference Schemes
for Multidimensional Problems
in Mathematical Physics
One of the serious developments in computational mathematics owes a debt
to economical difference methods available for solving partial differential
equations of several spatial variables. Recent years have seen the publica.-
tions of numerous papers on this subject for multiple equations of parabolic,
hyperbolic and elliptic types as well as the constructions of various
economical schemes. The general stability theory lies in the foundations of the
possible theory of economical methods which will be given special
investigation throughout the entire chapter. Two classes of admissible economical
schemes are of great importance: schemes with afactorized operator on the
upper layer and additive schemes generating a summarized approximation
in a certain up-agreed sense. These can depend on the range of situations
to be considered.
9.1 THE ALTERNATING DIRECTION METHOD
(THE LONGITUDINAL-TRANSVERSE SCHEME)
FOR THE HEAT CONDUCTION EQUATION
1. Some preliminary information on economical schemes. One of the most
important issues in numerical methods is the well-founded choice of
economical computational algorithms, the realization of which requires a min-
543

544 Economical Difference Schemes for Multidimensional Problems
imal execution time in giving an approximate solution with a prescribed
accuracy £ > 0. The total number of the necessary arithmetic operations
E(e) for doing so is the main characteristic of the algorithms in question,
since other characteristics such as the quality of the related software and
the availability of advanced-architecture computers are beyond our control.
In view of this, the economy requirement becomes rather urgent and
extremely important in numerical solution of multidimensional problems
arising time and again in mathematical physics,
To understand the nature of this a little better, we focus our attention
on the simplest examples serving to motivate what is done with
economical difference schemes and regarding to some preliminaries. The object of
investigation is the heat conduction equation in the space R^:
du sr^ d u
A) ^^ = Lu , Lu = } La U , LaU = -TT^ >
at ^—' ox'-
a=l "
Let G = G'op = {0 < Xa < I, cy = 1,2,.,. ,p} be a cube of the dimension
p; ujf^ = {{iih, , .. , i„h) £ G} be a cubic grid with step h in all directions
x^, a = 1,2,,,. ,p, and u>^ be the grid with step r = tg/rig on the segment
0 < t < to- At the next stage the operator
a
is approximated by the difference operator A„l/ = ?/.?„.,■„• so that A =
Yl'a=:i ^^"' leaving us with the weighted two-layer scheme
yt = A{ay + (l-a)y) , x£uj,^, Q < t =■ nr < tg ,
B)
2/1^, =0, y(x,0) = yo{x).
We recall from Chapter 5, Section 3 that scheme B) is stable with respect
to the initial data under the constraint
1 /i2
- 2 Apr °
By merely setting cr = 0 we have at our disposal the explicit scheme
C) yt=Ay oi- y = y+T Ay,

The alternating direction method 545
which is stable for r < 0.5/i^/p. If equation (I) contains the variable
coefficients, that is, it acquires the form
LaU = -—[k^{x,t)^—), 0 < ^„ < Cj ,
then
AaU= {a^y^J^ , A=^A„, 0 < a„ < c.
and the explicit scheme C) is stable for r < 0.5 Ir/i^pc^).
From here it seems clear that the admissible step in the explicit scheme
is yet to be refined along with increasing the maximum value of the
coefficient of heat conductivity. As a matter of fact, the last requirement is
unreal for the problems with fastly and widely varying coefficients. Just for
this reason explicit schemes are of little use not only for multidimensional
problems, but also for one-dimensional ones (p = 1). On the other hand,
the explicit schemes offer real advantages that the value y — y„,i on every
new layer t„_^.l = t„ + t is found by the explicit formulas C) with a finite
number of operations at every node of the grid ui/j^, so that the amount
of arithmetic operations required in passing from one layer to another is
proportional to the total number of the grid nodes and so it is a quantity
of 0(I//iP),
Being concerned with the implicit scheme for a = I, we may set up
the problem for determination of y"~^^:
2/"+i-rA2/"+'=t/, :y"+M^,, = 0, y{x,i)) = u,{x).
Numerical solution of this system containing l/h^ equations requires, for
example, during the course of Gaussian elimination 0{\/h?^~'^) operations
in connection with a special structure of the matrix E — tA.
Some consensus of opinion is desirable in this matter, since a smaller
number of operations is performed m the explicit scheme, but it is stable
only for sufficiently small values of r. In turn, the implicit scheme being
absolutely stable requires much more arithmetic operations,
What schemes are preferable for later use? Is it possible to bring
together the best qualities of both schemes in line with established priorities?
In other words, the best scheme would be absolutely stable as the implicit
schemes and schould require in passing from one layer to another exactly Q
arithmetic operations. As in the case of the explicit schemes, Q would be
proportional to the total number of the grid nodes so that Q = 0A/h'').

546 Economical Difference Schemes for Multidimensional Problems
Because of these facts, the number of the necessary operations at every
node of the grid is independent of the total number of the grid nodes. All
the schemes with the indicated property are said to be economical.
In what follows one possible example demonstrates for a system of
ordinary differential equations that there is an implicit scheme which is
rather economical than the explicit ones requiring the additional operations.
Example With this aim, it seems worthwhile giving the following system
of differential equations:
— +Au = 0, t>0, u@) = Uq ,
where u = (w*-"^•*(<), . . . , u'^'"\<)) is a vector of order m and A = (flj ■) is a
symmetric positive definite matrix. In passing from one layer to another the
explicit scheme y„^i = y„ —rAy^ requires 2m^ + 2??i arithmetic operations.
Furthermore, let A~ = (a~.) and A'^ = (a+.) be an upper and a lower
tridiagonal matrices with coinciding main diagonals a~ = «+ = 0.5 a^,,.
Both matrices (operators) are positive definite in the sense of the usual
inner product in the space R'", since A" = (A'^y and
{Ax,x) = {A+x,x) + {{A+)*x,x) = 2(A+x,x) = 2(A''x,x) .
Before going further, we initiate the construction of the scheme
E) ^^2n + 2-|/2n + l^^-^^^ ^^+^ Q^ n = 0,l,...,
T
in which it is necessary to perform the inversion of both tridiagonal matrices
(E + tA~) and (E + ryi+) in determining j/2,,,+1 and 1/2,1+2-
It is plain to show that the scheme in view is absolutely stable for any
r > 0, permitting one to eliminate 2/2n+i from the difference equations D)
and E). By subtracting equation E) from equation D) we find that
2 2/2n + l = y'2n + y'2n + 2 + T^"^B/2« + 2 " V^n) ■
Upon substituting the resulting expression into E) we obtain through such
an analysis the scheme

The alternating direction method 547
where the operator _B = {E+tA~)(E+tA'^) is a product of two self-adjoint
"triangle" operators, since yi+ — (A~)*.
Observe here that the factorized operator B also is self-adjoint. By
means of these operators a sufficient condition of stability becomes
S-0.5Bryi) = E + tA + t'^A~A+ ~tA>E
in light of the relations {A~A + x,x} = \\ A +x |p > 0 and A~ A+ > 0. This
supports the view that scheme F) is absolutely stable and it is of second-
order accuracy.
Let A be new tridiagonal matrices differing from A solely by the
zero elements on the main diagonals. While solving equation D) and
equation E) we should save in the storage the vectors ^~2/2n+i ^'""^ ^'''2/2n+2)
respectively, All this enables us to evaluate the number Qi relating to the
necessary operations in passing from the layer <2,i to the layer <2n+2' ^or
scheme D)-E) we have Qi — 'Im^ -\- 12m, while for the explicit scheme -
Qo = 4m^ -I- 4m, that is, Q\ < Qo for m > 4.
2. An alternating direction scheme. Further developments are concerned
with the heat conduction equation of two independent variables that can
serve as test vehicles for the difference schemes to be presented:
G) -^ = Lu+ f{x,t), x£Go2, t£{0,to],
u\^ = fi{x,t) , u(x,0) — Ug{x) ,
Q2„
L u = A u =: (Li + L2) u , La u = j^-Y , a = 1, 2 .
II
dx"''
Here 6*02 = Go = {0 < .t'^, < /„, a = 1, 2} is the rectangle of sides l^ and l^
with the boundary F.
As a first step towards the solution of this problem, it will be sensible
to introduce an equidistant grid w^ in the direction x^ with steps h^ — 1^1 Ni
and /I2 = I2IN2 and the boundary 7^ containing all the nodes lying on the
sides of the rectangle except its vertices, tj^ = o;^ -|- 7;^, We will use them
for later approximation of the operator Lq. by the difference operator
Aay = 2/^„,^■„ , A =: Ai -F A2 .
Recall that in the case of the one-dimensional heat conduction equation
a similar implicit scheme is associated on every layer with the difference

548 Economical Difference Schemes for Multidimensional Problems
boundary-value problem of the form
(8) A,y,_,^Ciy, + B,yi+,=~Fi, i = 1, 2,. .. , TV ~ 1,
yo = Ml > Vn = l-h^ ^i > 0, -Bi > 0 , C'j > Ai + Bi.
This problem can be solved by the standard elimination method requiring
0{l/h) = 0(N) operations, the amount of which is proportional to the
total number .V of nodal points of the grid Cjf^ = {x^ = i7i, 0 < i < A'^}.
With regard to problem G) posed in the rectangle, it is worth noting
several things. The grid tj;, at hand may be treated either as a collection
of the nodes along the row.s i-i = 0,i,... ,^¥2 or as a collection of the
nodes along the columns ii = {),l, . . . , Ni, thereby providing for subsequent
compositions the availability of A'^i + 1 nodal points along every row as well
as A'^2 + 1 nodal points along every column.
In trying to solve a typical problem like (8) by the elimination method
for fixed i^ (or ij, exactly 0(NiN'2) arithmetic operations are needed in
giving a solution at all the nodes of the grid. Their amount is proportional
to the total number of the grid nodes in the plane. The main idea behind
economical methods lies in successive solution of one-dimensional problems
of the type (8) along rows and along columns in passing from one layer to
another.
The scheme ascribed to Peaceman and Rachford provides some
realization of this idea and refers to implicit alternating direction schemes.
The present values y = y" and y = y"~^^ of this difference scheme are put
together with the intermediate value y = j/""*""^", a formal treatment of
which is the value of y at moment t = <,j_|_w2 = ^„ + t/S. The passage
from the «th layer to the (n + l)th layer can be done in two steps with the
appropriate spacings 0.5 r:
n + 1/2 __ n
(9) ^ ^, ^ =Aiy"+^/^+A2t/" + y",
0.5 T
n + l _ ,,n + l/2
A0) ^ ^ =:AiJ/« + l/2+A22/" + l+^".
U. 0 T
These equations are written at all inner nodes x — x^ of the grid lo^^
and for all < = <^ > 0. Let us stress here that the first scheme is implicit
along the direction x^ and it is explicit along the direction 2:21 while the
second one is explicit in the direction x^ and it is implicit in the direction
x^. Equations (9)-A0) are supplemented with the initial conditions
A1) y{x,^) = u^{x), xeoj,^,

The alternating direction method 549
and the difference boundary conditions, for example, of the special type
A2) j/"+^ ^ 1^1"+^ for i,_ = 0 and i^ = N2 ,
for ij = 0 and ij = N^ ,
A3)
where
A4)
y"+''' = fi
A=^(.-
The meaning of the boundary condition A2) is known to us. On the other
hand, condition A3), which assigns the boundary value y, needs certain
clarification. In this way, the difference boundary-value problem (9)-A4)
can be put in correspondence with problem G). The method for solving
this difference problem is mostly based on alternative forms of equations
(9)^A0):
A5)
To make our exposition more transparent, it is more convenient to introduce
the new members
and approve the following rule: when one of the subscripts is kept fixed, we
omit it for a while in relevant expressions. This should cause no confusion
and guides a proper choice of alternative forms of equation A5) for later
use:
~ y - An/ =
T
- 2/ - A22/ =
T
= F,
~~F,
F =
F =
■2
- - y + Any + 1^ ,
T
2 - A -
= - y + Aiy + f.
T
A6)
iyn--i~2K^ + ~h, + /^
2 y»i~i ~ 2 T^ + 71 yu + i7i y^+i = ~^>^ ■
A7)
ii = 1,2,.. . , A'l - 1 , y=/i for ii = 0,.'Vi
1. ofjLM- 1- - T?
i2 = 1,2,... ,7V2~ 1, y = fi for ij = 0, TVs

550 Economical Difference Schemes for Multidimensional Problems
Let a value y = (/"' be given. Starting from F, we move further
along the rows 12 — 1, 2,. .. , N2 — i to solve problem A6) by the standard
elimination method, whose use permits us to determine the values y at all
the nodes of the grid uif^. After that, we calculate F and move along the
columns ii = 1,2,... , A^i ~ 1 in an attempt to solve problem A7) and find
the values y ~ y"+i. In passing from the (n+l)th layer to the (n+2)th layer
the same procedure is workable, thus causing the alternating directions.
Since the elimination method requires several operations at one node,
the total number of which is independent of the grid step, the algorithmjust
established will be economical if one succeeds in showing that scheme (9)-
A4) is absolutely stable. The following sections place a special emphasis
on stability and convergence of the scheme concerned.
3. Stability. Subsequent considerations of stability of scheme (9)-A4) are
conducted with a priori elimination of the intermediate value y. This can
be done by subtracting equation A0) from equation (9) and re-ordering of
the relevant one as
A8) 2y = y + y^0.bTA2{y-y), xeto,^,
Substituting A8) into (9) yields
A9) y^^ _ i A2(^(/ ~ J/) = r Ai(.y + y) ~ 7 AiA2(y ~ y) + A22/ + y?.
With the relation y ~ y + ry^ in view, the intention is to use A9) in the
canonical form
B0) (^~0.5rAi)(^~0.5rA2J/i = A2/ + ^.
Under such an approach formula A8) should also be valid for Xj = 0 and
Xj = /j, since otherwise {Aiy)j refers to the undetermined values for ij = 1
and «j = A^i — 1. Knowing y = ft and y = fi for x^ = 0 and x^ = /^, we
deduce from A8) that
1 . r2
y = - [fi + fi) — A2/i( = /i for Xj = 0 and Xj = /j ,
by observing that these coincide with the boundary conditions A3)-A4).
Thus, we have proved that a solution of problem (9)~-A4) satisfies equation
B0) subject to the supplementary conditions
B1) ij\ =fi, y\y,=l^' y{x,0) = Uo{x).

The alternating direction method 551
On the other hand, a solution of problem B0)-B1) applies equally well to
problem (9)-A4). Indeed, by specifying y by formula A8) we deduce from
A8) that
{E-Q.hTK-2)y = 2y-{E + 0.bTA2)y
and insert then the resulting expression in B0). By minor changes we are
led to equation (9), which in combination with A8) gives A0). This
provides enough reason to establish the equivalence between problem (9)-A4)
and problem B0)-B1) with compliance of the boundary values y assigned
by formulas A3)^A4). Careful analysis of scheme (9)-A4) is accompanied
by more a detailed exploration of scheme B0)-B1) in "integer steps".
The general theory of two-layer schemes is quite applicable in such a
setting. By regarding the boundary conditions to be homogeneous we turn
to the problem
B2) {E-0.bTAi){E-0.bTA2)yt=Ay + 'P, i>0,
y{x,0) := Ua{x), y|^^^ = 0,
with further reference to the space H of all grid functions given on the set
uif^ and vanishing on the boundary jj^ under the inner product structure
(y,v) = 2, y(x) v(x) h^ h^ = 2, z, ?/('''i/in *V*2)'''('''i''-i^ *V*2) ^i'*2 ■
.TgW;, Jl = l 22 = 1
The associated norm is taken, as usual, to be || j/1| = v (j/, J/). We refer
to the operator A :=: —A :=: —(Ai -|- A2), which, by definition, is self-adjoint
and positive in the space H. The norm on the energetic space Ha is defined
either by
\\y\\\ = J2 J2 {vsAhf^i^hK)) KK+J2 J2{ysAhih'h'h)) KK
or by
B3) ll2/ll; = llfeJI? + lb.J^
When treating y = y{t) as an abstract function of the argument t £ ui^
with the values in the space H, scheme B2) admits an alternative form
B4) Byt + Ay:^ ^{t), 0 < t := nr < t, , y{0) = u, ,

552 Economical Difference Schemes for Multidimensional Problems
where B = {E + O.b tA^) {E - 0.5 t A2), Aa = -A„, A = Ai + A2.
Because of the rectangular form of the initial domain, the operators
Ai and A2 are self-adjoint, positive and commutative. It is straightforward
to verify the relations AiA-2 — A^Ai and A1A2J/ — A2A1J/ = yg^,i-^j-^^-^ at all
the inner nodes of the grid. In view of this, the strict inequality AiA'2 > 0
is simple to follow. From the form B4) it seems clear that
B5) B>E+0.bT,
thereby clarifying that scheme B4) is stable in the space Ha- Indeed,
2
B - 0,5 tA= (^E+^A + ^ A, A^^~\a
= E+ —AiA2> E .
From condition B5) it follows that for scheme B4) Theorem 7 from Chapter
6, Section 2 is still valid with e = 1, due to which a solution of problem
B2) satisfies the inequality
B6) ||2/(i + r)|U<||2/@)||^ + i=f^r||^(i')||^
Also, the a priori estimate holds true:
B7) II y{t + r) II < 11 y@) II + ^ f ^ r |l^(i')IL-i
1/2
1/2
To make sure of it, we apply the operator A '^ > 0 to both parts of equation
B4). The outcome of this
Bijt+Ay = ifi, A = E, ifi = A~^ip,
B8) , 2
B = A~' + -^E+-A~'Aa2.
Since the operators Ai, A2 and A"^ > 0 are commutative and self-adjoint,
the relations A~^AiA2 > 0 and B > A~^ +0.b t E occur. Applying
Theorem 10 from Chapter 6, Section 2 yields estimate B7).

The alternating dh-ection method
553
Theorem 1 Scheme B2) is stable with respect to the initial data and
the right-hand side. A sohition of problem B2) satisfies a priori estimates
B6)-B7).
4. Stability and accuracy. By utilizing the fact that scheme (9)~A4) is
equivalent to scheme B0)-B1) subsequent considerations of convergence
and accuracy of the first scheme will appear for the second one. Let u =
u{x,t) be a solution of problem G) and y = y(x{,t„) be a solution of
problem (9)-A4) and scheme B0)-B1). Upon substituting y = z + u into
B0) we may set up the problem for the error of this scheme:
B9)
B z^ = A z + ip , X £ uif^ ^ 0 <t — nr < tg ,
z\^^=0, z{x,0)=0,
where B = {E — 0.5 r A^) (£' — 0.5 r A2) and 0 is the error of approximation
equal to
C0) xp = if + All ~ B Uf =0.5 A(u + 11) — ii^ — r^Ai A2 Uf + ip .
From such reasoning it seems clear that
under the condition that the solution u = u{x,t) possesses in the region
Qt = Go X [Ojig] the derivatives
C1)
< M ,
d^u
9x2 gj,2 Ql
<M
d\
< M,
d X
5^
< M,
The final result is an immediate implication of the asymptotic relations
0.5 {u-\- u) = u + Oi^T") and u^ = u + 0{t"), where u = ii[x,t^ + 0.5 r)
and the quantity AiA^u is bounded. Since for problem B9) estimate B6) is
valid for the assigned value z@) = Zq — 0, we might formulate the following
assertion.
Theorem 2 Under conditions C1) scheme B0)-B1) cojiverges in the grid
norm B3) with the rate 0{t'^ + \h\'-).

554 Economical Difference Schemes for Multidimensional Problems
5. A scheme for the governing equation with variable coefficients. The
intention is to use the alternating direction scheme for the heat conduction
equation with variable coefficients
f9w
C2) -^=Lu + f^ ix,t)eQT,
m|p = fi{x,t), u{x,0) = Ug{x) ,
Lu = Liu + L2U , LaU= -—(k^{x,t) -—) ,
ox^ \ dx^y
kjx,t) >0.
In such setting, for any if, the operator La is approximated by the difference
operator
A„2/ = Aa{t)y = {a„{x,t)y^J^^, a= 1,2,
where a^ can be founded either by the formula a„ = ;fc(~0'5o) or by the
formula a^^ = 0.5 {k^ + ^(~^°^), a = 1,2, which guarantee a second-order
approximation provided by the operator A^:
A„u - LaV = 0{hl) .
Instead of (9)-(I0) we are trying to adopt another scheme
C3) y^ = A,{i)y + A,{tJy"+^"
l^-^^ = A,{i)y + A,{t„^,)y"+' + ^"
*' = *n+i/2, y{x,0) ^Uo{x),
with the boundary conditions
t/"+i =/," + i for i, = 0, 4 = N2,
C4)
y = /J. for ij = 0 , ij = N^ ,

The alternating direction method 555
where
[J^2H)t,n = (A2(i„)/^(i„))^ = ^ .
The value /i is put in correspondence with the expression
C5) y= ^ —{K2y)t,n,
arising from the statement of the difference problem C3) during the course
of the elimination of Aij/.
In this regard, it should be taken into account that scheme C3)-C4)
is equivalent to the factorized scheme B0)~B1) with the member A„'t/ =
{a^{x)yg ) , in special cases: either in scheme C3) the operator A2(i")
is involved at one and the same moment of time t in place of A2(i„) and
A2(i„_|_i) or h^[x) and, hence, A„ are independent of i. Recall that scheme
B0) provides on a solution ■ii[x,t) an approximation of 0{\h\^) + r^ if the
usual requirements of smoothness of k,^{x) in the variables x^ and Xj are
imposed in addition to conditions C1). The principal difference from the
case of constant coefficients is discovered in the stability analysis of scheme
B0) by observing that the operators Ai = —yVi and A2 = —A2 are positive
and self-adjoint. But, unfortunately, they are non-commutative:
where a,^ — a„(x,t). Just for this reason the product AivVo is not obliged
to be positive, thus causing some difficulties. In view of this, it is possible
to establish the stability only for sufficiently small values of r < ro(cj),
where Cj depends on the maximum of the derivatives of fe„ with respect to
Xj and X2- Let us stress here that r < T(,(Cj) is a severe restriction and, as a
matter of experience, it is connected with the available methods of special
investigations of stability. In what follows we will show that scheme C3) is
absolutely stable in another norm.
With this aim, we first write down the equation for the error by
introducing the new variables
z" =:y" - u" , z"+^ = y"+^ - m"+i , z = y-u,

556 Economical Difference Schemes for Multidimensional Problems
here u" = u{x,t,^), «" + '■ = u{x,t„_^_^) and u is calculated by the formula
wr
C6) u = — (A2tOi,„ ■
This is consistent with C5) as stated before. Under such a choice of u we
obtain the homogeneous boundary conditions for specifying z.
By substituting into equation C3) y" = z" + tt", y = z + u and
yu+i _ ^n+i _|_ yii+i ^g j^j.g jg(j i^Q ^}jg pi-oblem statement
0.5r
C7)
^ = Ai(t„+i/2)^ + A2(<„)z" + V'r
.^n+l
'05/ = Ai(i„,+i/2M +A2(Wi)^"^' +€ '
C8) ^17.= 0' z{x,0) = 0,
where -ip" and ■0" are the appropriate errors of approximation:
ft - ti"
C = ^iitn+112) u + A2(i„+i)«"+! + ^" - -^^^^
Before giving further motivations, it is worth noting here that i/i" =
i/)" This fact can readily be verified by substituting expression C6) into
the formula
^," ~r,= Mtn + l) «" + ' - Mtn) «" n. " = 0 ■
1O5^
At the next stage expression C6) is needed in the formula for the residual
■0«
,„n_^yn+l ^2
/ 7/" _L 7;"T^^ r \
^r = Ai(i„ + i/2) (^ ^(A2«).,n) +A2(iJ«"
+ y"-""^'~""+^(A2»),n-

The alternating direction method 557
with further reference to the following asymptotic relations:
= "(.c.^,+i/2) + O(t') ,
2
dt
'-'n-l-l/2
As a final result we obtain
.,,"+1 _ ,,"
V^r = (Ai « + A. u)\t=t^^^,, ~ + V"+ 0[t')
thereby justifying that scheme C3)-C4) generates a second-order
approximation:
r,=r,=o[T^ + \h\').
Further development of some a priori estimate for a solution of the
problem concerned is mostly based on an operator-difference analog of
problem C7)-C8) such as
+ A,{t„^,l,)z + A,{t^)z-^r,
0.5 r
C9)
^n-l-l _ 2
Q5^ + M{tn + ll2) 5 + MU + l) ^" + ' = C ,
n = 0,l,2,,,,, 2@) = 0,
where Ai{t) and A2{t) are linear operators in a Hilbert space H:
A, : H ^ H , An : H ^ H .

558 Economical Difference Schemes for Multidimensional Problems
In dealing with non-negative and, generally speaking, non-self-adjoint
operators Ai > 0 and A2 > 0, it will be sensible to omit for a while any
subscripts and superscripts
.42 = A2(i„,+i), A2 = A2it„), .4: = Ai(<„_^i/2),
allowing a simpler writing of the ensuing formulas:
{E+0.bTAi)z= {E~(}.bTA2)z + 0.bTlP, ,
{E + 0.5tA2)z = (i; - 0.5 rii)i-h 0.5 rV-a-
The triangle inequality gives
D0) \\{E + 0.5tAj_)z\\ < \\{E~0.5tA2)z\\ + 0.5t\\iP^\\,
D1) \\{E + 0.5tA2)z\\<\\{E~0.5tA,)z\\ + (}.5t\\^,\\.
An auxiliary lemma may be useful in the sequel.
Lemma 1 If .4 > 0 is a linear operator in a Hilbert space H, then
D2) \\{E~{\-(T)TA)y\\<\\{E+aTA)y\\ for cr > 0.5 .
This assertion is an immediate implication of the chain of the relations
\\{E + aTA)yf~\\{E~{\~a)TA)yf
= 2r(Aj/,j/) + 2(,T-0.5)r2||Aj/||2>0,
which are valid for a > 0.5 and the operator A > 0.
Collecting inequalities D0) and D1) and taking into account estimate
D2) with the value a = 0.5 incorporated, we proceed to the elimination of
z. Following established practice, we arrive at
II (i? + 0.5 ri2) ^ II < II (i^ + 0.5 rA2)z II+^A1^,11+ 11 ^,11),
yielding with the aid of inequality D2) either
II (i? + 0.5 ri2) f II < II (i? + 0.5 r A2) z 11 + ^ (II ^J| +II ^,11)

The alternating direction method 559
or
II {E + 0.5TA,it,^,)) z^-+^ II < II (E + O.^TA^it,)) z* 11 + ^ (II ^f 11 + 11 0,* II) ,
Summing up over fc =: 0,1, ... , n and inserting z° = 0, we find that
D3) \\z"+\^ = \\{E + 0.5TA2{t„^,))z"+'\\
1
where
<2E^(ii^fii + ii'^'ii)
D4) ||.||;;, = II (£^ + 0,5r.4:,):-|r-' = IU||^' + r(A2Z,.) + ^ 11.423 11^
Observe that estimate D3) remains unchanged if the norm H^IL i\ is replaced
either by the norm || z || or by the norm
vir
tue
of the
rela
II
lkllB) =
it ions
^lP<lh
= (lU
^llm
IP +
and
?"
Ik
A2Z
ii(=,
11=
<
y/2
ll^lla)
This type of situation is covered by the following assertion.
Theorem 3 Scheme C3)-C4) is absolutely stable (for any h^, /i2 and t)
an d con verges wi th the rate 0{\h\'''') + T^ m the norm \\ ■ ||, , given by formula
D4) under the conditions which guarantee a second-order approximation
on a solution u = u{x,t) of problem C2).
Recall that a second-order accuracy of scheme C3)-C4) is ensured
by making a special choice of the boundary conditions for the intermediate
value y ~ y such as
At" + ju"+i
r
2
y = fi for I'l =(},Ni, 11 = ^ (A2/^)( „ ,
It is pos,sible to demonstrate that the accuracy 0(|/ip) + r^ of this
scheme remains unchanged if we might agree to consider
n I n + l
D5) y = ^^—^ for i, =0,N,.

560 Economical Difference Schemes for Multidimensional Problems
This means that the second term 0G"^) in the available exppression for /i
will be excluded from further consideration. The proof of this statement
is omitted here. As a matter of fact, the stability of the scheme at hand
with respect to the boundary conditions is revealed through such a stability
analysis.
It is worth mentioning here several things for later use. Scheme C3)
with the boundary conditions D5) is in common u.sage for "step-shaped"
regions G, whose sides are parallel to the coordinate axes. In the case of an
arbitrary domain this scheme is of accuracy Odftp + r^^A). Scheme (9)-
A0) cannot be formally generalized for the three-dimensional case, since
the instability is revealed in the resulting scheme.
6. A higher-accuracy scheme. By minor changes we are led to a higher-
accuracy scheme such as
^11-1-1/2 _ y,i
= a,Aiy"+"^ + {l-a,)A2if+a,
^
D6)
...11 + 1 _ ..n + l/2
^ y = A - a,)A,y" + '/'' + a, A. 1)"+' + {l-a,)^\
xEojf^, ra = 0,1,2,..., y{x,0) = Uo(x), 2; 6 w,, ,
D7) j/"+i = /^"+' for ^2 = 0 , i,_ = N2 ,
j/"+i/2 = fi for I'l = 0 , ii = Ni ,
where
fi = a, /."+1 + A - ,tJ/." - r A2(<Tj<T,A«"+i _ A _ ^J A ^ a,) ^i")
1 h^ h'; hi ^
'n-i-1/2
It is plain to show that the scheme in view i.s stable and generates an
approximation of 0(r^-|-|/i|''). With this aim, it seems worthwhile to design
a factorized scheme for the same reason as before. The starting point is to
eliminate j/„_|_i/2 from equations D6) in the process of transformations
D8) Sij/"+i/2::.C'2j/"+.T,r^", Cy^+'l'' ^ B^,y-+'~ {\ ~ a,) T ^\

The alternating direction method 561
where _S„ = E ~ a^r Ka and Ca = E -\- [\ ~ u^) t Ka, a = 1,2.
Having multiplied the first equation D8) by 1 ~ (Tj and the second one
by (Tj and having added the final result, we find with the aid of the relation
A — (tJ5i + cr^C'i = E the intermediate value
D9) if + 'l^ = a, B^y"+' + A - <t,)C, y" .
Substituting this expression into the first equation D8) yields
a,B,Bry"+' = {C^ - A - <t,) B^C^) y" + <t,t^"
or
E0) Si B2 — = (Ai + A2 + A ~ (Ti - cr,) r Ai A2) y" + ip" .
T
By observing that
n ^ ^"^^"
this provides enough reason to rewrite the preciding scheme as
yn + i _ n
E1) B, B-2 '-^ ^ = A'j/" + /' , X 6 ^, ,
T
E2) y{x, 0) = Uo{x) for x 6 W;, , y" = yu" for x 6 7/,, ,
where
A'y = (Ai + A2) y + ' ^^ ' ^1^2?/.
Simple algebra gives
E3) ^P = A'«" + ^" - B, B2 ^ = 0{t' + \h\''),
T
thereby justifying that scheme E1)-E2) generates an approximation of

562 Economical Difference Schemes for Multidimensional Problems
0{t'^ + |/i|''). This is due to the asymptotic expansions
/ h^ h"^ ]? + h"^ ^
+ ^"+0(|ft|4)
/^; hi \ du hi , hi
12 12 ' dt 12 •' 12 -'
+ ip"+0{\h\'
BiB-2
n + i
T , ^ K hi
«i - o (Ai + A2) u, + -^ yViW( + -| A2M(
12
12
i = i„
0(r2 + |/^h
(9u r (9^u
T du h] du hi du\
^a7 + T2^^a7 + T2^^a7J
+ 0[T'+\h\*),
arising from the well-e.stablished relations
lIu= -L1L2U +La ^—- Laf , a =1,2.
dt
Further .stability analysis is connected with the homogeneous bound-
0
ary conditions. In preparation for this, we introduce the space H = Q ^ of
0
grid functions and refer to the operators Aa y = -j/g j. , (/ 6 fl/i, as
suggested before. All this enables us to concentrate on an operator-difference
scheme
E4) Byt+A'y = ^{t), 0<t = nT, y{0) = u„ ,
with the members
B = {E + a, TAi){E + a^_TA2), A' = Ai + A^ - {x, + x,_) Ai Ao ,
hl/12, aa = 0.5->cjT, a =1,2.

The alternating direction method 563
Here the operator.s Ai and A-j are supposed to be self-adjoint, positi\'e
and commuting:
Al=Ai>(}, Al = A2>0, AiA2 = A2A,,
implying that {Ai A2)* = Ai A2 > 0.
The simple observations that || Aa \\ < 4:/h''^ and x„ || Aa \\ < 1/3 may
be of help in verifying the stability condition in the space H^ : B > 0.5 r/!'.
Furthermore, with the aid of the inequality A„ < || Aa || E we obtain
B - - A' = E - H^Aj^ ~ x^M + 2 (^1 + ^2) MM
- E - H^Ai - ^2.42 + (^ + X1X2J MM
> E - H,\\A,\\E - H^\\M\\E <]^E ,
meaning B > 0.5 rA' + Ej'i and confirming the stability of the scheme in
view in the space Ha'■
In particular, Theorem 7 in Chapter 6, Section 2 states that scheme
E4) satisfies the a priori estimate
E5) iij/'+Niv<iiy°iL' + \/f (E^ii^'"in •
In trying to evaluate the accuracy of scheme E3) we set up the problem
for the error z"+i = jy"+i - w"+i:
Bz.^Mz^'i,^ 2@) = 0,
whose solution obeys estimate E5)
/ i \ 1/2
E6) lk^'+^IU<~(E^II'^''ll'
\i'=o
In passing from E5) to E6) we have taken into account that 2" = j/° — u" =
0 and that the operator A' must satisfy the inequalities
\ A or U\. > \ ||.||^

564 Economical Difference Schemes for Multidimensional Problems
which are immediate implications of the chain of the relations
2
A' = A- {x, + x^) A1A2 >A~x, II Ai II A2 ~ x^\\A2\\Ai> ~A.
Estimate E6) serves to motivate that scheme E1)-E2) and one of its
equivalent schemes, namely scheme D6)-D7) converge in the space Ha. with the
rateO(r2 + |/^|4),
Remark By analogy with the preceding section it is possible to evaluate
the errors of approximation for either of the equations D8) such as
r r
This can be done using formula D9) for the intermediate value it of artificial
character
Upon substituting this fictitious value into the above formula for '0j we find
that
■lb 1/" + ^ - 7/"
a, T
thereby clarifying that the quantity ^\cr^ coincides with the (jrror of
approximation for the factorized scheme E1) that is known to us from formula
E3).
By virtue of the relation
01 i>2
(Tj 1 "" "
we can be pretty sure that every of the equations D8) generates an
approximation of 0(| ft j*-^ r^).
9.2 ECONOMICAL FACTORIZED SCHEMES
1. Schemes with factorized operator. We now consider the two-layer
difference scheme
A) By^^Ay = ^f, 0<t = JT<t„, j = 0,l,..., y{(}) = y^ .

Economical factorized schemes 565
Knowing the value y = y^ on the jth layer, it is required to find the value
j/^ + '-. In preparation for this, we derive the equation related to j/^ + '- with
a known right-hand side F^:
B) BiJ + '=F^, F^ = {B-TA)/y^ +T^\ j = 0,l,...,.
As can readily be observed, 0{N) operations are needed in giving F-^ and
their amount is proportional to the total number of the grid nodes. This
is certainly so with any difference scheme, whose pattern is independent
of the grid. From equation B) it is easily seen that the stable scheme B)
will be "economical" once the users perform 0{N) operations while solving
equation B).
Let "economical" operators B^, a = 1,2,... ,p, be such that 0(N)
operations are necessary in connection with solving the equation
C) B,v = F.
Then scheme A) with a factorized operator B of the structure
D) B = BiB2 ■■■ Bp
will also be "economical" in line with established priorities: the
numerical solution to equation B) with operator D) requires 0{N) operations.
Indeed, a solution to the equation
Ej B,B2---Bpy^+' = F^
can be found by successive solution of p equations of the form C):
F) Biy^yi-F\ B,,yia)-y{a-i)< a = 2,3,. .. ,p ,
.so that j/^+i = j/(p). Here j/(ij = y^ + ^'P, ..., j/(„j = */•'+"'''', . . ., j/(p_i) =
yJ+ip-'L)/p stand for intermediate values arising in the process of
calculations. It follows from the foregoing that the stable scheme A) with the
factorized operator B being a product of a finite number of "economical"
operators Bi, .. . , Bp becomes "economical", so there is some reason to be
concerned about this. All the schemes with a factorized operator B are
called factorized schemes. It was shown in Section 1 that the
economical alternating direction scheme is equivalent to the factorized scheme with
the operator
G) B = B,B2, B^ = E~(}.5tA^, A^y^y^^^^, a =1,2.

566 Economical Difference Schemes for Multidimensional Problems
Especial attention is being paid to the factorized schemes with the members
(8) B = BiB2, Ba = E+TRc,,
where Ri and R2 are "triangle" operators with associated triangle matrices.
These fall within the category of explicit alternating direction schemes. Let
us stress here that the operators R,i and R2 may be, generally speaking,
non-self-adjoint, but they are always mutually adjoint to each other. In
that case a solution to equation C) can be found by "through execution"
formulas.
Undoubtedly, the reader comes across difference operators Ba of the
structure Ba = E — arAa, where A^ approximates the differential operator
La with partial derivatives of one argument x^- For example, if L„u =
d'^u/dx'^, then Kay — y^ ,,. is a three-point operator, who.se use permits
us to solve equation C) by the elimination method. It is worth mentioning
here that any difference scheme can be reduced to a sequence of simpler
schemes in a number of different ways. This is certainly so with scheme
A), implying that
y -I- 1 _ yj _|_ .J- y,7 ^
where w^ is a solution to the equation
(9) B1B2 ... 5p tw = $^', $^' = ^p^ - Ay^ .
The value iv^ can be recovered from the governing system of p equations
A0) _SiW(ij = $^ 5„ W(„^ = W(„_i), a = 2,3,...,p,
with further reference to
A1)
It is worth recalling here that the first economical schemes were intended
for the elimination of intermediate values with no problems. The main idea
behind this approach is to involve factorized schemes "in integer steps",
a key role of which is to relate the values y^ and ?/"'"'" in some or other
convenient ways.

Economical factorized schemes 567
2. The boundary conditions. As one might expect, stability and
approximation take place for the factorized scheme A). In this view, it seems
reasonable to adopt ec'iuations F) or (lO)-(ll) as a perfect computational
algorithm in designing the factorized scheme (]). But this equivalence can
be established only with consistent boundary conditions and needs certain
clarification.
In the forthcoming example the first boundary-value problem is posed
for the heat conduction equation in the rectangle Go — {0 < Xj^ < l^^, 0 <
'^2 ^ h} with the boundary F:
A2) -i^ = iLi + L2)u + f{x,t), u\^--=fi{x,t), u(x,0)^u,{x},
where L„w = d'^u/dx'^^, a = 1,2.
We begin by placing the factorized scheme A) with A„j/ = y^ x o^'^ ^
rectangular grid Qf^ = {(ij /ij,i2 /i2)} with steps h^ and h^ in the specified
domain Gq-
A3) 5i B, J/, = A J/ + V- , y^ \^^^ = /j.^ , j/" = u,ix).
Here and in all that follows 5„ = E — (TrA„, A =: Ai -|- A2 and ji^ is the
boundary of the grid W;j. In passing from one layer to another algorithm
F) is performed for problem A3):
A4) Bij/(i) = F, F' ={BiB2 + TA)ii +T^, B2 2/>i=y(^),
which is put together with the boundary conditions j/-'''"^| = p-'"'"^.
As far as the operator Bi B2 on lo^^ is concerned (including the
boundary Xj = 0, Xi = /j), the equation B^y^'^^ — y(y\ should be valid not only
for 0 < Xj < /j, but also on the l3oundary for x^ = 0 and x^ = /j. Since
the values j/^ + ^ | = /.t^ + ^ are already known, it follows from the foregoing
that
A5) j/(i) = (£-(TrA2)//^ + ' =/^^' + ^-(TrA2/^^'+i for x,^{],l,.
When specifying j/z^j for Xy = 0, /j by the preceding formula, problems
A3) and A4)-A5) become equivalent. This fact can easily be verified
during the course of the elimination of yiy-^ from A4), In the framework of
the second algorithm we accept
i3lW(i) = $^ $^ =: A ?/' -I- ^^',
A6)
^2 w,2, = W(| ,, j/^ + ^ = ;(/ -I-ru-'C2)

568 Economical Difference Schemes for Multidimensional Problems
and the boundary conditions are imposed to be
pj + i _ pi
cj/jN = (£■ — (TT A2) for a:;j=:0,/j
A7)
,(j + i _ pj
J/2) = for Xj = 0, 4
that is, W(i> = 0 and u)^--,^ = 0 on the boundary 7), if/,/ is independent of t.
Observe that in giving scheme A) in matrix (operator) form it is
possible to take into consideration the homogeneous boundary conditions by
rearranging the right-hand side ip at the near-boundary nodes. The design
of the factorized scheme also involves the homogeneous boundary conditions
(j/(.js =z j/^ =0, cjn\ = u>f2) = 0 for a; e 7;J, but the retention of the
approximation order necessitates imposing the extra member —(T^r^/i^^A2/^( on
the right-hand side of this scheme at the near-boundary nodes for I'j = 1
and ij — A^^ — 1.
3. Constructions of economical factorized schemes. Using the regulariza-
tion method behind, we try to develop the general method for constructing
stable economical difference schemes on the basis of the primary stable
scheme
A8) B^ — + Ay"=^"
T
with an operator of the structure
A9) B = E + tR.
The relation B > 0.5 tA is ensured by the stability property of this scheme.
In such a setting it is preassumed that i? is a sum of a finite number
of "economical" operators i?„, a — 1,2,... ,p:
B0) i? = i?i -F ■ ■ ■ -b i?p .
The operator B can be factorized by replacing B = E + t [Ri -|- • ■ ■ -|- Rp)
by the factorized operator
B1) B = Bi--Bp, Ba = E + rRa,
making it possible to ignore the primary scheme A) in favour of the
factorized scheme
B2) Bi---Bpyt+Ay = ^.

Economical factorized schemes
569
Sometimes such a passage permits one to justify the approximation being
ip instead of ^ near the boundary of the grid domain,
If the primary
torized
R2, ..
(Re >
a, P =
scheme
schenie
B2) in
A8) is stable,
then
the case, where the
. , Rp are self-adjoint (Ra =
0; and
1,2, .,
pairwise comiTiuiative
,P)-
K)>
(Ra
SO is
opera
JlOJl-
Rp =
the
factors Ry ,
negative
Rp R-a,
By virtue of the indicated properties of the operators Ra their
products Ra Rp, RaRp R-y, etc., will be self-adjoint non-negative linear
operators. This provides enough reason to conclude that
B =z BiB2= E + t{Ri + R2) + t'Ri R2
= B + t'^RiR2>B for p=2,
B = Bi B2 ■ ■ ■ Bp = E + T {Ri + R2 + ■ ■ ■ + Rp) + t'Qp
= B + T^'Qp >B,
where Qp = Qp > 0.
Thus, B ^ B > 0.5 tA, meaning the stability of the factorized scheme
B2). The operators i?„ are so chosen as to satisfy the condition of
approximation, too. The forthcoming example helps clarify what is done.
Example 1 We are looking for a solution of the first boundary-value
problem for the heat conduction equation with variable coefficients
B3)
-^ = Liu-\- L2U-\- f{x,t), x-eG, t>0,
w|p - fi.{x,t), u{x,0) = Wo(.-c),
LaU=-—(k^[x,t)-—j, 0 < Cj < A;^-, < C2, a = 1,
G = {0<a;„ </„, a = 1,2}.

570 Economical Difference Schemes for Multidimensional Problems
For this, we have occasion to use the two-layer factorized scheme
B4) {E+TRi){E + TR2)yt+Ay = ^,
X (z uij-,, t =■ nr , ?"» = 0, 1, .. . ,
y\-y^ - K^>t) , t -nr >0,
y{x, 0) = Ug{x), a; e W/j ,
where tD/j = {x^ = (ij h^, i^ h^), i„ = 0, 1,. .. , A^„, h^Na = l„, a = 1, 2} is
the grid in the rectangle G with the boundary ')\.
The operator L„ is approximated to second order by the difference
operator
Aaj/= (a„(a:;,i)j/j^) , Q < c^ < a^{x,t) < c^, a = l,2.
Let
A = -(Ai+A2), i?a = -crC2Aa, « = 1, 2 ,
0
where A „?/ = ?/f„r„-
Stability analysis is mostly based on the assumptions that the bound-
0
ary conditions are homogeneous and i/ = fl /i. is the space of all grid
functions given on the grid u)/j and vanishing on the boundary ■jj^. The inner
product structure is the same as suggested before for problem B2). A brief
survey of the properties of the operators A and i?„ as operators in that
space H is presented below:
0 0 0 0
A = A*>0, Aa<c^Aa, AaV^-AaV for any y e Qi, ■.
B5)
R„=z(TC.^Aa, Ra = K>0' ^ = [,2, Ri R2 = R2 Rl ■
As can readily be observed, the stability condition
B = [E + TRi)iE + TR2) > 0.5rA
will be ensured if cr > 0,5 or even if cr > 0.5 — 1/(t|| A ||), It seems clear
that the three-point difference operators Ba = E + tR^ with constant
coefficients are "economical", since the equations
BaLO =: [E + TRc)iO = Fa, « = 1, 2 ,

Economical factorized schemes 571
can be solved using Gaussian elimination along the rows for a = 1 as well
as along the columns for a = 2.
Under such an approach the factorized scheme is of no less than first-
order accuracy in r. In a similar way an economical factorized scheme can
be designed in the p-dimensional case when
aw ''
B3') ^ = E^«« + /
Q'=l
and the operator La is specified by formula B3), If so, we might agree to
con.sider
V
B= HiE+TRa).
a=l
Here the operators i?„ are of the same structure as was chosen for p = 2.
Example 2 The statement of the first boundary-value problem for the
parabolic equation with mixed derivatives in the parallelepiped Go = {0 <
Xa < L, a = 1,2,,,, ,p} is
du
B6) — = Lu + f{x,t), u\^ =: fi{x,t) , u{x,0) = Uo{x) ,
a /, , du ■
P
2
a=l a,/3=l a=l
As the operators Ra involved, we take once again the operators specified
by formulas B5) that are self-adjoint, positive (for cr > 0) and pairwise
° . - ,
commutative in the space fi/j, since Go is a parallelepiped. The scheme
with these members is stable for a > 0,5, As far as £?„ =: E -\- rRa,
a = 1,2, ,,, ,p, are three-point difference operators with constant
coefficients, the possible follow up is the algorithm suggested in Example 1 for
determination of y-''^^ on the basis of y^ . We will not pursue analysis of
this: the ideas needed to do so have been covered.

572 Economical Difference Schemes for Multidimensional Problems
Example 3 As a matter of experience, the intention is to use a higher-
accuracy scheme similar to A2) being used for the heat conduction
equation. As we have stated in Section 1, the factorized scheme
B7) (£-,TjrAi)(£-,T,rA2)j/, =A'2/ + ^,
a; G W/j, 0 < i = nr ,
yUf, - ft{t), t = nT, y{x,0) = u^{x), s G W/, ,
where
B8) A'= Ai + A2 + ^^^J^ Ai A2,
1 h''
^=(/+Y^A,/+^A2/
generates an approximation of 0{t'^ + \h\'^)-
Also, the scheme so constructed is equivalent to the alternating
direction scheme D6)-D7) from Section 1, Among other schemes, the users
prefer two alternating direction schemes, either of which is equivalent to
scheme B7).
The first scheme:
{E + a,TAi)ij= {{E - a, T Ai){E - a,^ T A2) + T A') y + Tif ,
[E ~ (T^T A'j)y = y .
Every equation refers to two-layer schemes with a common canonical form
B9) {E~a,TA,)^'^^^=Ayy+{l~a,)A2y
T
-I- (xj -I- ^2 -I- r (Tj(T2) Ai A2 yd- ^,
y-y
]<"
{E ~ G^T A2)- ^=G, A22/, ><„ = Tf , «=1,2

Economical factorized schemes 573
In turn, the supplementary boundary conditions become
J/"=y"" fori2=0,iV2,
C0)
y = {E-~a^TK2)n"^^ fori\=0,7Vi.
The second scheme:
C1)
T
{E~-a,TK2) y—^ = a,K{y-y)
T
with the supplementary boundary conditions for y:
C2) y = At" + r (£" - r cr, K-2) — for i\ = 0, Ni .
T
Then problems B9)-C0) and C1)-C2) can be solved by the
alternating direction method in just the same way as was done in Section 1
for problem D6)-D7) using Gaussian elimination along the rows as well as
along the columns of the grid Cb^^.
Thus, three different examples of interest bring out the indisputable
merit of the alternating direction method and unveil its potential. From
what has been said above it follows that the factorized schemes find some
range of applications solely in rectangles and parallelepipeds and no more.
The only exception is the case B — B1B2, Ba — E -\- tRc, where Ri and
i?2 are "triangle" operators, by means of which it is possible to generate a
lower-order approximation only under the condition r = O(ft^).
4. Three-layer factorized schemes. Being concerned with economical three-
layer schemes, we confine ourselves here to
Byo +T'^Rytt+Ay=::ip, (Xt^jrKt^, 5/@) = j/o, y{T) = y, ■
A first step towards the solution of this difference problem is to solve it
with respect to y^'^^, leading to
iB + 2 tR) :(/ + i =2t{2R-A) xi> -F (B - 2 tR) ;(/"^ -^ 2 r ^^'.
Because of this form, the operator B -\-'2 tR on the upper layer is yet to be
factorized for economical reasons.

574 Economical Difference Schemes for Multidimensional Problems
In what follows the weighted scheme
C3) J/o + A((TjJ/+ A - (Ti - (T2)j/+ (T2J/) = ^,
y@) = J/o> y{T) = yi,
IS treated as a primary one and it will be written in canonical form for later
use:
C4) {E+T{a, -(T,}A)yo + 0.5{(t, + a,) r^Ay^^ + Ay = ip ,
where the unknown operator B + 2tR =: E + 2(j-^tA is sought. Let now
A = Ay -\- A2. The factorized operator
B + 2tR= {E + 2a,TAi){E + 2a,TA2) = E+ 2(t^tA-V Au'lr'^AyA-i
will appear in place of the operator B + 2tR as one possible way of
connecting two operators B and i? by a unique condition and it may be of
assistance in designing many factorized schemes. Later we will expound
certain exploratory devices for obtaining them. For example, this can be
done using C4) in the form
[B + 2 tR) y, + {B-2 tR) yj +2Ay = 2'p
and replacing the operator B + 2 tR by the factorized operator B + 2 tR,
making it possible to write the scheme at hand in canonical form with the
aid of the well-known relations
Vt -Vi + 0.5 r j/j-j, yf = J/j - 0.5 r y^^.
The outcome of this is
C5) {E + T{a,-G^)A + 2GlT'AiA2)yo^
?/@) = ?/o, (/(t-)=?/i>
so that ^
B^B^2(y\T^AiA2, i? = i?-F cr^ rAj A2 .

Economical factorized schemes 575
Knowing the values y^~^ and it' we do follow the same algorithm for
determination of the value J/-'"'""'", whose use permits us to find that
Bi W(i) = F\ F = B r i? - B) j4 - 2 A ?/■ + 2 ^^',
Bi = £' + 2(TjrAi, B2 = E + 2(t^tA2.
A similar procedure works for the boundary conditions imposed for u>(i<. as
was done for a two-layer factorized scheme and for this reason it is omitted
here.
Stability of the factorized scheme C5) can be established on account
of the general theorems from Chapter 6, Section 3, due to which it follows
from the foregoing that the conditions
c^i >'^2. cTj-K G2 > 0.5 , Aa = Al>0, AiA2 = A2Ai
are sufficient for the stability of the scheme concerned. If (t^ > a^, (Ti+(T2 >
0.5, Aa = A*^ > 0, then the primary scheme is stable, since B > E and
4R > A. As far as the operators Ai and A2 are commuting, we deduce
that A1A2 > 0, meaning B > B and R> R, Due to this fact the stability
of the primary scheme implies that of the factorized scheme C5).
A particular case where R = a A, a — 0.5 (cTj-i-cr^), is showing the
gateway to the future research, whose aims and scope are connected with the
general method for constructing three-layer economical factorized schemes
by means of the regularization principle of difference schemes. A simple
example
(•36) yo+T^Rytt + Ay = ip, Q<t~JT<tg, y{(}) = Ug, y{T) = u^,
can add interest and help in understanding. Later we will elaborate on
this for rather complicated cases. Here the value 2/(r) = Uq for i =; r is so
taken as to provide a second-order approximation in r. Also, the stability
property guides a proper choice of the operator R.
That is why a reasonable form of the primary scheme is
C7) {E + 2TR)yt = ~F, F = {2tR~ E)y^ ~2 Ay+2^ .
Let R be a sum of "economical" operators such that i?. = Ri+R2 + - ■ ■ + Rp.
By replacing in C7) the operator
E+2tR = E + 2t Y. i?„
a=l

576 Economical Difference Schemes for Multidimensional Problems
by the factorized operator
p
E + 2tR^ '[l{E+2TRa) = E + 2tR + 4t^Qp, R= R + 2tC
■p.
a=l
where Qp - Ri R2 for p = 2 and Qp - Ri -R2 + -R1 -R3 + -R2 -R3 + 2 ri?i i?2 -R3
for p = 3, etc., the structure of the factorized scheme is
C8) Bi---Bpy, = ~F, Bc. = E + 2TRa.
The canonical form is given by
C9) Byo+T'Ry,-,+Ay = ip,
where B = E + 2 r'^Qp and R = R + rQp.
Let now '0o(^O be the error of approximation in the class of solutions
u = ti[x,t) of the continuous problem for the primary scheme C6) and
i/'j(u) be the same quantity for the factorized scheme C9). In this regard,
it should be noted that
xP,{tt) = il-oiv) + 4'*, V'* = 27
ip "i •
When ||Qp WjIIj-^.n = 0{1) is accepted in some suitable grid norm || • ||/,.s
built into stability theorems, we might achieve ||'0*|L,.s = 0{t'^) and in
passing from the primary scheme to the economical factorized scheme C8)
the error of approximation changes within a quantity of 0{t'^). Following
these procedures, we obtain economical factorized schemes of second-order
accuracy in r as stated before due to the extra smoothness of the solution
u. Such a stability analysrs of schemes C6) and C9) is mostly based ori the
further treatment of the operators R and A as linear operators acting from
0
the space H = Qh into the space H. In particular, this means that the
boundary conditions on 7/, are homogeneous for a scheme approximating
B6).
Under the natural premises
A = A*>0, /■?„,= i?;>0, a=l,2,...,p,
another conclusion can be drawn that the primary scheme C6) is stable,
provided the condition
D0) R>^^A, £>0,

Economical factorized schemes 577
holds.
In the case of a variable operator A = A[t) it is required, in addition,
that A{t) is Lipshitz continuous in t, allowing the operator R to be constant.
In the situation when the operators R^, are pairwise commutative, the
stability of the primary scheme implies that of the factorized scheme C9),
since Q,, > 0, B = E + 2 t'^Qj-, > E and R= R + rQp, giving
r>'-±1a.
4
Let us stress that the well-founded choice of the regularizer R is in
complete agreement with an approved principle governing what can happen
aud depends on the range of situations to be considered. This is certainly
so with two-layer and three-layer schemes and, therefore, one and the same
regularizer R could be useful and perceived to be useful for different
operators A.
Example 4 By having recourse to problem B3) associated with the heat
conduction equation with variable coefficients for the same choice of the
operators A, Ri aud i?2 as in Example 1 for the two-layer ecouomical
scheme B4) we concentrate ou the primary scheme C6)
yo +T^Ryi;t + Ay = ip, R = Ri + R2,
which generates an approximation of order 2 and possesses the residual
'0 = 0{t^ + \h\'''). This scheme is stable for cr > A -|- £)/4, Observe that
the factorized scheme of the type C9) with the members
B := E + 2T-R1R2, R^ R+TRiR2
turns out to be absolutely stable for cr > A -|- £)/4, £ > 0. This is clue to
the fact that the operators i?i and R2 are commuting.
Also, the operator A[t) will be Lipshitz continuous if we agree to
consider
KflcJJ < ^3«ai C3 = const > 0 , a = 1, 2 .
Since every operator £?„ — E -\- 2TRa is "economical", the same property
holds true for scheme C9).
In an attempt to solve the system of difference equations
BiB2yt- ~F , Ba - E + 2TRa, y\^^-^,

578 Economical Difference Schemes for Multidimensional Problems
the following algorithm is recommended for the possible applications:
_S2 ^(-2) = W/jN, X^U)/^, CJ(.2\ = /,tj for X2 = 0,l2!
j/^ + 1 = J/J +rwB)-
In this context, we should focus the reader's attention on the process of
specifications of boundary conditions by doing Gaussian elimination along
the rows for cj/j) and along the columns for i^Jfoy When //. happens to be
independent of i, that is, /,« = i-i{x), the quantities cJ(^) and lo^o) satisfy the
homogeneous boundary conditions.
Example 5 In tackling problem B6) the operator L is approximated by
the difference operator
Ay =0.5 Y, [(^"a/3(^>^):fej^,,^^ + (^a/3(»^0j/xJ^,J
with the coefficients k^a still subject to the following conditions with
constants Cj > Cj > 0:
p p p
'i 12C< E i^c/i(^''.0e„^,i<c,YlC foi'all■'■ eGo. t>o.
a=l «,/3=l "=i
Having stipulated the condition \{k^a)f\ < Cg, where c,^ = const, the oper-
0
ator A[t)y = —A(t)y is Lipshitz continuous in the space H = fi/,,. Also, it
will be sensible to approve the same selection rule for the regularizer as in
the preceding example:
p
EO 0
Ra, Ray = -crc^ Aay, Aay = ysi„x^-
a = l
In that case the factorized scheme C9) is stable under condition D0)
or, what amounts to the same, for cr > A + £)/4, £ > 0, and it is of
second-order accuracy with respect to all the variables.

Economical factorized schemes 579
In this direction, the intention is to construct in a similar manner the
factorized scheme related to another primary scheme
p
D1) {E + T''R)y-,,+Ay = ^, R = J^ Re,
a = l
where the operators i?„ are pairwise commutative, self-adjoint and
positive. This scheme is certainly associated with the hyperbolic type equation
d'^u/dt^ = Lu + f. There are several mechanisms for passing from scheme
D1) to the factorized scheme with the accompanying factorized operator at
the members y, y^^ or y^. The usual way of covering this is connected with
further replacement of the operator E -\- t'R by the factorized operator
_ p _
E + T^R=ll{E+T^R^), R^R + T^'Qp, Qp = Q;>0,
leaving us with the scheme
{E + T''R)yit+Ay=^-,
which is stable only if the primary scheme is stable, since R = R* > R.
The resulting scheme differs from the primary one within a quantity 0{t''^).
Example 6 To avoid generality for which we have no real need, the
object of investigation is the equation of hyperbolic type in the rectangle
Go = {0 < a:;„ < /„, a = 1, 2}. The boundary conditions of the first kind
are specified on its boundary F. The complete posing of the problem is
described by
D2) _ = (Li + L2)^,, + /(a;,i), L^u=-^, a=l,2,
a;eGo, ie@,T],
u\-^ = li(x, t) , t > {), u[x,0) = Ug[x),
-g-{x,0) = Uo{x), xGGo.
We contrived to do the necessary factorization in a number of different
ways. First, we initiated the construction of a primary weighted scheme on
an equidistant rectangular grid
^/i = {^i = ihlh^hlh), ''a = 0,1,. .. , 7V„, h„N„ = /„, a =1,2},

580 Economical Difference Schemes for Multidimensional Problems
on which the statement of the difference problem is
D3) ijit = A ((Ty+A-2,7) y+(TJ/) +^(a;,i),
i = JT , J = 1,2,... , a; e w;,,
?/l7,, = fi{x,t), / = IT , ,i > 0,
ij{x,{)) = u„(x) , iJt{x,0) = Uo{x), xeto^,
where iio{x) = Uo(a:;)+ 0.5 r(Luo +/(a:;,0)), A = Ai + A2 and A„j/ = y^^^^^.
Assuming the primary scheme D3) to be stable, that is, granting
4G > 1 + £, £ > 0, we rewrite scheme D3) in the canonical form
{E-a r^A) y^^ =Ay + ip,
whose use permits us to turn to the economical factorized scheme
D4) [E-a t''Ai){E - a r^As) j/,-, = A j/ + ^ .
Another way of proceeding is to reduce scheme D3) to
{E-aT''A}yt = F, F = {E - a T''A)y^ + t {Ay + ^).
Having completed the factorization of the operator E — a r^A at the member
t/j so that
{E-aT^A,){E-aT^A2)y, = F,
we deduce that
D5) a'r^ AiA2yo + {E - a t''A + 0.5 a'r''Ai A2) yu = Ay + ^ .
Both factorized schemes D4) and D5) generate second-order
approximations in r for any a and they are stable under the condition 4G > 1 + £,
£ > 0, since the operators Ra — — A„ are self-adjoint, positive and commu-
0
tative ill the space H = fi /j of all grid functions given on the grid cj^j and
vanishing on the boundary 7/j of the grid.
In an attempt to recover ij = y^^^ from the difference equations just
established, we shall need the boundary conditions for an intermediate value.

Economical factorized schemes 581
Witli tliis aim, tlie equation Bi B'zVit = Ay can be solved in sucli a way
tliat
■SiW(i) = Aj/ + ^, W(i)|^.^ = 52/^A , x,=QJ,,
52^B) = ^(:), ^B)l7;j =/'« . a'=0,/2.
In so doing tliree sequences of tlie values J/-'"'", i/-' and lui^) ^i'g Y^t to be
saved in tlie storage in tlie process of calculations oiy'^^. One trick we liave
encountered is connected with a prior reduction of the preceding scheme to
BiB2%=$, $ = r(A«/ + ^j + BiB2j/r,
thereby increasing the total volume of computations, but saving only two
sequences of the values y^ and 1^B) ■
The algorithm of solving equation D5) was demonstrated before and
so it remains only to construct economical factorized schemes associated
with problem D2) by means of the operator La acting in accordance with
the rule
LaU=j.— {k„{x,t)~—), 0 < Cj < ^„ < C2 ,
OX^ V OXa '
and adopt as a primary one in that case
(£ + r2i?):(yf( = Ay+^,
where
2
El \ a a
The parameter a is so chosen as to satisfy the stability condition
iRy,y)>~^{~^y,y), f>0,
0 0 _
for any y (z H = Q, where fi is the set of all functions given on the grid
u)/^ and vanishing on the boundary j/^ of the grid. True, it is to be shown
that the choice cr = A + £)c2/4 is sufficient for doing so.

582 Economical Difference Schemes for Multidimensional Problems
Having replaced the operator E + r^(i?i + R2) by the factorized
operator {E + T^Ri) {E + r^i?2), where i?„J/ = —cr j/j.^^.^, a = 1,2, we obtain
under such an approach the economical diiTerence scheme
{E + r'^Ri) {E + t''R2) %-, = a j/ + ^ ,
y\-y=^^, y{x,0) = Uo{x), tjt{x,0) = uo{x),
with iig = iig + 0.5 t{Lu + /)|j_q incorporated. This scheme is of second-
order accuracy in r and \h\. What is more, it is absolutely stable and
appears preferable in practical implementations.
5. Economical schemes for a system of equations of parabolic and hyperbolic
types. Let G = {0 < a:;^^ < /„, Of = 1, 2,. . . ,p} be a parallelepiped in the
space RP,
Qt = Gx[0<t<T], Qt = G x{0<t< T]
and k = {kao) — {k"g), s,m = 1,2,... , n, be a matrix of size p x p with
square blocks of size n x n satisfying the condition of symmetry
D6) kl^{x,t) = k'^^{x,t) for all (a-,i)eQT
as well as the condition of positive definiteness
D7) =.EE(o'< E E ^^a;(-.oe;;c<c.EE(C)'.
5 = 1 a=l 5,771 = 1 a,/3=l 5 = 1 q;=1
where Cj and c, are positive constants and ^„ = (^^, . .. , ^^, . .. , ^^) is an
arbitrary real vector. The positive definiteness of the matrix k is equivalent
to being strongly elliptic of the operator L with the values
V^ (9 / du \
D8) Lu = > Lau u , Lau u = ~— k^^i^(x,t)j— ,
a,/3=l '« ^P
where u = (m\ . . . , u^, . .. ,u") is a vector of order n. The meaning of this
property is that we should have
D9) Cj(-L(°)u,u)<(-Lu,u),

Economical factorized schemes 583
where
(u, v) = 2_^ / "■"i^) v^{x) dr. , dx = di\ . . .dx^,
^^°^-Au=EH
a = l
dxi
and u is an arbitrary sufficiently smooth vector-function vanishing on the
boundary F.
The problem here consists of finding a continuous in Qt solutioi] of
the system of parabolic equations
E0) ^ = LM + i{x,t), (a;,i)eQT,
u = p(a;,i) for i; e r, ie[0,T],
u(j;, 0) = U(,(;c) , a G G'.
Before going further, it will be sensible to introduce in G a grid
^ft = {^i = ih^u--- ,-iphp) 0 < i„ < 7V„, ft„ = la/Na, a= 1,2,... ,p}
arid on the segment 0 < i < T a grid oj^ = {i ■ = jV, j = 0,1,...}.
The operator Lag is approximated by the difference operator Aq,^ acting
in accordance with the rule
E1) Aa/3u = 0.5 [{k^pM,,^)^^ + (^„^u,.^)_J
with the well-established notation
p
E2) Au= ^ A„^u.
a,/3=l
Along these lines, we obtain for (] = a
0
The inner product in the space Qh of all grid vector-functions given on the
grid uji^ and vanishing on the boundary jf^ is defined by
n
(y-v) = ^(/,r'), (y\vn= J2 /(.c)!''■(.«)/h-'/v

584 Economical Difference Schemes for Multidimensional Problems
On the strength of D6) the operator A is self-adjoint, meaning
(Ay,v) = (y,Av).
From D7) we derive the inequalities
E3) c, (-A(oV,y)<(-Ay,y)<c,(-A(o)y,y), yGfi^,
where
A^°V = E y..... (-A<°V.y) = E E A. i:ylf] ■
a=l q; = 1 5 = 1
As further developments occur, the operator
p
E4) i?=E-^«' -^ay =-crys„xv ' a = 1.2,...,p,
will be declared to be the regularizer R. Here cr is a numerical parameter,
the choice of which is stipulated by the stability criteria in the sequel.
In view of this, a reasonable form of the primary two-layer economical
scheme is
[E + tR) y; = Ay -F 95 ,
where
V, = F+0(|/i|2 + r-).
Upon replacing E -\- tR = E + t Yl'a=i -^" '-'y '^'^'^ newly formed factorized
operator
p
11{E + tR^) = E + tR,
a = l
R=R+tQp, Qp = E -^"-^'3 +■•■ •
a</3
we are led by exactly the same reasoning as before to the economical
factorized scheme
p
E5) Yl{E + TRa)yt= Ay+<fi, xeu>h, teu>r,
a-1
y{x,t) = f.i.{x,t}, x^j^, teui^,
y{x,0) = Uo{x), x^Qj^.

Economical factorized schemes 585
In what follows the computational algorithm capable of describing
p
<^(i) = Yl {E + T Rp) fit for a;j = 0,/j,
/3 = 2
p
'^(„) = n (-^ + ^-^/3)Mi for 2;„ = 0,/„, a= 2,3,... ,p-1,
/i=:a4-i
will be performed for determination of a vector-function y-'"'"^ = y. Scheme
E5) is absolutely stable for a < 0.5 C2 and converges with the rate 0{t +
The three-layer scheme
E6) yo +T'^Ry^, = Ay + v
is of second-order accuracy in r, so there is some reason to be concerned
about this. Rewrite it in the form
{E + 2TR)y,=F, F = 2{Ay-ip)-{E-2TR)yt
with further replacement of the operator
E + 2tR= E + 2t J2 R»
a=\
by the newly formed factorized operator
p
Jl (£ + 2 r i?„) = £ + 2 r i? + 4 t'Q^ .
a-Y
By these changes we are led to the economical factorized scheme
p
E7) Y[{E + 2TR„)y^ = F,

586 Economical Difference Schemes for Multidimensional Problems
whose canonical form is
E8) (£ + 2r''Qp)yo + r^(i? + rQp) y,-, = Ay+ ¥>,
y = A* for a; e 7/>, i G tD^ ,
y{x,0) = u„{x) , y^{x,0) = Uo{x) , x^Cbj^,
where Up(i') = Lug + f(i',0).
The following algorithm is performed for the recovery of y = y-'"'"'
from scheme E8):
= 1,2,.,. ,n,
V
/3 = 2
(£+ri?„)cj(^„^ = cj(^^_j^, a = 2,3,...,p, s = l,2
, . . . , "- ,
(a)
H {E + TRp)iil for a;„ = 0,/„, a = 2, 3,,.. ,p - 1
,e=a+i
yJ+1 _ yj +rW(p).
As can readily be observed, the components wf x can be found
independently. The primary scheme E6) becomes stable by merely setting
(T = —^ , £ = const > 0 .
4
The operators Ra are positive and pairwise commutative, assuring us
of the validity of the inequalities Qp > 0 and R > R, where R = R + rQp
is the regularizer of scheme E8). This supports the view that scheme E8)
0
is absolutely stable in the space Qh-
Let y be a solution of problem E8) and u be a solution of the original
problem E0). Upon substituting y = z + u into E8) we establish for the

Economical factorized schemes 587
error the following conditions:
E9) {E+2t''Qp)z. +T\R + TQ„)zt, = Az + V^,
z = 0 for a; e 7;,, / £ tD^ , z[x, 0) = 0 for a; G W/j ,
Zj(a:, 0) = i^{x) for a- £ tD^j ,
where
F0) ij) = Au + ip ~uo ~T^Ru^-^-2T^QpU^ = ij^o -2T^QpUt,
V'o is the error of approximation of the primary scheme E6) and
V = U = Tl( = 0{t) .
Since B = E + 2r^Qp > E, Theorems 6 and 9 in Chapter 6, Section 3 are
still valid for scheme E9), on account of which the error of approximation
V to the second initial condition can be most readily evaluated in the norm
||i/||^, where
\\u\\l = {Du,u) = t~{Ru,u) + tHQpU,u) = 0{t^) ,
||i/||^ = O(r^), since u = 0[t) .
From F0) it is easily seen that ij) =■ 0{t'^ + |/ip). The smoothness
properties due to which we would have ij) = 0{t'^ + |ftp) and Wi^Wj-, = C'(r^) will
become more stronger in connection with increasing the number of
observations p. These restrictions can be relaxed with the aid of the chain of the
inequalities
2 r (V', z=) = 2 r^ (Q^v, z=) = 2 r^ {Qf\,Zo) + 2 r^ ^ r^-2(QWv, Zo)
s = 3
<r||z.||^ + r^||Q(%||^ + f:r^(Q(/)z,z.) + X:r^+^(Q(/)v,v)
s=3 s=^
r
< r (Bz,, z=) + r^ II Q^py f + ^ r'+''{Q\;\,v} ,

588 Economical Difference Schemes for Multidimensional Problems
which are put together with a priori estimates for equation E9) with the
right-hand side
p
s = 2
Two last summands in the preceding inequality are some quantities of
0{t^), so there is some reason to make their contributions to the accurate
account of the accuracy z. Thus, scheme E8) converges in the grid space
W2 with the rate Oir"^ + j/ip).
The object of investigation is the system of hyperbolic equations
F1) -^ = Lu + i{x,t), {x,t)eQT, L= Y, ^«/5.
with the supplementary conditions
u~ti[x,t) for a; G r, ie[0,T],
(^^^ dulx 0)
u{x,0) - u„{x) , —-^— = u„{x) for .X- e G ,
under which it is required to find a continuous in the cylinder Qi' solution.
Here the operator L — X]a/3=i ^"8 ''^ specified by formula D8). Observe
that a system arising from elasticity theory such as
-^ - /« A u + (A + ju) grad div u + f , Au = ^ —y ,
where A = const > 0 and yU = const are Lame's coefficients, u = (u^tr,
. . . , tfP) is a vector-function of order p, can be viewed as one particular case
of the system F1) for n = p and
k'Jp = 1^ ^ap Km + (^ + /^) [9 Kehm + (^ " ^) Km^ps] .
Here 6 is an arbitrary constant. In this context, several things are worth
noting. Condition D6) is automatically fulfilled. Condition D7) continues

Economical factorized schemes
589
to hold with constants Cj = /.t and C2 = A + 2/.t, when providing current
manipulations:
j= E E ^a^ce;
^^ E (C)' + (^ + /^)
0 E cc + (i-^) E cc
a,5=:!
Indeed, accepting here ^ = 1, we find that Cj = /.t, since
p
^= E i^c,3L,Cp)
a,0-1
P f '' \^
= /'E ieai' + (A + /o EC
a = l
P
Va = l
> M E 1^0
a=l
For ^ = 0 we are led to C2 = A + 2 yU by virtue of the relations
j = /^Ei^«i' + (^+'") E cc
</'
a=l
Ei^o
a=l
' +
A + //
2
E(C)^+E(C
aN2
■,."■ = 1
a,5 = 1
/^Ei^«i' + (^ + '")E(E(C)
a = 1 5 = 1
(A + 2^0E l^"l'.
a-l
Thus, Cj = /.( and Cj = A + 2/.t, The same operator R will be adopted as a
regularizer in the further development:
p
-R = E -^«' Ray = -^ Vscx^ ■
a=l

590 Economical Difference Schemes for Multidimensional Problems
Observe that the primary scheme y^^ + r'^Ry^^ ^ Ay + y) is stable
under the restriction
(T = - ,—'- , £ - const > 0 .
By replacing
V
E + T^R = E + t'^J^ Rc>
a=l
by the factorized operator
p
Z)= HiE + T^R^)
we obtain the economical factorized scheme
p
Yl{E + T-Ra)ytt= Ay + V for .cew^, teu>r,
a = l
y = M foT .re 7ft, i G w^ ,
y(a:;,0) = U(,(i') , y((a;, 0) = Uo(a:;) for a; G W/, ,
where Uo(a:;) = Uq + 0.5 r (Lu^ + i[x, 0)).
It is plain to show that the scheme concerned is absolutely stable and
it generates an approximation of order 2: i/; = 0[t'^ + |/ip), i^ ~ 0{t''^).
Whence the convergence with the rate 0{t'^ + |/ip) immediately follows.
The search for y-'"''^ amounts to successive solution of three-point
equations of the form [E + T'^Ra)yv — F„ by the elimination method for every
component of the vector w with the index account from a to a -|- 1. One
possible way of covering this is connected with the performance of the
following algorithm:
P
{E + t'Ri}w^,^^F, F= ll{E + T'Ra}yi + T{Ay + v),
a = l
(£'-br-i?„)w^„, = w,„_i) , n = 2,...,p, yJ + i = yJ -^ r w^p) ,

Economical factorized schemes 591
which includes as part the supplementary boundary conditions for the
vector-functions w,qN , cv =; ], 2, . . . , p — 1, in the case x^^ = 0, l^^:
W(i) = {E + t''R2) ■■■{£ + r'^Rp) m* , x,= 0,1, ,
From here the components of the vector w,^^, a = 1,2,... ,p, can
be recovered independently, since the operators £)„ = E + t'^Ra possess
diagonal matrices of coefficients with diagonal blocks.
9.3 THE SUMMARIZED* APPROXIMATION METHOD
1. The problem statement. First of all, it should be noted that it i.s impo.s-
sible to generalize directly the alternating direction method for three and
more measurements as well as for parabolic equations of general form.
Second, economical factorized schemes which have been under consideration in
Section 2 of the present chapter are quite applicable under the assumption
that the argument x = (Xj, X2, ■ ■. , x ) varies within a parallelepiped.
Because of this, there is a real need for designing the general method,
by means of which economical schemes can be created for equations with
variable and even discontinuous coefficients as well as for quasilinear non-
stationary equations in complex domain.? of arbitrary shape and dimension.
As a matter of experience, the universal tool in such obstacles is the method
of summarized approximation, the framework of which will be explained a
little later on the basis of the heat conduction equation in an arbitrary
domain G of the dimension p with the boundary F
du
A) —~Lu + f{x,t), X = {Xi,x^,. .. ,Xp} eG, i>0,
Y^ 9 / \ 9u \
Lu=} Lali, LaUziz.-- ik^{x,t)- , «„ > Cj > 0 ,
^—' ox„ V ox^J
azrl ^ "
provided that the conditions hold:
B) ii\^ — i.i{x ,t) , i>0, u(x,0) = itg{x), x^G.
*Editor's note: Summarized = summed.

592 Economical Difference Schemes for Multidimensional Problems
The quasilinear heat conduction equation reproduces the case when k^ =
k^[x, t, u) and / = f{x, t, u).
Of course, the words "arbitrary domain" cannot be understood in a
literal sense. Before giving further motivations, it is preassumed that the
boundary V is smooth enough to ensure the existence of a smooth solution
u = u[x,t) of the original problem (l)-B). In the accurate account of
the approximation error and accuracy we always take for granted that the
solution of the original problem associated with the governing differential
equation exists and possesses all necessary derivatives which do arise in the
further development.
A common algorithmic idea behind available economical methods is
connected with further reduction of numerical solution of a
multidimensional problem to the process of solving a few simpler problems. In order
to understand the nature of this a little better, we focus the reader's
attention on second-order equations of hyperbolic and parabolic types for which
the "basic algebraic problem" is related to a three-point difference problem
(a second-order difference equation). The three-point difference problem
obtained through such an approximation can be solved by the elimination
method and it can be treated, as a rule, as a difference approximation to
the one-dimensional (in x^) differential equation. Some consensus of
opinion is to create on this basis a chain of simpler algorithms which constitute
what is called an economical algorithm for complex problems. This idea
lies in the origin and terminology of many economical methods available for
solving multidimensional problems. Among them, the alternating direction
method permits us to solve at every stage a one-dimensional problem along
a fixed direction x^, the method of "fractional steps" necessitates placing
in the storage intermediate (non-integer) values at every stage of a complex
computational procedure, the method of separation of variables in a
common setting of the problem reduces to a number of particular simpler tasks,
etc. All these terms reflect one of the real advantages and the essence of
economical methods.
However, throughout this book, the classification of difference
methods is mostly based on the origin of difference schemes rather than on a
possible way of constructing them and a perfect tool for solving this or that
difference scheme (equation).
2. The notion of summarized approximation. In the preceding sections and
chapters the basic fundamental property of difference schemes is to generate
approximations on a solution to the governing differential equation. In what
follows we get rid of the classical notion by introducing a more weaker
condition of summarized approximation, expanding our possibilities and

The summarized approximation method 593
leading to additive schemes. The describing schemes of this type will be
studied in more detail in Section 10, but a great deal of work still needs
to be done in adopting those ideas. It is worth mentioning here their main
peculiarities:
• the passage from the jth layer to the (j+l)th layer can be performed
through the use of a sequence of the usual (two-point, three-point,
etc.) schemes;
• the error of approximation provided by an additive scheme is taken
to be a sum of the residuals of all auxiliary schemes, that is, any
such scheme generates a summarized approximation.
In this regard, we should take into account that auxiliary schemes are
not obliged to approximate the original problem; the approximation here
is ensured by summarizing all the residuals obtained.
In Chapter 2 we came across the necessity of generalizing the notion
of approximation in the real situations when a difference scheme cannot
provide on the grid W/j local approximations with a desired order in the
norm of the space C, but it does the same in one of the negative norms,
that is, in a certain sense of summarizing.
Likewise, it may happen that a scheme on the grid u)^ cannot provide
local approximations in t, but at the final stage the approximation will be
achieved once we bring together the residuals over several time layers. The
notion of summarized approximation needs certain clarification. It seems
worthwhile giving simple examples.
Example 1 The Cauchy problem, being the most familiar one, comes
first:
— +au = 0 , i > 0 , it{0) = Uo .
Common practice involves for solving it the difference scheme
C) ^ L + «^y;=0, ^ ^ +a,j/+i/2=,o,
r r
i = 0,1, 2 iP = iig , a^ + a, = a ,
which consists of two explicit schemes with residuals i/'j and i/^j, respectively.
Within more compact notations

594 Economical Difference Schemes for Multidimensional Problems
we find that
ri + l/2 _ j .4 + 1 _ ^J + 1/2
+ a,zi = -v'V , : : + «,^J + i/2 ^ _^^y ^
r ' r " -
where
iJ + '^-u^ ■ , uJ + ^-tt-?' ui+ui + i
tpi = ?, h Hitf' , i/'2 = ?, ^ «2 7] ■
/ r / r /
By substituting here the expressions
„; + l^yi + l/2 I,„i + l/2 r!,;., + l/2.
2 8
„i ^ „J + l/2 _ I -, + 1/2 , ^ ••i + 1/2
0G
0G
j + 1/2 /^ , n r ^ ■ ** ■■ "^ "
■* c/i d/'
we are led to
i .,'/ J- n 11] — L n T i'/J + l/2 .
tP,= {^u+a,u] - ^a,TuJ + ''' + 0{
i + 1/2
r
Whence it follows that 'ip^ = 0A), ■i/-'2 = '^A) ^^^d 'if^^ +'4'2 = C'(''')i meaning
that none of the auxiliary schemes concerned provides an approximation,
but the triplex composition generates a summarized approximation of 0(r),
Example 2 Of special interest is the heat conduction equation
du d'^u
0 < a: < 1 , u{{),t) = u^{t), ii{\,t)-U2{t).
In dealing with the grid Cof^ = {xj = ih, i = 0,1,. .. , N, Nh —I] and
the operator Aw = ri.j.,. ~ Lt/, it seems reasonable to employ the explicit

The summarized approximation method 595
scheme on the odd layers and the implicit one on the even layers. The
outcome of this is
D)
„2; + l _ 'ij
,,2;+2 _ 2i + l
y- ^ = 2,tAj/2;+2_ j = 0,1,2,
where cr > 0 is an arbitrary parameter. In order to calculate the residuals
,,2i + i_,,2i ,,2i+2_ 2i+i
T T
we insert in them the expressions
„2i+2 ^ ^2; + l + ^y2; + l ^ q 5 ^2 ^.2; + l ^ ^(^3) _
W^i = „2; + l ^ ^ ,^.^2; + l ^ q 5 ^2 ^.2; + ! ^ ^(^3) ^
Am = L u + (9(/i ), Lu = u ,
implying that
0 0
V-i = (BG- l)u + {l.b-2a)Tuf^^\
4,.^ = {-{2a-l)u+i0.5-2a)Tuf^^\
From such reasoning it .seems clear that -ip^ = 0A) and -ip^ — 0A) for
a j^ 0.5, but for any a there is no doubt that
^ = ^^+^P^^O{{(T- 0.5) T+h^ + T^).
Collection of equations D) permits us to eliminate the value J/^"*"^, leaving
us with the weighted scheme with step 2r:
2;+ 2 _ 2i
^ ^^ ^ = A {at/^+' + {l-a)y'^), j = 0, 1,. ..

596 Economical Difference Schemes for Multidimensional Problems
An alternative form of writing includes the intermediate value yi+'^l'^ and
a smaller step r in two times;
,,i + i/2_,,i . yi + i _ ui + i/2 .
l L = (i_^)A2/, l y- = aK'i/ + K
T T
Here the passage from the jth layer to the (j + l)th layer is carried out in
the following two steps: the first one involves the explicit scheme and the
second one - the implicit scheme as suggested above.
Of course, this example is not of global character and can serve mainly
as an illustration of such theory. It should be noted that in the general case
the elimination of intermediate values with further reduction to a scheme
including the values of y only at integer steps may be impossible and even
meaningless in the theoretical research.
3. Reduction of a multidimensional problem to a chain of one-dimensional
problems. The multiple equation we must solve is
du
'dt
E) —^Lu + f{x,t), 0<t<t,,
u{x,0) = Uo{x) , X = {x^,x^, ■■■ ,Xp),
where L is a linear differential operator acting on u[x,t) as a function of
X, X = (xi, X2,... , x„) is a point in the p-dimensional domain G with the
boundary F, on which proper boundary condition.s are imposed in one or
another convenient way. That does not matter for us in subsequent
discussions. An effective tool in designing economical methods is the accepted
decomposition
L = Li + L2 + ■ ■ ■ + Lp
with simplified operators. For example, if Lti = Aw, and /.„« = d''u/dx''^,
then La is the operator of the second derivative with respect to the
argument Xa (operator of one variable).
Furthermore, to problem E) there corresponds the first chain of "one-
dimensional" equations by reducing either E) or
Vu= —-Lu^!{x,t) = 0
to
^ I du .
^-' p at

The summarized approximation method 597
where fa{x,t), a = 1,2,... ,p, are arbitrary functions with the same
smoothness properties as the function f[x,t), satisfying, in addition, the
normalization condition
/i + /2 + • ■ ■ + /p = / •
In worliing on the segment 0 < t <to with an equidistant grid
^T = {'^j = JT , j = 0, 1,... , jj
with step T it seems reasonable to divide every interval into p parts by
recording the points tsj^^i = i- -|- ar/p, a = 1,2,... ,p — 1. Half-intervals
^i-l-(Q._i)p < t < tjj^^i denoted by A„ are made up, as usual, by those
points. By successive solution of the equations starting from « = 1,2,...
F) 'Pa^'(a) = 0, xeG, ieA„, a=l,2,...,p,
under the additional assumptions
G) «(i)(a;,0) = Uo{x), ■"(„)(a;, ij+(„_i)/p) = ?'(„_ j)(a;, ij+(„_i)/p) ,
a = 1,2,. .. ,p,
we find the values v[x,tj) = V(p\[x,tj), j = 0,1,2,... ,jg, with further
treatment as a solution of the problem concerned. For the sake of simplicity,
it is supposed that we have on the boundary the homogeneous boundary
condition of the first kind and this should confine no generality of further
motivations.
Every equation 7'at'a = 0 or
1 dvr^s
F') - -^ = ^a^(a)+/a> a=l,2,...,p,
is replaced by the newly formed difference scheme in which du/dt and L„
are approximated by the appropriate difference expressions of the general
form on a grid W;, with steps h^,h^,... , hp
(8) n„y(„, = 0, a = 1,2,..., p.
In the simplest case the resulting two-layer scheme is aimed at
connecting the values
j/(„) = j/' + ^/P and J/(„^i) = 2/'+(«-^)/r

598 Economical Difference Schemes for Multidimensional Problems
For example, the weighted scheme
')' _ ,J+("-i)/p
= Aa (t„2/' + "/^^ + A - (tJ j/^+(«-i)/p) + ^+»lv
yj + a/p
suits us for doing so. Here A„ ~ La and (t„ is an arbitrary parameter.
The governing equation VaV(a) = 0 is approximated by scheme (8) in
the usual sense so that
(9) *„ = n„w^+«/p - {Vauy+"^"
tends to zero in some suitable norm as r ^ 0 and /i„ —;■ 0.
The difference equations (8) constitute what is called an additive
scheme. Indeed, let ■0„ = n„u^ + "'P be the residuals of the same scheme
(8) with the number a attached.
Arranging -0^ as a sum
A0) •0„ = (Pa«y + "^"+*a
and taking into account that
we deduce that
and \\'il>* II ^ 0 as r ^ 0, ft ^ 0, where || • || is some suitable norm on
the space of all grid functions given on the grid LOf^. It follows from the
foregoing that
P 0 II '^ I
l]^a = 0, ||^||=|^^„| ^0 as r^O, \h\^{),
that is,
scheme (8) generates a summarized approximation if either
of the schemes (8) with the number a approximates the
corresponding equation F) in the usual sense.

The summarized approximation method 599
As a matter of experience, the estimation of the nearness of a solution
of the difference problem amounts to the proximity between a solution of
the original problem E) and a solution of the chain of problems F)-G).
The main idea behind this approach is connected with the obvious relation
||2/--«^||<||2/--.^„^|| + ||.^„^-«i||.
In addition to F)-G), the second chain of the equations
A1) -^ == ^« "(a) +/« > x^G, tj<t<tj^^,
under the additional conditions of conjugation complements subsequent
studies
*'(«](«, ^i) = *'(a-l)('^'.^j + l). « = 2,3,... ,p,
A2)
V(i){x,tj) = v{x,tj) , j = l,2,..., V(^^-^{x,0) = Uo{x) .
The solution of this problem is the function v[x,t) = V( s[x,t).
In contrast to F)-G), every equation with the number a is solved
here on the whole interval t: < t < tjA.i. It is interesting to note that in
some particular cases solutions of problem F)-G) and problem A1)-A2)
will coincide. This is certainly true in the situation when both /„ = 0 and
La are independent of i.
Along these lines, both chains generate approximations on the solution
u = ii[x, t) of the original problem E). Indeed, it is straightforward to verify
for problem F)-G) with the aid of the relation Va u =■ {Va uy + '^/^ + 0{t)
that
'0„ = ^ „ + i'l, where 4',, = (P« uY + 'Z'' , d'l = 0(t) ,
where ■0„ = Vciu{x,t) is the residual for equation F) with the number a.
In view of this, it follows from the foregoing that
p p p
it being understood that the system F)-G) approximates equation E) in
a summarized sense.

600 Economical Difference Schemes for Multidimensional Problems
Thus, the attainable summarized approximation of the additive
scheme (8) owes a debt to the simultaneous usual approximations and a
summarized approximation. In accordance with what has been said above,
equations E) are approximated by the chain of the difference equations
F)-G) in a summarized sense and every scheme (8) with the number a
approximates the corresponding equation involved in collection F) in the
usual sense.
All this enables us to obtain through such procedures a perfect
approximation. Let us stress here that the summarized approximation is ensured
by the following conditions:
• the operator L is representable by a sum L = Li -\- L^ -\- ■ ■ ■ -\- Lp\
• the right-hand side / contains exactly p functions such that / —
These conditions will be relaxed once we consider
V P
Lu-J2l,,u^0{t), f-Y,fa = 0{T).
a=l a=l
If the operator L„ includes the derivatives with respect to only one variable
x^, we call it a one-dimensional operator and the equations VaVra) ~ 0
refer,correspondingly, to equations of one variable. The additive scheme
(8) is termed a locally one-dimensional scheme (LOS).
Section 5 of the present chapter will be devoted to such schemes
relating to the heat conduction equation,
4. Examples of reduction of multidimensional problems to chains of one-
dimensional ones. It is apparent from that discussion that some class of
problems for which a solution of problem F) or problem A1) coincides on
the grid u>^^ with the exact solution of the multidimensional problem E)
plays an important role.
Example 1 The Cauchy problem
du
— + au{t)^0, i>0, m@) = Wo,
where a > 0, is good enough for the purposes of the present section. With
the representation a = a^ -\- a^ in view, we may set up the problem
^ + a,^«(ij(i) = 0, t.^i)@) = «o, 0<i<r,
^ + «2V)(^) = 0' ^B)(o) = ^(i)(i*), o<i<r.

The summarized approximation method 601
where i* > 0 is an arbitrary number. By resolving these equations we
obtain
V(i-^{t) = Uoexp( —flji) and
^B)@ = V(ij(^*) exp (-%<) = Uo exp(-a,t - a^f) ,
yielding v^it*) = u{t*).
Example 2 Recall now the statement of the Cauchy problem for the
transfer equation
du du
-^ + Li u + L2 u - 0 , L„u - -— , a = 1,2,
at dx„
—00 < 2;^^ < 00 , i > 0 , u{x, 0) =: ii{x),
whose solution u[x,t) = ii{x^ — t.x^— t) is a "travelling wave" if ^[x) is a
twice differentiable function. Since the operators Li and L2 are commuting,
it is plain to show that
u{xj*) = v^2){x,t*) ,
where Vf'2){x,t*) is a solution of the system of equations
-^ + ^ = 0, (}<t<t\ v^,^{x,(}) = ^i{x),
dVB\ dVB^
^+^ = 0, (}<t<t*, VB){^X,0) = V(,^{x,n.
Indeed, a solution to the first equation acquires the form
^(:L)(a;,0 = A'l'Ci - t,x^,_) .
On the other hand, we find from the second equation that
implying that Vi2){^yi*) = u{x,t*).

602 Economical Difference Schemes for Multidimensional Problems
Example 3 We learn from Ladyzhenskaya A968) that a solution of the
Cauchy problem for the heat conduction equation
A3) ^ = (Li + L2 + is)" = 0, LaU=-^, a =1,2,3,
—00 < a:;„ < 00 , i > 0 , u[x, 0) = Uq[x) ,
is given by the formula
A4) u{x,t) = u[x^,X2,x^,t)
00 00 00
LtI^Xj , Xj, Sg', 4i , ^2! S3! ^j '"oIm ; 42> S3J ''m ''42 ''sa !
— 00 —00 —00
where G'(a;j, a:;2, a'g; ^j, ^2. ^3; 0 i^ a source function such that
'-''l"^l!"^2!'^'3!M!S2!S3!^J — Ltq\X-i,C^^,Ij LTQ(^a:;2, S2!^J'-''ol"^3!S3!^j!
Go(^-„,e.,i) = exp(-(a;„-e„)VD0)/BV^), a =1,2,3.
Here Go(£^,^„,i) is a function of the heat source of the Cauchy problem
associated with the one-dimensional heat conduction equation
—^-LaVf^^-), i>0, ■«(„)(a:;,0) given.
The general methodology provides proper guidelines for the selection
rules in studying one-dimensiooal heat conduction equations
dt
= ^i^(i)>
3^B)
0<t<t*,
dt
= L
3^C)
'«(l)(a-,0) = Mo('^") . ■"B)(*'0)= '''(I)(«.^*) . ''^'C){^>^) = i'B)ixj*)^

The summarized approximation method 603
It is plain to derive for this chain oof the equations the following expressions:
oo
^(i)B;,0= / Go{x^,^^,t)v(^^{^^,x2,x.i,0) d^^ ,
— OO
OO
VB)ix,i)= / Go{x2,^2,t)v(^-^{x^,^2,x,^,0) d^2,
-OO
OO
— OO
Upon substituting here
■y(j-,(<^i, x-2, ;c,,, 0) = «o(^i. a;-2. •■C3) ,
we obtain formula A4) for V(^\[x,t*) at moment t = t*, meaning
Vr2,){^,'t*) = u{x,t*) for any /* > 0 .
The property of this sort is an immediate implication of the
representation G{x,^,t) = G{x^,X2, x.j;^^,^2!^3>'^) ^s a product of the functions
of one variable of the special type Go{x„,S,„,t). The well-established
representation of the source function is still valid for equation A3) with zero
boundary conditions of the first kind u = 0 for x^, = 0,/„, a = 1, 2, 3,
relating to the boundary-value problems in the parallelepiped {0 < a;„ < /„,
Q = 1,2,3}. For this reason identity A4) is certainly true in that case.
What is more, it seems clear from the preceding examples that v,p^
coincides with u{x,t) at all nodal points.
Among other things. Examples 2 and 3 show the ways of expanding
some spatial process into a sequence of processes being one-dimensional
ones and continuing along the coordinate axes. Following established
practice, the three-dimensional heat conduction problem in a space or a
parallelepiped with the zero temperature on the lateral surface reduces on the
same grounds to the model problem concerned. If the initial distribution of
temperature is known at moment t =^ tg, then the heat conduction will be

604 Economical Difference Schemes for Multidimensional Problems
possible at the same moment only along one direction k, without concern
of others. At moment t = tg + At the same procedure work.? along one
direction x.^ and at moment t = tg + 2 At - along another direction x^. As
a final result we obtain at t = tg + S At the same temperature distribution
as in the three-dimensional case at moment t = tg + At. As a matter of
fact, this is a way of reducing the three-dimensional process to a sequence
of one-dimensional ones of prolongations a.s well as of a real physical process
in three times.
Generally speaking, proper guidelines in such matters are not so
obvious as the users might expect. For example, in isotropic media the operator
La involved in F') is taken to be
ox a V "' ' dXa
In spite of the fact that Vf'ps{x,t*) does not coincide with u{x,t*), the
asymptotics reveals itself as
"(P)
{x,t*)-u{x,t*) = 0(t*).
5. A locally one-dimensional scheme (LOS) for the heat conduction equation
in an arbitrary domain. The method of summarized approximation can
find a wide range of application in designing economical additive schemes for
parabolic equations in the domains of rather complicated configurations and
shapes. More a detailed exploration is devoted to a locally one-dimensional
problem for the heat conduction equation in a complex domain G = G +T
of the dimension p. Let x = (Xj, Xj,. .. , x ) be a point in the Euclidean
space RP.
The problem statement for the heat conduction equation in the
cylinder Qio = G X [0 < i < tg] is
A5) -^ = Lu + f{x,t), L=J2La, {x,t)eQt,,
ii]^ = f-t.{xj), t>0, ii(x,(}) = Uq{x) , xeG.
Here F is the bouudary of the domain G and L is a second-order elliptic
operator. For the sake of simplicity we agree to consider L = A, that is,
L„u = d'^u/dx'^, a = 1,2,... ,p (Laplace operator). Also, we take for
granted that

The summarized approximation method
605
• problem A5) has a unique sufficiently small solution;
• the intersection of the domain G with any straight lines Ca parallel
to the axis Ox^ may consist only of a finite number of intervals;
• it is possible to compose in the domain G a connected grid W;, with
steps /)„, a = 1,2 p (for more detail see Chapter 6, Section 1).
To make our exposition more transparent, it is supposed that the
intersection of the domain G with the line Ga is a unique interval.
Thus, the set W/j of all inner nodes contains the points a; = [x^,X2, ■ ■ ■,
X ) G G of the intersection of the hyperplanes x^ — i^h^, z„ = 0, ±1, ...,
a = 1,2,... , p, while the set j/^ of all boundary nodes consists of the points
of the intersection of the lines C'a, a = 1, 2,.. . , p, passing through all inner
nodes x G W/,, with the boundary F.
Here we retain the same notations as was done in Chapter 6, Section
1: 7;j „ is the set of all boundary nodes in the direction x^; Ji^ is the set of
all boundary nodes a; G F; w* is the set of all near-boundary nodes in the
direction x^; ui*^ is the set of all near-boundary nodes; >jJ**^ is the set of all
irregular nodes in the direction x^] w^* is the set of all irregular nodes and
uif, is the set of all regular nodes.
The difference approximation of the operator La at a node x is
constructed oil the pattern consisting of the three nodal points x^~^"\ x,
a;(+^»). In view of this, the difference operator Aa ~ L^ is taken to be
A6) Aaij = yg_^
at the regular nodes and
ft2
(y(+i.)„2j/ + j/(-^"))
A7)
Aa y = y.i-
1
If^'"'-
ha
y( + l.)_
-y
-y
y-~y
(-Ic)
y-y
.(-Ic)
,(-!«)£
7h,
.( + lc)
eTft,
at the irregular nodes, where h*^ is the distance from an irregular node x
to one of the boundary nodes: .c'- + '°) or .c'-"''"'. But it may happen that
both nodes x^'^^"^ and x''-"^'-"'), which are neighboring to x G w* ^, belong
to the boundary, that is, x^ "' ^ 7h a- ^^ ^^^t '-^se
A8)
A„j/ =
y( + lo) _ y
y-y
.(-lO

606 Economical Difference Schemes for Multidimensional Problems
where h*^^ is the distance between x and a:;^*^") C^ai ~ ^a")' '^^'^ preceding
expression for .'V„j/ is in common usage. If x is one of the regular nodes,
then h*^, = h*^_ = h^, yielding formula A6).
Observe that A^ provides a second-order approximation at the regular
nodes A^m -~ L^tt = 0(h'^), while A^u — L„m = 0A) at the irregular ones.
The intuition suggests some things in an attempt to write down the
locally one-dimen-sional (LOS) scheme in conformity with available
constructions of Section 3. In working on the segment 0 < t < tg we introduce
a grid tD^ = {tj = jr , j =; 0, 1, .. . ,jg} with step r = tg/jo and involve
arbitrary functions f^^ subject to the normalization condition
E/«-/-
In line with established practice we replace the governing multidimensional
equation by the chain of the one-dimensional heat conduction equations
1 "^^ra)
A9) - -g^ = Lc. t'(tt) + /„ for tj+^a-l)lp < i < ij + alv -
a =: 1, 2,... ,p, a; e G ,
with the supplementary conditions
B0) t;(i)(a;,0)=: Ug{x) , ■«(„)B;,i;+(a-l)/p) = ■"(a-l)B-'> <i+(a-I)/p) ^
cv = 1, 2,.. . ,p , V(^<i = fi(x,t) for a; G Fq ,
where ij+„/p = {j + a/p)T.
Here the boundary conditions for tv„^ may be imposed only on some
part F„ of the entire boundary F consisting of the points of the intersection
of F with possible lines C'a parallel to the axis Ox^ and passing through
any inner point x G G. The nodal points a; G 7/j „ belong to that part F„.
If, for example, G = {0 < a:; < /„} is a parallelepiped, then F„
comprises the planes a;„ = 0 and a;„ = /„.
Through the approximation of every heat conduction equation with
number a on the half-interval tj_^/^_iy < t < t-j^^, by the standard two-
layer weighted scheme we arrive at the chain of p one-dimensional schemes

The summarized approximation method 607
with arbitrary numbers a„:
^ ^ := A„ {a, y^+"'P + A - a J y^+("-^)/P) + ^^+"l\
a = 1, 2,. ,, ,;9, X ^Lo
h
The describing scheme is termed LOS. In what follows we confine ourselves
to the applications of purely implicit LOS's with (t„ = 1:
B1) ^ ^ = A„2/+«/P++^+«/p^
T
Q- =: 1, 2,. . . ,;j , X eoj^ .
In preparation for this, the preceeding is put together with the boundary
condition
B2) yJ+^/p^^j+c/p fo, xej,^^, j = 0,l,...,j,,
a= 1,2,... ,p,
and the initial condition
B3) y{x,0) = u,{x).
As we will see a little later, the right-hand side i.pi+°'lp and the boundary
value j/^+"/P| can be expressed through the functions /,.,,(x, /) and i.i{x, t)
taken at arbitrary moments T^ and i" from the segment [ij,/j_|_J, so that
^^"Iv - f^{x,t*^) and ^^+"/p = ij{x,t"), thereby retaining the accuracy
order. For the sake of definiteness, we accept
<+"^^ = /a(^,ii+0,5), f^'^"'''=K^,tj+a/p), a=l,2,...,p,
by regarding tf' to the known values. The value j^ + ^ can be found on
every new layer from B1)-B2) by successive solution of p equations of the
form B1) with the boundary conditions B2) in a step-by-step fashion for

608 Economical Difference Schemes for Multidimensional Problems
a — f, 2,. .. ,p. In determining y^'^"/P we must solve the boundary-value
problem
B4)
with varying subscripts only. The difference equation is written on the
segment A^ £ Ca with the endpoints belonging to the boundary 7^ ^^ and
it can be solved by the elimination method along all the segments A„ for
fixed a. For doing so our expenses are not considerable, since the number
of arithmetic operations required at every node of the grid uj/^ are 0A)
solely in connection with successive determination of the values y^'^^'^,
yj+'^/p^ _ ^ yj+clp^ , _ ^ j/i + 1 by setting « = 1,2,,.. ,p and changing the
directions of the eliminations. Thus, the locally one-dimensional scheme
B1)-B3) falls within the category of economical schemes.
6. The error of approximation of a locally one-dimensional scheme. Upon
raising the question regarding the error of approximation provided by one
or another LOS it is straightforward to verify that every separate equation
B1) with the number a does not approximate equation A5) in spite of the
fact that the sum of the residuals «/' = i/ij -|- Vs + ' ' ' + 0n involved tends to
0 as r ^ 0 and |/i| — 0.
Let n ~ ii{x,t) be a solution of problem A5) with the operator LaU =
d'^u/dx^ and yi+"lp^ a = 1,2,... ,p be a solution of problem B1)-B3).
The accuracy of LOS is characterized, as usual, by the difference «/ + ' —
„J+1 = ,3+\
The intermediate values yi+"lp need to be compared with u^^"Ip =
u{x,tj_^^, ) by making the choice z^^"Ip — y^+"/p — u^+"Ip. Upon
substituting y3+"IP - z3+»/p + ui+civ into equation B1) we may set up the
problem for the error z^'^'';
J+a/p _ yj+{a-l)lp
B5) = A„ 2^+"/P + ^iPl+'-'IP ,
i = 0,1,... ,io, a = 1,2,... ,p,
.j+alv^^ for ie7,,,„, z(,c,0) = 0,
where
.,,3+alp _ ul+{a-i)lp
B6) i'^ +"/P = A„ M^ +"/P + ^/"/P

The summarized approximation method 609
Further analysis relies on the quantity with omitted superscripts:
;„ = (...../;.-i|r'
A simple observation that
P 0
in the case where
p
E /«= /
a-Y
serves as a basis for the representation of tp^ = ^^+"/p as a sum
0
where
wl = (A„ t^+-'" - L„ u^+"') + {^[+-if - fl+^r^)
\ T p ^dt
From the very definition of A,,, and 99q, it follows that
i>* = 0(/).^ + r) at the regular nodes,
■0* — 0A) at the irregular nodes.
Thus, we might have
''^=E^« = EC = 0(r+|/ir^
at the regular nodes, that is, LOS in question generates a summarized
approximation of 0{t + |ftp) at the regular nodes of the grid uif^, while
■(/> = 0A) at the in-egular nodes.

610 Economical Difference Schemes for Multidimensional Problems
7. Stability of LOS. The main goal of stability consideration is to establish
that the uniform convergence with the rate 0{t + |/ip) follows from a
summarized approximation obtained. This can be done using the maximum
principle and a priori estimates in the grid norm of the space C for a solution
of problem B1)-B3) expressing the stability of the scheme concerned with
respect to the initial data, the right-hand side and boundary conditions,
Recall that in Chapter 4, Section 2 we have proved the maximum
principle and derived a priori estimates for a solution to the grid equation
of the general form
B8) A{P)y{P)= Y. B{P,Q)y{Q) + F{P) for Pen,
Qeiii'iP)
y{P)^fi{P) for Pes,
where P and Q are some nodes of a connected grid Ci + S and III'{P) is
a neighborhood of the node P except the point P itself. The coefficients
A{P) and B{P, Q) must satisfy the conditions
B9) A{P)>Q, B{P,Q)>Q
D{P) = A{P)- Y. B(P,Q)>0.
Qeiii'iP)
Applying theorems of Section 2, Chapter 4 to problem B1)-B3) yields
the following result,
Theorem 1 The locally one-dimensional scheme B1)-B3) is uniformly
stable in the metric of the space C with respect to the initial data, the
right-hand side and boundary conditions and a. solution of problem B1)-
B3) admits for any r and h the estimate
C0) IIj/'IIc < 11^0lie+„ max, llp(^.0llc.
+ max h'Mx,t')\\^, + X: - E lb{+"^^ll"
J-1 P
j' = 0 0 = 1

The summarized approximation method 611
where
h=maxh^, ||y|| = max |y|, ||y|| = rnax |y| ,
||<^|L. = max \(p\ , \\(p\\ o = max \(p\ .
h xEtJJ /i
To prove this assertion, we represent a solution of the problem
concerned as a sum
y = y + v + w ,
where y is a solution to homogeneous ec|uations B1) with boundary and
initial conditions B2)-B3) and v and w are solutions to nonhomogeneous
equations B1) with the homogeneous boundary and initial conditions:
j+a/p _ ,J-|-(a-l)/p , 0 . ,
C1) =A,v^+-/P + ^i+-/p, xeoj,,,
a = l,2,...,p, v{x,0) = 0, v^+"/P = Q for a; 6 7ft_„ ,
■,,J+"/p _ „J+(«-i)/p . , . ,
C2) ~ = A„ w^ +"/" + ^*J +"IP , X 6 CO,, ,
T
a = l,2,,,,,p, w{x,0) = 0, w^+"'P = Q for xejh,a-
0
Here 'fi „ and ip* are specified by the formulas
j ip^ for a; 6 cj,, , ^ 10 foi-xEi^h'
[ 0 for X Ei^,, , [ If a loi" ^ G ^/j '
so that
0
thereby clarifying that ip* differs from zero only at the near-boundary
nodes.
For convenience in analysis, the grid to' is made up by
c^; = {0 , t^+^,p = (i + a/p) r, i = 0, 1, 2, , , , , i„ - 1 , a = 1, 2, . . . , p} ,

612 Economical Difference Schemes foi' Multidimensional Problems
containing not only the nodes tj = JT of the grid lo^., but also fictitious
nodes tj_^_^i , a = 1,2, . .. ,p— 1. Let co'^ be a set of the nodes of the grid
ui'^, for which ^ > 0. Finally, let P{x,t'), where x € cj^, t' G cj^, be a node
of the (p+ l)-dimensional grid SI = cj^ x cj' ; S be the boundary of the grid
SI containing the nodes P{x,0) for x G ^^ and the nodes P{x,tj_^^,) for
ij+a/p G ^T ^^'^ ^ S '^h.a fo^' <^H ''"^' = li 2, ■ .. .p, i = 0,1,. . . , jg; fi*, be a
set of the nodes P{x,tj_^_^i ), where ,f; G i^^ is a iiear-bouudary node in
the direction x^ of the grid a);,.
With these, we proceed to the complete posing of problems related to
y and w. The intention is to use the equation for y in the canonical form
B5) in combination with expression A8) for the diiTerence operator Aa not
only at the regular nodes, but also at the irregular nodes:
C3)
11/1 1
I ^ -/ + «/P , 1 ,ri+(«-l)/p
+ ,* , i/i„-i + y >
where yl ^^^' = t/(a;f^^°',/•_,_„;,). From here it seems clear that conditions
B9) are satisfied and D{P) = 0. Because of this, Theorem 2 from
Chapter 4, Section 2 asserts that for a solution to equation C3) the estimate is
valid:
max |w(_P)| < max |w(_P)|
Pen+s '^ ' - Pes '^^ '
Taking into account that
max \y{P)\ = max \\y{x,t')\\
where ||y(a;)|L. = max \y{x)\
.rew,,
max |t/(_P)| = max (max ||//(a;, ^')|L , llMnlU)
we eventually get
where ||p(a;)|L, = max |p(a;)|
xejh
C4)
y°\\c<\K\\c + .?}^^ M^'i'
0<i'<ioT
ic,

The summarized approximation method
613
With regard to problem C2) for lu, it will be sensible to rewrite
statement C2) in the canonical form B8)
1 W 1 J_
T h„ \h*+ h*_
+
L_ ^i+"/P I 1 „;i+(«-i)/P I ^*i+«/P a = 1, 2
h* _ /k
,P,
■w^+»Ip = 0 for X 6 7ft,„ , iv(x, 0) = 0 ,
meaning w = 0 on the boundary S of the grid fi:
iu(P) = 0 for PES.
The right-hand side ip* differs from zero only at such nodes {x,t'),
where x G Lot. The trace of the homogeneous boundary condition w = 0
should be clearly seen in
D{P) > min
1
1
Ki ^c, h"^
I'here h = max h.
Applying Theorem 4 from Chapter 4, Section2 yields
p*ix,t')
C5) max |y(-P)| < max
n+5 t'£uj'
D
< max /i ||<^*||^-,.
In the estimation of the function r we write down equation C1) in the
canonical form B8) by regarging P — x to nodal points of the p-dimeusional
grid cj ft!
11/1 1
r h^\ h* ^ h* _
,J+a/p
1
^*+^o
j+a/p
"ic + l
+
1
h*_h"'^'-'
P + C^/P , pj + a/p
;■ -1 ^ ^ a '
pi+"lp
,^j + {a-i)lp 0
+f
j+a/p
yj+a/p ^ Q foj. ^ g ^^ ^ ^(j,^ 0) = 0

614 Economical Difference Schemes for Multidimensional Problems
In that case D(P) = 1/r and Theorem 3 in Chapter 4, Section 2 states
that
C6) \w^"i'\\c< ^v- =^\\n''""\\c
^ c
The first summation of C6) over a = 1,2,.,, ,p gives rise to the relation
\W^X<\W\\c + rY.\\"vi^"
Ip\\
c
a = l
which IS followed upon summing up over j = 0,1, 2,, , , , , jg — 1 by the
estimate
C7) lk1lc<E ^Ell^i+"^1lc'
J'=0 Q = l
Estimate C0) for a solution of problem B1)-B3) is an immediate
implication of the collection of relation C4), C5), C7) if we might invoke
the arbitrariness in the choice of the number jg.
8. Uniform convergence of LOS. This type of situation is covered by the
following assertion.
Theorem 2 Let problem A5) possess a, unique solution u = u{x,t)
continuous in Qt and there exist continuous in Qt„ derivatives
d^u d'^u d^u (Pi
a p a ex
Then scheme B1)-B3) converges uniformly with the rate 0{lP + r) (it is
of first-order accuracy in r and of second-order accuracy in h), so that
y -u^\\c<Mih^ + T), i = l,2,...,
where h — max /i^, M — const > 0 is independent of r a,nd h^.

The summai-ized approximation method 615
Proof Let z,^\ = z^+"''\ As usual, we may attempt a solution ?/^) =
j/(„) — u^+"/P of problem B5) as a sum z,-^) = tij-^^-, + yy,-^) with i^j-^) still
subject to the conditions
C8) =V'a for a;6cjft+7; a = 1,2,... ,p,
T
il{x,0) = U.
From here we deduce for j = 0,1,. . . , Jq that
0 0 0 .
since 7y° = 0. With regard to 1],^^ it is plain to show that
0 0
'?(«) = T ii> 1 + Ip 2 + ■ ■ ■ + ■'P a) = -T {Tp a + 1 + ■ ■ ■ + 'P p) ■
The function Vf^i is yet to be recovered from the conditions
C9) ^ ^^^^^^ = A„V(„) + '(i„, xEoJ,,,, a=l,2,...,p,
■y,
(a) = -??(«) for a; e 7ft,„ , v(a;, 0) = 0 ^
where ■tp^ = iP^ + A„7y(„).
A solution of problem C9) can readily be evaluated on account of
Theorem 1 in Section 7 with v = 0 incorporated for ^ = 0, stating that
D0) ll^^'ll,. < max , {h'U^'^^'nic + \W'^"''\\c )
0<j'+a/p<j ''
+ ErJ2\\i>^+"/''\\^.
j' = 0 a = l
If there exist for a ^ fi the derivatives (9^w/(9j;^ 'dx^ continuous in the
closed domain Qy, then at all the nodes ,); G cj;,
0 0
Aa??(a) = -rA„(i/>„_^i + .. . + ,/,p) = 0(t),

616 Economical Difference Schemes for Multidimensional Problems
since the quantity yy/^N is recovered from equation C8) everywhere on cj;, -|-
7;j „. On the other hand, t/i* = Oih"^ +t) at the regular nodes of the grid coj^
and 0* = 0(/»^ + r) at the irregular nodes. Therefore, /i^||i/'||p. = 0{h'^ + T)
and \\^\\o = 0(/2^ + ■'')> so that estimate D0) in such a setting gives
c
\\z^\\c = \\v^\c<Mih'+T),
since rj^ = 0 for all j = 0, 1, .. . , j^.
Observe that the stability with respect to the right-hand side and
boundary conditions implies that the moments t* and t** can be arbitrarily
taken from the interval {t:,t:_^_^).
9. LOS for equations with variable coefficients. One way of covering
equations with variable coefficients is connected with possible constructions of
locally one-dimensional schemes and the main ideas adopted for problem
A5). It sufficies to point out only the necessary changes in the formulas for
the operators La and A„, which will be used in the sequel, and then bear in
mind that any locally one-dimensional scheme can always be written in the
form B1)-B3). Several examples add interest and help in understanding.
1) A linear equation of parabolic type.
Let in the statement of problem A5) involves
d f , . du \
LaU- j—lk^(x,t) -—I, 0 < Ci < fc„ < C2 ■
Minor changes in the complete posing of problem B1)-B3) are based on
the formula for the difference operator A^ acting in accordance with the
rule
Aa y^a) = (aa(^'. 0 VsJ^^ ' 0 < ^i < «„ < ^2 , i= tj+1/2 ■
A second-order approximation provided by the difference operator A« on a
regular pattern
A„ M - L„ M = 0{hl)
is ensured by a proper choice of the coefficient a„. This can be done using,
for example,
X — [X^, ■ ■ • , ^a-l 1 i^a ~ U.O ft„, Sq._|_j , . . . , Xpj ,

The summai-ized appi'oximation method 617
allowing Theorems 1-2 to hold.
2) A quasilinear equation of parabolic type.
The complete posing of problem A5) includes
if / {/7I \ r)tc
LaU=-—[kjx,t,u)-— , 0<q<fc„, -~>Ca>0.
ox^ \ Ox„' ou
Two possible ways of approximating the operator La are:
In the first case a nonlinear equation is aimed at determining j/(„) that
can be solved by one or another iterative method, Every iteration can be
found during the course of the elimination.
In the second case we obtain a linear equation related to yr^) and then
use the elimination method for solving it. The uniform convergence with
the rate 0{t + K^) takes place under the extra restrictions concerning the
boundedness of the derivatives d'^k^I'du^^ d'^k^/dx„du, 'd'^k^jdx'^^.
Locally one-dimensional schemes find a wide range of applications in
solving the third boundary-value problem. If, for example, G is a rectangle
of sides /j and /j or a "step-shaped" domain, then equations B1) should be
written not only at the inner nodes of the grid, but also on the appropriate
boundaries. When the boundary condition 'duj'dx^ — a~u-\-v\ is imposed
on the side Kj = 0 of the rectangle {0 < x'„ < /„, a = 1, 2}, the main idea
behind this approach is to write for a — \ equations B1) at the node x^ — 0
as well. This can be clone by setting
, y(i>. - ^^^^A) y\~'^
0.5/zi ^' 0.5/ij '
assuring the uniform convergence of the describing locally one-dimensional
scheme with the rate 0{t -\- |/i|').
10. Additive schemes. The general fomiulatioiis and statements.
Considerable effort is devoted to a discussion of additive schemes after introducing
the notion of summarized approximation. With this aim, we recall the
notion of the n-layer difference scheme as a difference equation with respect
to t of order n — \ with operator coefficients:
n-l
Y. Gp(t:^)y(t:^^, - /?r) = /(^^.), {n-\)T< t^ < t, ,
13 = 0

618 Economical Difference Schemes for Multidimensional Problems
where Cp are linear operators acting in a vector normed space Hh- VVe shall
need yet n — 1 initial vectors t/@) = j/q, t/(r) — y^, . . . , y({n - 2) r) = Un-'j.
for its numerical solution.
By the n-layer composite scheme of period m (of order m) we
generally mean a system of differential equatioins with operator coefficients
D1) Y. ^«/3(^i)y^h + /?^) = E Dap{tj)y{tj-/?r) + f„(t.),
13=1 j3 = 0
a = 1, 2,, .. , m , (n — 1)t < tj Ktg ,
with known initial values y{k t), k = 0,1, .. . ,n — 2. Here t; takes on the
values
t. = [n- l)T + kmT , A; = 0,1,...,
and the total number of layers is equal to the amount of the initial
conditions.
With knowledge of the values j/j_.,^, ,d = 0, 1, . . . , /). — 2, where t: =
[m + 11 — 1) r, it is possible to find y[tj -\- inr) — y.^j^^ in the process of
solving a system of equations with the operator matrix C = {Cap) of size
m y. m.
Several particular cases will be given special investigation. For m — \
the composite scheme D1) falls within the category of standard n-layer
schemes. For n = 1 the describing scheme is termed a two-layer
composite scheme of period m
D2) Yl ^'^-P^h) y(^j + /?r) = /?„0 y{t,) + fa(tj),
1 <Q <m, y{0) = y^ .
If for the composite scheme D1) the error of approximation tp is adopted as
a sum of the residuals ■(/'„ of separate equations, that is, V' = V-'i + ■ ■ ■ + U'm <
the composite scheme D1) is called an additive scheme.
By replacing r by r/m scheme D2) admits an alternative form
D2') Y ^'"d^yi^j +Prlm) = Daoy(tj) + Ia(h)'
,3m
a = 1,2,... ,m .

The summarized approximation method 619
A two-laver additive scheme can alwavs be written in the canonical form
D3) B^- y~ + j2A^f3y'+^''" = ^i,
'^ 13 = 0
Q- = 1,2,... ,m,
where B and Aa/d are some linear operators. It is straightforward to verify
that all of the available economical methods with canonical form D3) can
generate a summarized approximation.
Furthermore, let u{t) G Hq be an abstract function of the argument
t G [0,^o] with the values in a normed space Hq and Uf^ = VhU E H^ be
the projection of u onto Hk,
j-\-a/m j-\-(a~y)/m in
;3=0
is the residual for equation D3) with the number a and u^+"/"^- = «(/. -|-
ra/m). Assuming this to be the case, the sum
^viK) = Yl "^"("'i
is of our initial concern.
By definition, the additive scheme D2) provides a summarized
approximation on a function u{t) G Hq if
max ||V'(w{)|| —>0 as r ^ 0 , /i ^ 0 ,
0<3<3o (^h)
where || • ||,„ > is some suitable norm on the space Hi^. In conformity with
Section 2 the additive scheme D2) is said to be economical if the operator
(matrix) C is economical. That is to say, the work and storage necessary
in the numerical solution of the system of operator equations
m
D4) Y.C^pi/+^/'^ = '^i
require a minimal number (in some up-agreed sense) of arithmetical
operations. For example, it may be proportional to the dimension A' of the space
Hh (it is equal to the total number of the grid nodes cj^).

620 Economical Difference Schemes for Multidimensional Problems
If (Cajs) = C~ is a lower triangle matrix and all the operators C««
are invertible, then the procedure of solving equations D2') can be reduced
to successive solution of the equations
a-l
13 = 1
Such a triangle additive scheme will be economical once we involve
economical diagonal operators C««, a = 1,2,... , m. Economical schemes arising in
practical implementations of multidimensional mathematical-physics
problems turn out to be triangle additive schemes (usually lower, but sometimes
upper), whose matrices are of a special structure. As a rule, nonzero
elements of the matrix (C'a/j) stand only on one or two diagonals adjacent to
the main diagonal. With this in mind, the scheme
C„„y^+"/" + a„„i y^+("-i)/" ^D^oiJ +fi, a = 1, 2,... , m ,
may be of assistance in achieving the final aims. In particular, when Dao =
0 for all tlie values a = 1,2,... , m, the preceding reduces to
n .J+n/'n I r; , ,J+("-i)/n'' — fj
^a ay T L'tt a—1 y ~~ J a '
one special case of which is the weighted scheme
yj+a/m _ j+(a~l)/m. . . ^,
^~ '^ + A, (a„ :v^+«/" + A ~ ^J y+(«-!)/"
^i
Such locally one-dimensional schemes were investigated before, all the triks
and turns remain here unchanged. The following issues are yet to be
answered in the possible theory:
1) the estimation of stability and accuracy of an additive scheme;
2) the design of an economical additive scheme for a multidimensional
problem in mathematical physics.
11. Methods for the convergence rates of additive schemes. So far we have
established many times that approximation and stability of a difference
scheme provide its convergence. For additive schemes we shall need stability
with respect to the right-hand side so that it follows from the condition of
summarized approximation
Y. '"^^
0

The summarized approximation method
621
that a solution of the relevant difference problem supplied by the zero initial
condition approaches zero. Such a priori estimates hinging on the
summarized approximation properties hold true for additive schemes associated
with systems of parabolic and hyperbolic equations.
We outline in what follows the general theory for an additive typical
scheme in a Hilbert space Hh such as
D5)
J+a/p __ j+(a-l)/p
13=1
a 13 ^
i^i
«= l,2,...,p, i = 0,l,..., --" = 0.
Theorem 3 If B = B* is a positive definite constant operator and tlie
matrix-operator A is non-negative A = (Aap) > 0, tliat is, for any vectors
L,^i3eH
V
D6) Y. (^«/3^'a.^,3)>0.
then a solution of problem D5) satisfies the a priori estimate
D7) \\z^
< max
~ 0<k<j
E^l
+p
^EUt
The proof of this formula is omitted here. It should be noted that from
such reasoning it seems clear that due to the summarized approximation
ill the space Hg-i the convergence occurs in the space Hb- That is to say,
the conditions
D8)
E^<^
,-1 ^0
o(i)
0, \h\^0
guarantee the convergence Hz-'H^ ^ 0 for all j = 1,2,... by observing
that estimate D7) is valid under rather mild conditions: B is a positive
definite self-adjoint operator and A is a non-negative matrix-operator. But
in a Banach space Hk another method of further derivation of a priori
estimates is employed for scheme D5).
In what follows scheme D5) is supposed to be stable so that
D9)
p
'-'IL, < M max y^
('-) ~ o<k<j ^—^
a=:l
2'
^aMB)'

622 Economical Difference Schemes for Multidimensional Problems
where || • |L and || ■ |L,,, are some suitable norms on the space Hi-,. The
usual trick we have encountered is to represent the residual ijj^ by
P
E0) ^a = ^a+i'*a, SO that ^^„ = 0.
a = l
By setting z^+"/p = r]^+"/P + ^i+«/p_ where r]^+°/P is determined from the
conditions
j+a/p __ j+(a-l)/p 0
B^ '- ^^Pi, a=l,2,...,p, if^Q,
T
it is plain to show that
B i+"l^ = B,f +7^2°^^' ^ ^f^^ = Bt]^ = ■■■ = Bif =0,
3 = 1
giving if = 0 and z^ = v^ for all j = 1, 2,.. . and
a p
i+"l^ = TY^B-'lp^ =~T J2 B-'ii, a=l,2,...,p~l.
13 = 1 I3 = a + 1
In turn, t;-?+"'P satisfies ec|uation D5) with the right-hand side
a p
13:^1 P'=I3+1
and the initial condition v° — 0. Having stipulated condition D9), the
following estimate is valid:
\W\\^,^=\W\\^,^<M m^^ J2\K\
o<j'<j
B)
The reader is invited to prove this assertion on his/her own. The
summarized approximation condition means that
1) the residual i/'a admits representation E0),
2) ||04||B^ ^ 0 as r ^ 0 and \h\ -^ 0.

The summarized approximation method 623
The second condition is ensured if
IICIIB)^0. \\A„pB-'°^l\\ = 0{l) as T^Q,\h\^Q.
The second method of special investigations with concern of additive
schemes was demonstrated in Section 8 in which convergence in the space
C of a locally oae-dimensional scheme associated with the heat conduction
equation was established by means of this method. Let us stress that ia such
an analysis we assume, as usual, the existence, uniqueness and a sufficient
smoothness of a solution of the original multidimensional problem under
consideration.
Let, for example, u be a solution of problem E) and y = t/^ be a
solution provided by this or that additive scheme, t/^j G Hh, where Hh is
the set of grid functions. Following established practice, the difference z^ =
j/^j — m'Ij , where m^ = Vh m and Vh is a linear operator from the space Hq into
the space H^ (u G Hq, Uj^ G Hh), needs investigation. To be more specific,
we are interested in the possible estimates of the quantity ||j/jj — wj, ||,, >.
in some suitable norm || ■ ||,. s on the space Hh' The traditional ways of
covering this are to set up the relevant problem for the error z^, calculate the
0
residual ip^ — V' a + V** and then adapt one of the well-developed methods
for determination of z^.
12. All approximation of the "multidmieiisioiial" abstract Caucliy problem
by a chain of the "one-diniensioiial" Caucliy problems. It is to he hoped
that the forthcoming reduction helps clarify what is done. The problem
statement involves problem E) with the homogeneous boundary conditions
on the boundary V under the agreement that the function u(x,t) as a
function of the argument x can be treated in a common setting as an element
of some vector normed space Hq. Then L refers to a linear operator in that
space and u = u(t) may be viewed as an abstract function of the argument
t with the values Hq, it being understood that u(t) G -ffo for all t G [0,io].
In this view, it is possible to write in problem E) the usual derivative in t
in place of the partial derivative, making our exposition more transparent.
As a final result we get the abstract Cauchy problem
ciu
E1) —+Au = fit), 0<t<t,, uiO) = u,eHo,
where yl is a linear operator in the Banach space Ho. The domain V{A) C
Hq of the operator A is everywhere dense in the space Hq and comprises
all the functions satisfying the homogeneous boundary conditions on the
boundary V and its range A(A) belongs to the space Hq.

624 Economical Difference Schemes for Multidimensional Problems
Also, the operator A will be taken to be
E2) yl= £ yl„,
where linear operators Aa are so chosen as to provide the relation
p
f] V(Aa)=V(A),
making it possible to reduce the solution of the Cauchy problem E1) to
solving successively the Cauchy problems of the same type, but with
operators Aa standing in place of the operator A. We confine ourselves here to
two possible ways of such a reduction.
For later use, it will be sensible to introduce on the segment 0 < t < t^
a grid co^ = {t, = jr, j = 0, 1, . . . , jg} with step r and to attempt the
function / in the form
p
/ = E /a ■
The first reduction (for more detail see Section 3). The object of
investigation is a chain of the equations
1 dv^a)
p dt
E3) Z —JT+^aV(^a)'^ fa, a = l,2,...,p,
h+{a~l)lp < ^ < ^j+a/p .
with the supplementary initial conditions
E4) •(;(i)@) = «o , ^{i)(^j) = ^(p)(^j)' i=l,2,...,
^ia)(ij+ia-l)/p) = '^(a-l)(^j+(a-l)/p) '
i = 0,1,. .. , a = 2,3,... ,p.
The function v(tJ_^_<^) = Vf^p\(tj_^_<^) refers to a solution of this problem
for t = tj^i- In the general case we miight have
\\v(t^)~u(t.)\\^OiT) for all i=l,2,...

The summarized approximation method 625
The second reduction of the Cauchy problem. On the whole segment
-^'7 < ^ < ^7+1 we must solve sequentially p Cauchy problems
E5) -^ + Arit) •«(,)(<) = f\{t), t^<t< t^^, ,
^ + A^t) ^(„)(i) = fjt), t^<t< i^ + i ,
'^"^ +Apit)v^^,^it) = ^t). t^<t<t^^,
dt
with the initial data
E6)
\a)(ij) = ^"(a-i)(ij + i), i = 0,l,...,, a = 2, 3,...,p.
By definition, an element
vitj + i) = V(^p)itj + i), i = 0,1,2,.., ,
gives a solution of problem E5) for t = ^7+1- For i = 0 we agree to consider
E7) V(,p) = u{0) = u, .
Knowing v{tj), it is possible to determine v,^^{tj_^_^) from the first equation
entering the above collection with Vr-^^(tA = v{tj) incorporated. At the next
stage i^cJ-)(iJ_|_I) is taken as the initial value of V/2-)(t) for t — tj, allowing
to solve the second equation for a = 2, etc. The outcome of solving all
the p problems is v^ Jtj_^_j) = v(tj_^_^), giving a solution of the system of
equations E5)"E7) for t = ^,-+i-
When the operators Aa happen to be independent of t and / = 0,
problems E3)-E4) and E5)-"E7) become equivalent. With this in mind,
we are going to show that problem E1) is approximated by problem E5)-
E6) in a summarized sense. To that end, the differences
^{a){^) = ^{a)(^)~'u(t-^^) for a = 2,3,...,p, te[tj,tj + i],
^(i)ii) = ^ii)(i) ~ u{t) for te[tj,tj^^]

626 Economical Difference Schemes for Multidimensional Problems
will be given special investigation, where u{t) is a solution of the Cauchy
problem E1) and v^^^-^{t), a = 1,2,... ,p, is a solution of problem E5)-
E6). By substituting here ii(„-)(i) = ^(a)@ +«■''''\ '"•' + ^ = u(i^_,_J, a =
2, 3,... ,p and lij-j-) = z^(t) + u{t) into E5)-E6) we are led to
-\i)ih) = \p)^h) ' i = i,2,..., z,{Q) = Q,
where
0„(i) = ^A^At) «^ + ' + i\S), « = 2, 3,... ,p ,
dii
i^,(t) = ~A^it) u(t) - — + /^(i), te [t,,tj^,].
From such reasoning it seems clear that
^0 = ^0j(i) + -.-+0p(i)
a = 2
With the aid of the relations m-' + ^ = ■u(i)+0(r), valid for any a = 2, 3,... ,p
on the whole segment t G [tj,tj^i], we finally get
0 0 fj^i
i^a = 0 . + C . C = Oir), V. „ = ./;(i) ~ A„{t) u{t) - 5„^, - ,
where S^ j is, as usual, Kronecker's delta. By the same token,
a —1 o-—1 a~y
yielding
p
a=l
E^''': = o(r)

The summarized approximation method 627
This means that the system of differential equations E5)-E6) generates
an approximation of order 1 in a summarized sense to the Cauchy problem
E1) under the extra restrictions on the existence and boundedness of the
derivative A{t)d?u/dt'^ in some suitable norm.
Further comparison of the solution v{ii) of problem E5)-E7) with the
solution u(ti) of the original problem allows to cite without proofs several
interesting remarks.
a) Let / = 0 and all the functions /„ = 0. If constant operators Aa
are pairwise commutative: AaAp = Af}Aa, «', /? = 1,2,... ,p, then for any
r the equality holds:
E8) v{tj) = u{tj) for all j = 0,1,... ,io ,
where ?; is a solution of problem E5)"E7) and u is a solution of problem
E1).
When the operators Aa = Aait) happen to be dependent on the time
t, equality E8) is still valid for commutative operators Aa{t') and Apit"),
a :/: j3, taken at different moments t' ^ i", so that
Aait') Ap(t")^Ap(t") Aait'), a, 13^ 1,2,...,p,
for any i',i" 6 [O,^^].
In this regard, we refer the readers to a few examples of Section 4 in
which equality E8) holds true for commutative operators AaAp = Ao Aa-
b) If the operators Aa(t) and Apit) are non-commutative, then
estimate
E9) \\vit.^)^uit.^)\\ = OiT), J = 1,2,...,
will be valid under the additional condition of smoothness:
\\AaA0u\\<M, «,/?= 1,2,... ,p.
Is it possible or not to improve the accuracy in r without essential
modifications of the compo.site Cauchy problem? In an attempt to give
a definite answer to this question, the composite Cauchy problem E5) is
designated by the symbolism
Ai —^ A2 —' •' Ap.
The symmetrized problem consists of 2p Cauchy problems such as
Q.hAi —-'Q.hA2 —' •' Q.'oAp
—-'Q.hAp —-'Q.hAp^i —^ ^Q.hAi.

628 Economical DifFei-eiice Schemes foi- Multidimensional Problems
This chain can be associated with the representation of the operator A as
a sum
^ , , ( O.bAa for 1 < a <p,
A=Y^ A', where A' = \ - - /^ -
t'l I O.bA.p-a+i for p<a<2p.
The problem we have posed above is of second-order ticcuracy in r:
provided that some smoothness property of the initial vector u^ of the type
\\AlA^ru„\\<M , a,P=l,2,... ,p,
holds together with the smoothness property of the operators Aaii) in t.
In such a way, the procedure of solving problem E1) reduces to a
sequence of simpler problems E5)-E7), whose solution can be obtained by
means of exact or approximate methods. In particular, the finite difference
method suits us perfectly for doing so. If the operators Aa are pairwise
commutative, the accuracy of an approximate method available for
solving problem E1) depends on how well we are able to solve every auxiliary
problem with the number a from sequence E5). The above exploration is
still valid for the case of the homogeneous boundary conditions. In dealing
with the nonhomogeneous boundary conditions the accuracy of the
composite Cauchy problem E5)-E7) depends .significantly on the possible ways
of specifying the boundary conditions for v^^y The same remark applies
equally well to difference analogs of problem E5)~-E7),
The difference approximation of every auxiliary problem from
collection E5) through the use fo the simplest two-layer scheme with weights
leads to an additive scheme. If either of the auxiliary schemes with the
number a is economical, then so is the resulting difference scheme.
Remark The accurate account of error zj^ = y^ — U/^ can be done in
a number of different ways. In concluding Section 11 the usual way of
proper evaluation of the error Zf^ was recommended for an additive scheme.
Another way of proceeding is connected with the triangle inequality
II^h II = II Vh -«/, 11 <\\yh~ -"h II + Ihft - "ft II.
where ?; is a solution of the locally one-dimensional problem E3)"E4) or
E5)~-E7). From such reasoning ir seems clear that the further estimation
of the error z^ amounts to evaluating the proximity between ijj^ and Vf^, u^
and u^. Some progress in such matters can be achieved by the subsidiary
information about the smoothness of the functions u and v, thus causing
some cumbersome exposition in connection with more a detailed exploration
of the properties of the solution v of the composite Cauchy problem E3)-
E4) or E5)~-E7).

The summarized approximation method 629
13. Alternating direction methods as additive schemes. The starting point
for special investigations is the alternating direction method developed in
Section 1;
F0)
^^ ^ = 0.5(Aij/J+i/2+A2j/) + 0.5 9^,
^ ^ - = 0.5(Aij/+i/2+A2 2/^ + ') + 0.5^^'.
r
As can readily be observed, this system of equations is equivalent to
'- '— = 0.5 A2 y^ + 0.5 <p^
,J+i/2 _ yJ + 1/4
F1)
r
.,J+3/4„ j+1/2
^^ ^ = 0.r3Aa/+'/-
„J+l„„J+3/4
= 0.5 A2 y^+^ + 0.5 ^^,
relating to locally one-dimensional schemes. Both intermediate values
jyj+i/4 j^jjfj ,(yi+3/4 |,j^j^ i^g eliminated in the usual way without any
difficulties. It is straightforward to verify that scheme F1) generates a summarized
approximation such that
,P = ^;^+ '02 + 1^3 + '04 = 0(r2 + |/f) .
The alternating direction scheme ascribed to Duglas and Rachford
and associated with the difference operator AaV = y^ x comes second;
F2)
,J + i/2„„J . ,
yj + i „.(,J + l/2 ,
r
Simple algebra gives the residuals
u + u u ~~ u u — u
■iPi = Ai h A2 u , V-'a = A2 (u - u)

630 Economical Difference Schemes for Multidimensional Problems
where u = u^ and ii = 1/.^+-^. From here another conclusion can be drawn
that w^ =0(\) and tJ)^^ = 0[l), but
thereby clarifying a summarized approximation provided by scheme F2).
Upon eliminating the intermiediate value y^+^f'^ we have at our disposal the
factorized scheme containing the values y^ and y^'^^ and approximating the
heat conduction equation to 0(r + |/ip) in the usual sense, But this scheme
is stable only for commutative operators Ai and yV2. No restrictions of this
sort is made for the additive scheme concerned.
14. LOS for a multidiinensional hyperbolic equation of second order. The
method of summarized approximiation offers miore advantages in
designing absolutely stable and convergent locally one-dimiensional schemies for
equations of hyperbolic type. The object of investigation is the equation
a = l " "^
k^{x, t) > Cj > 0 , Cj == const ,
where x = (x-j,... , x ) G R^ and G is an arbitra.ry domain in the space R^
with the boundary F, G = G + V and
Qt = G x[0<t <T], Qt = 6' X @ < / < T].
The problemi statemient here consists of finding a continuous in the cylinder
Qx solution to equation F3) satisfying the boundary condition
F4) u = ij{x,t) for xer, 0<t<T,
and the initial conditions
ouiX 0 I -
F5) u{x, 0) = Uq{x) , —-^— = 'Uo{x) for x EG.
As usual, it is preassumed in a common setting that the problem
concerned is uniquely solvable and its solution u = u{x,t) possesses all
necessary derivatives which do arise in all that follows. The domain of
interest G is still subject to the same conditions a,s we imposed in .Section 5
for parabolic equations. Also, let lo^ — {t- = jr, j = 0, 1,.. . } be a uniform

The suininarized approximation method 631
grid in r on the segment 0 <t < T and a grid ulf^ in G remains unchanged
(for more detail see Chapter 5).
If G is a parallelepiped in the space TV, problem F3)~F5) can be
solved through the use of an economical factorized scheme with accuracy
Oir"^ + |/ip). The design of such a scheme was made in Section 2 and it
was investigated there in full details. Applying the same procedure serves
ti motivate that, first, the operators
F6) Vau= -^'--^-{L^u + fJ, a=l,2,...,p,
with /q, still subject the normalization condition
p
E fa = I
are approximated successively with step r/p.
Second, the difference expresions
F'^) ^fcfc
7^ 4 [inr^
J+(a-l)/p
are aimed at approximating to the derivative 'd^ul'dt^ with step tj]), where
"(q) — U , W(a) — ^' 1 '■'@) — M — tl , U(^2) — ^ I
_ ^(a)-"(«-l)-«(a-2)+f'(«) 2 d^U
a = 1,2,3 for p = 3 ,
with the members Uf^i-) = "B) = u^^~^')+'^/^ and W(_2) ~ ^{i) = u^^"^!^.
Third, the operator Lc,u + /„ is approximated to second order by the
homogeneous difference operator A„t/ + i^„ on the grid cj^ in the space R^.
The coefficients at the member A„ and the righ-hand side <^„ are taken at
the moment
t'^ = 0.5(i!j+„/p +ij_i+„/p) = ij+„/,;_o.5 = ^i +(a/?>-0.5)r,
so that Aa = Acv(i^) and ^„ = ip^{x,t'^).

632 Eoonomical Difference Schemes for Multidimensional Problems
"Locally one-dimensional" schemes for hyperbolic equations acquires
the form
F9) yf„f„ = o-p A„(y(„)-F^(„))-F2o-p^„, a=l,...,p, p=2,3,
where
1/4 for p=2,
1/3 for p=3,
and the left-hand side t/f j- is given by formula F7) for p = 2 and by
formula F8) for p = 3. With the detailed forms in mind, the final schemes
refer to three-point additive ones (p = 2) and four-point ones (p = 3).
The principal difference from the case of parabolic equations lies in the
dependence on the number of measurements p.
Equation F9) can be rewritten as
G0) (S-o-pr-yV„)(y(„)-F^(„))
__ J 2y(„_i)-F2o-2rVa for p = 2,
\ 2/(^-1)+2/(a-2)) +2 o-grVa for p = 3.
The solution yr^\ is sought from the three-point equation
(£'-o-pr'A„)y(„) = Fa
along segments parallel to the axis Ox^ with the boundary condition
G1) y(a) = K^,'tj+a/p) for j;6 7ft
with further reference to the elimination method.
First of the initial conditions u(x, 0) = Uq{x) is approximated exactly:
G2) y(x,0) = u,(x).
The intermediate values y^/'^ = y(x,T/2) for p = 2 and y^f^ = y(a;,r/3),
y2/3 ^ y(^.j^^ 2r/3) for p = 3 are found, respectively, from the equations
G3) ^E ~ - Ai) y'r^ = F, ,
Fi=u„ + ^ u„ + T_Aiu, + T^ (/; ^ - (A M + /))
4=0

The summarized approximation method 633
if p = 2 or
G4) (^E-^Ai)y''^ = F,,
We are now interested in the more detailed designs of LOS for p = 2:
G5) uix,0) = Uaix), I^E - —Aiji//^~ = Fi for t = Q.bT,
4
_L 1 / 9 .-^ ; ; _ 1 / 9
' ^^^^^-^^— = J Ai (j/+j/-^/^) + ^^^
G6)
^^ % ^ = ^A,(j/^ + '+y) + i^^,, i=l,2,..,.
In that case the boundary conditions become
2/^"^^^^ =A'(^',^j+i/2) for xEjI,
G7)
y^+^ = ^(x.ij+i) for a; 67J] ,
and the function yi+^l"^ can be recovered from the equation
j^>i/2^^Ai2/^-+i/2^$;
with the right-hand side ${ and the known boundary conditions G7), In
turn, the function y^'^^ is found fromi the ec|uation
y'+'-'^A2y^+' = ^-
^ ■ j + l ^ fl>J+l/2
with the right-hand side $2 ^^'^ the known boundary conditions G7),
Either of these equations can be solved by the standard elimination method.

634 Economical Difference Schemes for Multidimensional Problems
With the basic tools in hand, we proceed to carry out the accurate
account of the approximation error z,^-. = z^+^l"^ ■=. yi+"l'^ — u{x,tj_^^^i2)
for scheme G5)"G7), where u is a solution of problem F3)"F5) and y is a
solution of problem G5)"G7), by inserting the value y^^-, = z,^-^ + u-'+"/^
in equation G5). As a final result we obtain
G8) 2t-^t-„ = - K{\a) + ^(a)) + V'a foi' i>T,
2
f£- — Ai)^ = '01 for i = 0,5r,
z{x,Q) = Q, x^LO,^, 2j^^ = zJ+"/2 = 0 for x 6 7^ , «=1,2,
with the member
G9) ij,^ = - Aa (m„ + ii„) - Mj-^j^^ + 0.5 ^„ ,
serving as one possible representation for the residual of equation F6) with
the number a = 1,2. The error of approximation for LOS of the form G5
)"-G7) is viewed as a sum
(80) ij ^'4', + i^2 ■
Further progress in this area will be achieved by utilizing the fact that
scheme G6) approximates problem F3)~F5) in a summarized sense and
'(/' = 0{t + |/ip). Indeed, taking into account that
0.5 yV„ (u„ + u„) = (L„ «)>+(«-^V2 + 0{hl) for xE^h,a,
0.5A„(n„ + Mj = (L„n)^'+(""i)/2^O(/jJ for x^uj^^^,
1 /(92My-+(«-i)/2
^„ = /^+(«-i)/2+o(,2)_
0
we find that ip^ = ip „ -\- ip* ^ where
1 / 1 d''u , Ni+('«-i)/2
''I' a = 2 [LaU - ~ j^ + f,^] , a = 1, 2 ,

The summarized approximation method 635
and
<*_ i ^^^l + ^'■^) foi' ^' e ^ h,a ,
"~\0{h,+T^) for xe^l,.
Whence it follows that
= 0.5 ((Li +L2)u~u + f, + f,y + 0.5 r ^ 2f, •
With equation F3) in view, the first summand becomes zero:
(Li + L2) u - w + / = 0 , / = ,/; + f,_ ,
yielding
(81) Jj+^J=0.
By definition, this means that scheme G5)-G7) generates a summarized
approximation. The accepted view is to involve the sum
•02 + 2^ + i;, = 0.5 [(L2 u - u + f,y+'f'' + (L2 u -«+ /;)^'"'^']
+ (Li w - ii + /j )^
= ((Li + L2)«- ii + /; + f,y + {i>{+'f^ - 2^^^ + Jr'^')
making it possible to deduce that
(82) i;, + 2i>, + i>,=T^i>,) =0(t').
No wishing to load the book down with full details on this point, we
cite here only final results: LOS of the form G5)-G7) converges in the grid
norm of the space 14-^2'^ with the rate 0{t + |/ip) if the solution u = u{x,t)
has in Qx continuous derivatives of the first four orders, the derivatives
d'^u/dx'^^ satisfy the Lipshitz condition in t and the right-hand side / is
twice differentiable in t.

636 Economical Difference Schemes for Multidimensional Problems
15. Additive schemes for a system of equations. Later in this section we
will survey some devices that can be used in trying to produce additive
schemes for systems of parabolic equations. With this aim, problem E0)
we have completely posed in Section 2 will serve as a basis for the up-to-
date presentation of tools and techniques, their theory and applications. In
this connection we may attempt the operator L in the form L = L~ -\- iy+
with "triangle" operators L", L+, the associated matrices fc„„ of which
arrange themselves as sums A,' „ := k~ -\- fc+ , where k~ = (fc"*'") and
fc+ =: (A,'+^'") are triangle matrices with entries
aa ^ aa ' ^
aa aa ' aa aa ' aa aa ' aa '
k"'"' = 0 for 77? > s , k+'"' = 0 for m <s.
a a a a
Observe that the matrices k~ and fc+ are symmetric each to other, since
f^-sm _ i^+ms whence it folloWs that"'
a a aa '
Laa - i^„„ + l^aa , ^aa - q^^ (^^„ q^^ \ ■
By introducing a few auxiliary operators with the properties
a — l a
La " = -^aa " + X] ^^ ^ "^ Zj ^'c'P " ' ^aj3 ^' = ^afi U for /? < a ,
13 = 1 13 = 1
P P
;3 = a-|-l /3 = a
and representing the operator L by a sum
p p
(83) L = L-+L+, L-u=Y,L-u, L+u = Y, L+u,
a=l a=l
the solution of the system of equations E0) or
■ ^ = 0,
(84)
where
^p r 1 du
a=l '^
-(L-u + L+u)-(t-+i+}
E(C+f+) = f,

The summarized approximation method 637
reduces to successive solution of the system of 2p equations
1 (9v __ _,_ ^
^ "jg^ - -^« "^ + *a ' ^J + l-a/Bp) < ^ < '^;+l_(a-l)/Bp) .
a = 1,2,... ,p.
The usual starting procedure is connected with approximating the operators
Z.„ by the difference operators A„ acting in accordance with the rules
A« = X] A„^ , A+ := ^ A+^ ,
P=l j3 = a
In this regard, it seems clear that the operator L^ is approximated to
second order by the difference operator yVj, the structure of which involves
the coefficients k^g taken for all a and /? either at one and the same moment
t = tj,ii2 or at any another moment t* G [i,-, i,-_|_j]. With these members,
the additive scheme in question acquires the form
(86) ^^ ^ = Y. A;^y^+'5/Brt + (<^;yWBp)_
yj + ai/Bp) _yi + («i-l)/Bp)
a = 1,2,... ,p,
;3 = a
where a^ = 2p + 1 — a and /?i = 2p + 1 — /?. Also, we should indicate the
direction of index account: aj is being increased from p + 1 to 2p along
with decreasing a in reverse order from p to 1.
The usual boundary conditions are imposed on the boundary x E J^'-
yi+«/Bp) ^ ^j+a/Bp) foj. a; 6 7,t, a = 1, 2,. , , ,p,
(87)
yj+ai/Bp) ^ ^^j+„,/Bp) fQj. ,^ g ^Qi ^ ai = p + 1, . . . , 2p .

638 Economical Difference Schemes for Multidimensional Problems
Tlie initial condition is satisfied exactly:
(88) y{x,0) = u„{x).
The following system of equations emerges in determining yJ+"/Bp) = y,
and yJ+°^i/Bp) = y/^ N during the course of the elimination:
(E - r yV-Jy^„, = F; , (E - r A+Jy^^,) = F+
a-l
P
Fa, =^ Y^ A+^y(,30+^'^« +y(".-l)
I3 = a + 1
Having completed the elimination, we must solve the system of
equations
(^-^A;„)y(„) = F- ,
111 light of the special structure of the diagonal matrices k~ , whose blocks
are lower triangle, the components t/f s, 5 = 1,2,... ,n, of the unknown
vector y(Q-) are to be determined successively by the elimination method in
passing from a to a-|-1 and from s to s+l. By the elimination formulas for
a three-point equation we constitute in a term-by-term fashion the vectors
y. X, a = 1,2,... ,p. Moving in reverse order from a -|- 1 to a and from
s -|- 1 to s the vectors y(-p_|_n, . .. ,yBp) ^^^ recovered from the system
(i?-rA+Jy(,,) = F+,,
Upon terminating this process the resultant vector yBp) is just the solution
y^"^' = yBp) on the layer t = t^_^_y.
Since the system of differential equations E6) approximates equation
E0) in a summarized sense in compliance with approximating equation
(85) by equation (86) with the number a, the additive scheme (86)~-(88)
generates an approximation of 0{t + |/ip):
p
E
a=l
if- + ^^+) = Oir + \h[').

The summarized approximation method 639
In concluding this chapter it will be sensible to introduce the space
0
Q of all grid vector-functions given on the grid i^f^ and vanishing on the
boundary 7/, of the grid under the inner product structure
n
5 — 1 X£LOh
Having involved the operators
p
A-=Y^A-, A+ = Y^At
Q- n I Q- — J.
a p
^^« y = - X] K0 y ■ -4+ y = - ^ A+a y , y 6 fi ,
13=1 /3 = a
we are going to show that these operators A" and A~^ are mutually adjoint
each to other:
(A-y,v) = (y,A+v) for all y,v6fi,
if the matrix k = (k^'^) is symmetric, that is, under the condition of
symmetry fc'"? = k"^\ Indeed, kZ = k'^a and, because of this, we might
■^ a/3 I3a ' pa ap ^ j o
have
p a
(A-y,v) = o.bY,Y^ [(^a;3y.«,.^.^J + iKpy^.'^.J]
a=l 13=1
P P
I3 = l a = l3
P P
a = l fj = a
- 0.5 x: E [iKf3^'^,^y^..) + (^■c!;3v„^..y..J]
a—I I3~a
^{y,A+v).

640 Economical Difference Schemes for Multidimensional Problems
It follows from the foregoing that
(^"y,y) = {A+y,y) = 0.5 (Ay,y) > 0.5q (A y,y),
where
p
a = l
yielding
Thus, the operators A" and .4+ so defined are positive definite:
A->6E, A+>6E, 6 = 4c,J2l-\
a=l
Stability of scheme (86)-(87) with zero boundary conditions is asserted
by Theorem 3 in Section 11, due to which a solution of the auxiliary problem
^U = Y^ Kp %i) + ^a . ^f„, = Yl ^"/3 ^(/30 + '"^t '
P = \ I3=a
^(a) = 0' 2(aO = 0' 2(^.0) = 0 for ^e7ft>
where
„ Z(a)-Z(«-1) _ Z(a,)-Z(a,_i)
satisfies the a priori estimate
||z^- + Ml <M, ^max, \\J2{i^'j'^"'^"'''+ i^lY'^''^"''^'^"''')
P
This supports the view that the additive scheme (86)-(88) converges in the
grid norm of the space L2 with the rate 0[\/t + |/ip)-

The summarized approximation method 641
We will pursue the discussion of additive schemes further with
regard to problem F1)-F2) capable of describing the system of hyperbolic
equations
What has been done i.s to reduce this system to successive solution of simpler
equations moving from a to a + 1:
(90) -^ = L-u + L+u + f„, a=l,2,...,p.
One possible additive scheme
-, a V
(91) — yij. = Y. '^«/3 y(/3) + E ^^«/3 y(/5) + '^" ■
f f}=l I3 = a
a = 1,2,... ,p, {x,t) Euj,^ Xio^ ,
y(a) = AtG-c.C)' ^<y = OJa, a=l,2,...,p,
y(a;,0)=: Uo(a;),
can be obtained through the usual approxnnations of p equations, where
Va = ^ai-^'J'a) ^^1^ t[^ = ^j+(„/j,_0.5)r> ^he Coefficients A'^^, are taken at
moment t[^, Vi^i^ is determined by formula F7) or formula F8), Cp = 0.5
for p — 2 and o- r;; 1.5 for p = 3.
The second initial condition is approximated by setting
y«/P = vi,{x) + ^ u„(;s) + ^ (Luo + Ux, 0)) ,
a = 1,2,... ,p-- 1 .
Because of these facts, the describing scheme generates a summarized
approximation
^ = Y^^,^=0{T+\h\
a=l

642 Economical Difference Schemes for Multidimensional Problems
With regard to y^"*"^ = ytp-^ we obtain the system of equations
where the right-hand side F^ is expressed in terms of the vectors yf/j\,
/3 < a. Such a system can be solved successively moving from a to a + 1
and from s to s + 1 during the course of the elimination. By interchanging
the positions of A~ and A+ we arrive at the second scheme
1 a P
(92) — Yuu = Y. ^^a/3 yw + Y. '^tp y{P) + '^« ■
P 13 = 1 P = a
In this case the system can be solved successively moving from a + 1 to a
and from s + l to s. Alternation of schemes (91) and (92) leads to the third
scheme, which interests us.
By means of the energy method the reader can derive on his/her own
an a priori estimate for the error z = y — uhy exactly the same reasoning as
before with further reference to the property of summarized approximation.
On this basis the convergence of additive schemes can be obtained through
such an analysis.

10
Methods for Solving Grid Equations
In this chapter economical direct and iterative methods are designed for
numerical solution of difference elliptic equations.
In Section 1 we confine ourselves to direct economical methods
available for solving boundary-value problems associated with Poisson's
equation in a rectangle such as the decomposition method and the mathod of
separation of variables.
The general theory of iterative methods is presented in the next
.sections with regard to an operator equation of the first kind Au — f, where
A is a self-adjoint operator in a finite-dimensional Euclidean space. The
applications of such theory to elliptic grid equations began to spread to
more and more branches as they took on an important place in "real-life"
situations.
10.1 DIRECT METHODS
1. Direct and iterative methods. Recall that the final results of the
difference approximation of Ijoundary-value problems associated with elliptic
equations from Chapter 4 were various systems of linear algebraic
equations (difference or grid equations). The sizes of the appropriate
matrices are extra large and equal the total number N of the grid nodes. For
643

644 Methods for Solving Grid Equations
example, for any grid with steps h along all the directions x^jX^,--. ,Xp
(/ij — h^ = ■ ■ ■ = hp = h) the amount of nodal points is N = 0{h~^),
where p is the total number of observations. In the two-dimensional and
three-dimensional cases N ss lO'^ — 10®, for example, for h = 1/100. More
specifically, such sparse matrices are of the special band structure and the
characteristic rations between their greatest and the smallest eigenvalues
are high-order quantities (~ 10^— lO"* of order 0(/i~^)).
Numerical solution of elliptic grid equations necessitates, in view of
their peculiarities, creating special economical algorithms, because direct
economical methods are applicable only in some narrow classes of grid
equations. We will elaborate on this later.
When solving difference boundary-value problems for Poisson's
equation in rectangular, angular, cylindrical and spherical systems of
coordinates direct economical methods are widely used that are known to us as
the decomposition method and the method of separation of variables. The
calculations in both methods for two-dimensional problems require Q
arithmetic operations, Q = 0(N'^\og^N), where N is the number of the grid
nodes along one of the directions.
As a matter of fact, the first method is some modification of Gaussian
elimination relating to the odd-even elimination with the accompanying
factorization, the second one is mostly based on the algorithm of the fast
Fourier transform. The third method of the matrix elimination seems to
offer more advantages in the domains of rather complicated configurations,
but the work during the course of the matrix elimination is done with
Q = 0{N'^) arithmetic operations and the extra storage in connection with
emerging intermediate values.
Iterative methods of successive approximation are in common usage
for rather complicated cases of arbitrary domains, variable coefficients, etc.
Throughout the entire section, the Dirichlet problem for Poisson's equation
is adopted as a model one in the rectangle G ~ {0 < x^ < /„, a = 1,2}
with the boundary F:
d 11 d u
X = (xj, x'J 6 G , u|p = ij(x),
and a rectangular grid in G with steps h^ and h^ is taken to be
^ft = {^iiio = (h^i'hK) 6 G, l„ = 0, 1,. . . , Na, h^Na ~ la, « = 1,2} .

Direct methods 645
The statement of the difference Dirichlet problem associated with
problem A) is
B) Ay=-f{x), xei^h' J/I7,, = A'(^),
A = Ai+A2, AaV = y^^,^^ , a =1,2, yi^i^ = y{i^h,^,i^h^),
Problem B) was the object of special investigations in Chapter 4,
2. The decomposition method. Common practice involves the reduction
of problem B) to the system of vector equations
C) -Y,-_i + CY,'-Y,- + i=F, , j = l,2,...N-2-l,
Yo = Fo , Yw^ = Fftfj ,
wliere Yj and Fj are vectors, whose components are the values of the
solution y^j = y(ihj,jh^) and the right-hand side /^ ■ = f{ih^,jh2) on the
jth column of the grid cj^ and the difference operator C will be specified
in the sequel. By re-ordering of the riglit-hand sides of equations B) at the
near-boimdary nodes we miglit agree to consider t/j • = 0 at the boundary
nodes for i = 0 and i = Ni.
An alternative form of writing equation B) may be useful in the
further development:
D) -y,,i~i + B;!/-/'?y:g.,rJij -y,,j-n = '^Iv^j -
1 < i < A^i - 1, 1 < i < A^2 - 1,
% = yN,j = 0' 0 < i < A^2,
where
^,^. = fi^ for 1< i < 7V1 - 1, l<j <N2-l,
_ , 1 _ , 1

646 Methods for Solving Gi-id Equations
Also, it will be sensible to deal with the newly formed vectors
E) Yj = (yij.,y2j,--- ,J/Wi-lj), i = 0,1,... ,//2,
_ I 2 K 2 2 2 K \
J=.l,2,... ,N2~l,
and a difference operator C acting m accordance with the rule
2 t J
From such reasoning it seems clear that problem B) in view is tantamount
to the system of vector equations C).
Let N2 = 2" for the clarity only. The main idea behind the
decomposition method is the further successive elimination from the governing
equations of the vectors Yj with odd numbers and, after this, with even
numbers divisible 2, 4, 8 etc. Other ideas are connected with setting the
following equations for j = 2, 4, 6,. . . , ^^2 — 2, where N2 = 2":
- Yj~2 + CYj_i - Yj = Fj_i,
- Yj_i + C Yj - Yj+] = Fj,
- Yj + C Yj + i - Yj+2 = Fj + i .
Applying the operator C to the second equation and summing up three
resultant equations yield a revised "short" system
F) -Y,_2 + Cf^'Y, -Y,+2 = F;.'\ i=2,4,6,... JVo-2,
Yo = Ff), Yw, = FiY, ,
containing only the unknowns with even numbers and involving the
members
^.A) = [cW] '^2E, Ff) = Ff}, + C(°)Ff) + Ffl,,
CW = C, Ff = F, .

Direct methods 647
Having recovered from the system F) the vectors Yj with even numbers,
we succeed in finding these with odd numbers from the following equations:
C(o)Y, =Ff^+Y, + i+Y,_i, i=l,3,5,...,iV2-l.
The same procedure works albeit with obvious modifications for the vectors
Yj subject to the system F), whose subscripts j are oddly (but unevently)
even, etc. As a final result we obtain a proper system of equations with
regard to all the unknowns:
G) C(^~^)Y,- =Ff~') + Y,_2.-.+Y,+2.-.,
k = n,n — 1,. .. , 2,1,
Yo = Fo, Yw, — Fwj,
where C^*'-' and F) must satisfy the recurrence formulas
(8) C^*')= [C(^--^)]''-2^, fc= 1,2,... ,n-l, C^^'^^C,
„{k) _ ^{k-D ^(J;_i) (A-~l) p,(i~l)
i = 2^ 2 ■ 2*, 3 ■ 2^ . .. , 7V2 - 2^ A; = l, 2, ... , 71 - l .
The decomposition algorithm necessitates performing the
factorization of the operator C^*) of the special structure
(9) C/W = J](C-M,i?), A'; = 2 cos H
B/- lOr
;=r
2^+1
making it possible to reduce the further inversion of the operator C'^^^ to
successive inversion of difference operators by the elimination method. In
what follows one simple equation
C(^-)t;
V
.serves to motivate what is done. With representation (9) in view, the
sought function v — lA" ^ will appear as the outcome of successive solution
of three-point difference equations
(C-/..ii?)t;(^) = ^, {C-ii,E)v^')^v^'-^\ / = 2,3,...,2^

648 Methods for Solving Grid Equations
or, what amounts to the same,
2 vC) - hi v'^l^ - ij, ^(" = v^'-^\ h,<x^<l^-h,,
yi''@) = t'^"(/i) = 0.
The preceding typical equation can be solved by the elimination method
(k)
With regard to three-point equations. In giving F^ the intention is to use
the formula
A0) Ff)=C(^)pf) + qf)
with new vectors py and q^- -^ still subject to (8):
(U) C(^)pf) + qf) = C^'-'^pfJl +p^%ll + qf ~^)]
+ (C(^-))^f~^^q(^;iL+q;^E..
Granting the decomposition
1+2'-'
A2) qr^ = 2p)'^'+q)':;ii,+ q;:
and taking into acount that
we recast equation A1) as
(ci^-ny (pw _ p(^~>)) = c(^~^) {pfsfi. + p^. + qf-^))
with further elimination of C^^~^\ The outcome of this is
with q^-'^ still subject to A2). Substitution of formula A0) into G) yields
C(^-^)[Y,-pf-^)]=qf~^) + Y^^2->+Y,
+ 2*

Direct methods 649
showing the new vectors to be sensible ones. Summarizing, the numerical
solution of problem B) can be done using only two operation: inversion of
the operators C'^^'^^ and summation of relevjint vectors in the process of
calculation of the right-hand sides of these equations. Thus, the
computational procedures include the following steps:
• specifications of the initial values p^ ' and q^ ' so that
• for all ^ = 1, 2,.. . , n — 1 solution of the ecjuation,?
and calculations of the vectors p^. and q^ ' by the recurrence
formulas
(k) _(k-l) ,a(k-^) Jk) - '->r.O') ,Ak-l) ,(k-l)
p)- = pr~- + S- -, qr = 2p- + q-i.+q
3-2"-' ij+2
k~l
J+2"-'
for all i = 2^ 2 ■ 2*^, 3 ■ 2^ ... , //2 - 2*^;
• the solution of the equations
C'(^~^)s|.'~^' = q;.'~^' + Y, ,.-. +Y.
J J J -^ J
Yn = Fo , YiYj = Yn^ ,
for determination of the unknown vectors by the formula
Y^3=pf~^) + Sf~^^
for alli=: 2*^~\3-2^--^5-2*~\... ,//2-2*^~^ fc = n,n-l,... ,1.
During the course of the decomposition method the users will perform
Q — 0{Ni N2 log2 A'^2) arithmetic operations with the extra storage about
1.5A'^, where N is the total number of the unknowns. Some modification of
the preceding algorithm with insignificant prolongations in time may be of
assistance in mastering the last difficulty involved.

650 Methods for Solving Grid Equations
3. The method of separation of variables. The problem we must solve is
problem B) with the homogeneous boundary conditions
A3) Ay=-^, y|^, =0,
where the function ip differs from the right-hand side / of problem B) only
at the near-boundary nodes in the following way: within the quantity —-
h
for 11 = 1, ij — Ni ~-\ and within the quantity —r for i^ = I, i, =^ N-2 — l.
h
2
Before giving further motivations, it will be sensible to introduce the
eigenfunction ji^kiih^) and the eigenvalue \f. with the number k of the
problem
A4) A2/U.J.-F Aj. A«j; = 0 , h^<x^_<U_-h^_, ^i^{0) = ^^{12) = 0 .
We learn from Chapter 2 that
^.^ /2.fc7rj , i . ^ k TT h„ , ,„
,,(,/^,)=^-sm—, A. = ^sm-^, k=l,2,...,N,^l,
and may attempt a solution of problem A3) in the form
N2~l
A5) Vij = Y^ Ck(ih,)iJk{jh^),
k = l
t=l,2,... ,Ni-l, j = l,2,...,N2~l,
where the Fourier coefficient Cj. depends on x^ = ih^.
Upon substituting representation A5) into equation A3) we obtain
A6) Ay = Aiy + A2y
N2~l
k=i
- Y fk{>-K) l^^kijK) :
k = l

Direct methods 651
where ipf.{ih^) is the Pburier coefficient of the function f(x):
N2~l
Due to the problem statement A4) and the orthogonality of the functions
/Uj. we derive from A6) the problem statement for determination of the
numbers Cf. for all fc = 1, 2, .. . , yVa — 1:
A7) KiCk-XkCk^ ~fk, h, <x, <l^~- h^, Cfc(O) = cj^(y = 0.
Because of this form, the applications of the elimination method for
N2 — I times to Cf^(ih^) as a function of the argument x^ = ih^ for fixed k
permit us to find a solution of problem A3) by means of formula A5). As
can readily be observed, the calculations of the Fourier coefficients i^j. and
solutions Uij can be carried out by the same formulas related to common
sums of the special type
N- 1 ^_ ^ _.
"i = Y^ ''^k sin -—, i = 1, 2,...,//- 1 .
k=\
Omitting more details on this point, we refer the readers to the well-
developed algorithm of the fast Fourier transform, in the framework of
which Q arithmetic operations, Q ~ 2iVlog, yV, A' = 2'\ are necessary in
connection with computations of these sums (instead of 0[N') in the case
of the usual summation), thus causing 0{n\N-:i\og^ N2) arithmetic
operations performed in the numerical solution of the Dirichlet problem B) in a
rectangle.
In such matters some progress can be achieved by combinations of
the decomposition method and the method of separation of variables. For
example, this can be done using the method of separation of variables
for the "reduced" system F) upon eliminating the unknown vectors with
odd subscripts j. This trick allows one to solve problem B): liere tire
expenditures of time are Q ss 2nin2^og^ N^ arithmetic operation, half as
much than required before in the method of separation of variables.
4. The method of matrix elimination. The system of equations C) is one
particular case of the following problem;
A8) ~A,Y,.i + QY, ~B,Y, + , =F,, i = 1,2,... , A^ - 1,
6\, Yo - Bo Yi = Fo, -An Yw_ 1 + Cn Yw = Fw ,

652 Methods for Solving Grid Equations
where Yj and Fy are vectors of the same order Mj, C'j is a square matrix
of size Mj X Mj, Aj is a rectangular matrix of size Mj x Mj^i and Bj is a
rectangular matrix of size Mj x My_|_i. As usual, we may attempt a solution
of this problem iu the form
A9) Y,=a^^,Yj + ,+p^^,, j = N-l,N~2,...,l,Q,
where a ■ is a rectangular Mj^i x My-matrix and /?.• is an My_i-dimensional
vector. Following established practice with Gaussian elimination, we derive
from A8)"A9) the recurrence relations for finding a- and /?■ both:
«, + i = iCj ~ Aj a/r'Bj, j=l,2,...,N~l,
"i = C'G Bo,
fij + r = (Q - Aj a^.)""HF; + Aj fij), j = 1,2,. .. ,N - l,N ,
A = C'o Fo,
Yn = {Cn - An a^r)~^(Fw + An Pn) ~ Pn+i >
Y, = a^.+i Y, + i + /3^-+i, J = iV - 1, yV ~ 2,... , 1, 0 .
For a complete and rigorous treatment, it is required that
\\Co'Bo\\<l, \\C],'An\\<1,
||67^A,|| + ||C7^B,||<1, l<i<A^~l,
and, moreover, at least one of these inequalities should be strict. This
provides the sufficient background for the stability of the matrix elimination
method with respect to random errors, meaning || a- || < 1, j = 1, 2, . . . , N.
In the case C), which interests us, the members become Aj = Bj = E,
Cj = C for 1 < i < // - 1 and Bq = An = 0, Co = Cw = E, by means of
which the ensuing formulas can be written in simplified form:
«y+i=(C~a.)-S a, = 0, i=l,2,...,iV2-l,
/3y+i =«^-+i(F, +/3,), A=Fo, i = l,2,...,iV2-l,
Yj = ay+i Yy+i + /3y+i, Yn,=Fn,, j = N2 - I,.. . ,2,1 .

Direct methods 653
Here Yy and /3,- are vectors of order Ni — I, a.- and C are square matrices
of the same size (A^i ~ 1) x (Ni ~ 1). Then tlie unique sufficient condition
of stability is
||C-MI<0.5.
This is certainly true, since C > 2E.
Proper evaluation of the necessary actions in solving problem E) by
the matrix elimination method is stipulated, as usual, by the special
structures of the matrices involved. Because all the matrices »■ are complete in
spite of the fact that C is a tridiagonal matrix, 0{Ni) arithmetic
operations are required for determination of one matrix Oj^i on the basis of »■
all of which are known to us in advance. Thus, it is necessary to perform
0{Nl N2) operations in practical implementations with all the matrices
aj, j = 1,2,... ,N-2. Further, OiN"^) arithmetic operations are required
for determination of one vector ,/?_|_j with knowledge of Pj and 0{N^ N2)
operations for determination of all vectors /?■.
With regard to all the Y^'s, the same number of operations suffice
and so the total volume is still Q = 0{Nf N2).
10.2 TWO-LAYER ITERATION SCHEMES
1. Two-layer iteration schemes. The problem statement. In what follows
it is required to solve a first kind equation of the form
A) Au^f,
where A \ H t-^ H is a linear operator in a finite-dimensional real space H
of the dimension N with an inner product (, ) and associated norm \\y\\ =
\/iy, y). In a common setting it is supposed that A = A* > 0, where f E H
is an arbitrtary vector. From the viewpoint of possible applications, any
iterative method provides proper guidelines for successive determination
of approximate solutions t/i,j/2>--- 12/^ ^ J/t + ii ■ ■ ■ to equation A) with the
.starting point (the initial approximation) y^ G H. Any such approximation.s
are known as iterations with relevant iteration numbers k = 1,2,.... The
essence of these methods is that the value yt^i can be obtained through the
preceding iterations j/j._j, y^.,. ... An iterative method is called a one-step
method or a two-step method if only one or two preceding iterations
are needed in finding every value y/^^i- As we will see later, these methods
fall within the category of two-layer and three-layer methods, respectively,
on the same footing as in Chapter 6. What is more, the one-step iterative
method coincides in form with a two-layer scheme designed m C'hapter 6.

654 Methods for Solving Grid Equations
Before going further, we recall that any linear one-step iterative
method can be written as
B) Bky,+y = Cryk + Fk, fc = 0,1,2,...,
where Bk and Ck are linear operators from the space H into the space
H depending, generally speaking, on the iteration number k, Fk E H is &
known function of A; and t/j. is the ki\i iteration, under the agreement that all
the inverses B^ exist. A natural requirement in the further development
is that the exact solution u to equation A), not depending on k, should
identicall}' satisfy equation B);
{Bk-Ck)u=F,,.
But it is possible only if {Bk — Ck) A~^ f = Fk, implying that
• the inverse operator [Bk — C'k)~^ exists;
. f = AiBk-Ckr'Fk.
This is acceptable if we agree to consider
T;;l,{Bk-Ck)^A, Fk = fT,^„ k = 0,1,2,...,
where r^.,j > 0 is a numerical parameter. Under such an approach the
canonical form of two-layer iteration schemes is
C) ^^yfc+i-y^ +Ay,=f, k = Q,l,2,...,
n+i
where the initial approximation y^ E H is free to be chosen in any
convenient way. Since the inverse B^T' exists, it follows from the foregoing
that
D) yk+i = Vk - T-,+1 B- ^ (A y, ~ /)
or, what amounts to the same,
Vk+i = Vk ~ Tk+i Bk ^k = Vk ~~ Tk+i ''-'^k >
where r^. = Ay^. — f is the residual and W). = B^ r^, is the correction.
With knowledge of i/j. the value of t/j._|_j can be recovered from equation
D). Knowing y^, it is plain to determine successively j/j ,y,^ .. .. Of course,
it is meaningful only for convergent iterative methods:
E) l|yfc-«ll—^0 as /~*oo.

Two-layer iteration schemes 655
The usual practice involves the numerical solution of problem E) with
a prescribed accuracy e > 0 (a relative accuracy ||i/j. — «||/||i/o — w||), it
being understood that the calculations should be terminated if
F) llyt-"«ll < £||yo-m||-
In connection with inconvenience caused by the unknown vector u, it seems
reasonable to replace this condition by the inecjuality for the residual
G) \\Ay,-f\\<e\\Ay,-f\\.
In the general case the accepted view is the termination condition of
the type
(8) \\yk-u\\^<e \\y,-u\\^,
where D = D* > 0 is some operator. By merely setting D = A'^ we deduce
from (8) inequality G).
We are now interested in the governing equation related to the residual
Zj, = yi. — u. Since Au = /, we might have
(9) B,'''+'~'' +Az,=.Q, A: = 0,1,2,...,
'''k + l
where z^ £ H is known. As far as Bk = B is independent of the subscript
k, the correction iVj,, = B'^r^. satisfies the homogeneous equation
B -^^^ + A w^. = 0 .
Indeed, D) implies that
Vk+i -Vk = -n+i B~^ rk = -n+i iVk ■
Applying the operator A twice to both sides of the preceding equality and
taking into account that
Ay^+i - Ayj, = (Ay^ + i ~ f) - [Ay^ ~ f) =^ rj,_^i - r^,
rk+i -rk = B{B-^rk^i - B'^k) = ^ (^^fc+i - Wi) ,
we establish the homogeneous equation for the correction W/..

656 Methods for Solving Grid Equations
Also, it seems clear from (9) that
zj,j^i= Sk + yZf., Sk + i = E - Tk+iB^ A ,
where St + i is the transition operator from the layer k to the layer ^ + 1.
Having completed the elimination of Zf., Zj.^^, ... , Zj, we find for k = n — I
that
where T„ is the resolving operator of scheme (9). This serves to motivate
the estimates
W^nWo = WT^'n-oWo < ll^"L ■ \\~o\\d oi'
From such reasoning it seems clear that the condition of termination is
ensured if q„ < e, thereby reducing the question of convergence of the
iterations to the norm estimation of the resolving operator T„.
Scheme C) generates an exact approximation on a solution u of the
equation Au = f for any operators {-Bn} and any choice of the parameters
{Tk^i}, but the quantity g„ depends on {B„} and {rj,_|_|} both. Some
consensus of opinion here is that {Bn} and {rj._|_j} should be so chosen as
to minimize the norm ll^^nll^, = Qn of the resolving operator T„ of scheme
C) and to minimize the total number of arithmetic operations which will
be needed for recovering the value j/j._|_i from the equation
Bk Vk+y = Fk , Fk = Bk iJk - Tfc + i [A y^ - /) ,
with a known value j/j._|_i.
In accordance with what has been said above, any iteration scheme
C) can be treated as a two-layer .scheme being used for solving the nonsta-
tionary problem
«^ + - = /-
where the parameter Tj,,! regards to one possible step in a nonreal time
^k+i — '}Zm=i'''m- The main differences between iteration schemes and
available schemes for nonstationary problems are:
• the iteration scheme C) approximates exactly equation A), since
a solution u to equation A) satisfies equation C) for any Bk and
'''k + l')

Two-layer iteration schemes 657
• proper choices of the parameter rj-.j and the operators Bk are
caused only by the necessity of convergence of the iterations and the
economy requirements in trying to solve the original problem with
a prescribed accuracy, while the restrictions on the steps for non-
stationary problems are connected with the approximations which
do arise in such matters.
Let now Q{€) be the total number of arithmetic operations necessary
for obtaining a solution to equation A) with a prescribed accuracy e > 0
regardless of the initial approximation in the iteration scheme C). Its
ingredients Bk and T/. should be so chosen as to minimize the quantity
E(e). If the desirable accuracy can be attained in a minimal number of the
iterations n = n{e), then
n(£)
Qis) = J2 Qk = Qnn,
k = l
where Q^ is the number of the necessary actions during the course of kth
iteration. Thus, the minimum problem for Q{e) reduces to the minimum
problems for n[e) and the number Qk, which depends on Bk.
In this context, if Bk = E is the identity operator, then scheme C)
refers to explicit iteration schemes of the structure
A0) yiS+l^ll + Ayk = f , -fe = 0,1,2,..., for any y, en.
If Bk ^ E, then scheme C) is termed an implicit iteration scheme.
2. A stationary scheme. The main theorem on the convergence of
iterations. Quite often, the iteration schemes such as
C') B?^i+l^i^ + Ay, =/, ^ = 0,1,...,
r
with a constant operator B and constant r are called stationary
methods of iterations. In particular, the upper relaxation method and Seidel
method fall within the category of such methods. In that case equation (9)
related to the error of approximation Zj. = i/j. — u takes the form
(9') 5!t±l^i^ + Az, = 0, ^ = 0,1,..., z, = y,-u,
T
and it remains valid for the correction Wk = B~^{Ayk — f). The operator B
is, generally speaking, non-self-adjoint and possesses the own inverse B~^.
This type of situation is covered by the following assertion.

658 Methods for Solving Grid Equations
Theorem 1 If A is a self-adjomi operator [A = A* > 0), then
A1) B>-rTA or iBx,x) > -T {Ax,x) for all xeH
is a sufEcient condition for the convergence of the method of iterations C')
in the space Ha with the rate of convergence of a geometric progression
A2) IU.+,IU<p|Ui.lU, k = o,i,..., p<i,
where p = A - 2t 6^,6/\\ B \\^)^f'^ is its denominator and 6 = mini Aj.(A),
6^ = minfc Aj.(_Bo ~ tA/^), Bq = (B + B*)/2 is the symmetric part of the
operator B.
Proof Knowing from C') Zi._^_^ = Sz-^. with the operator S = E — tB~^A,
we find that
W^k+i Wl = {Az,^_^_i,Zk+i) = {ASz^,Szi^)
= {A{E - tB-'A)z,, (E - tB-'A)z,))
= II z, \\l - r [iAB-'Az,,z,) + (B-'Az^Az,)]
+ t\AB-'Az„B-'Az,).
With the relation A = A* in view, we deduce upon substituting here Az). =
—Bvj. and Vj. = -~B~^Az^., where V). = - {zk-\-i — ^/t), that
A3) II ^,+, 11^ = II z, ||2 _ 2 r {(B - r A/2) v,,v,) .
Because of A1), by utilizing the fact that the operator P = B — tAJ^ is
positive we establish its positive definiteness in a finite-dimensional space
H (for more detail see Chapter 2, Section 1):
A1') B- -tA>6^E, <5, > 0,
where (^„ is the smallest eigenvalue of the operator Pq = Bq — -^ tA, so that
A3') 2Ti(B-^TA)v,,v,)>2TS,\\v,\\\

Two-layer iteration schemes 659
On the other hand, it follows from the foregoing that
\\-^t\\l = {Az,,Zk) = {Bv,,A-'Bv,)
<\\A-'\\ ■WBv.W'
<\\A-'\\-\\Bf -Wv.f
12
Vk
yielding
A4) \\vA?>wirj^\\-'^k
|2
By inserting A3') and A4) in A3), it is possible to show that the bilaterial
estimate
becomes valid with p^ =: 1 — 2t66^/\\ -B|p < 1, assuring estimate A2) and
the inequality || £-„ ||^ < p" \\ z^ ||^ and justifying the convergence of the
iterations, since p" —- 0 as n -^ cx>. The same estimate is certainly true
with the correction w^ = _B^^(.4j/„ — /),
Remark Condition A1) for fixed B may be viewed as a selection rule for
those values of r for which the iterations converge. For example, for the
explicit scheme with the identity operator B = E condition A1') is ensured
if all the eigenvalues are subject to the relation
or, what amounts to the same,
l~\r\\A\\>Q.
Thus, the iterations converge for any r < 2/||.4||. Let us stress here that
the estimate obtained for p is too rough for determination of the total
number n{e, N) of the necessary iterations and indicates mainly the true
order in n as N —^ oo.

660 Methods for Solving Grid Equations
3. The explicit scheme with optimal set of Chebyshev's parameters. In
what follows the intention is to use the explicit scheme A0) without concern
for how the parameters Tj , r2, . .. , r„ will be chosen in trying to minimize
the total number of iterations n = n{s). Also, under the agreement that the
operator A is self-adjoint and positive we operate with its smallest 7j > 0
and greatest 7^ eigenvalues:
A5) A = A*>{}, 7iE<A<j,E, % > 0.
The meaning of this is that we should have
7i II ^ 11^ ^ i^x, x) < 72 II 2; 11^ for any x £ H .
If the parameter r = const is independent of the subscript k, that is,
Tj = r, =:■•• = r,j =: r, scheme A0) is called the simple iteration scheme:
A6) yk+i^yk~T{Ay^.~ f).
In Section 1 of the present chapter we have established that the residual
Vf. = Ay). ~ f satisfies the homogeneous equation
A7) !1+1ZL!1 + Ar, = 0, k = Q,l,2,..., r, = Ay„ ~ f e H ,
'''k + l
or rj._|_| = Sk+iri;, Sk+i = E ~ Tf-^iA, showing a way of relating r„ and r^:
A8) r„ = T„ra , t„ = Si S2 ■• • 5„ .
Here T„ is the resolving operator being a polynomial of degree n with
respect to the operator A:
A9) T„ = V„iA) = {E^ T,A) {E - T,A) ■ ■ ■ {E ~ t„A) ,
so that r„ = ■p„(j4)rg. On this basis the residual r,-, obeys the estimate
B0) lknll<l|7^n(A)|| -llrJI^gJIrJI.
The next step is to evaluate the quantity || Vn{A) \\ of interest in terms
of 7j and 7,, making it possible to extract those parameters Tj , Tj, .. . , r,-,,
for which the minimal value of g„ = ||'P„(A) || is attained. The preceding
polynomial
n n
P4A) = J] (i? - r„A) = ^ c, A^ Co = l, P„@) = 1,
m = l k-0

Two-layer itei-ation schemes 661
refers to self-adjoint operators, since any degree of the operator A also is a
self-adjoint operator: A™ = (A'")*.
Let {A,,.J,} be eigenvalues and orthonormal eigenfunctions of the
operator A:
A^.^X.^s, s= 1,2,... ,iV, 0 < Aj < A2 < ■■• < A^,
where N is the dimension of the space H and Aj = minj A^ = 7i, A^, =
maxj Aj =72. By definition,
meaning that Aj' is one of the eigenvalues of the operator A''. Thi.s serves
as a basis for the representations
n / ^ \
'PniA)^, = ^ C, A' e» = E 'k A,' C = Vn{K)^s
k=0 \k=0 J
and, on the same grounds, X{VniA)) = P„(A(A)).
Thus, the eigenvalues of the polynomial 'Vn[A) are equal to the
polynomial ■pra(A) of the eigenvalues A = X[A) of the operator A. With the
relation {Vn{A))* = Vn[A) established, we find that
B1) \\Vn{A)\\< max \Vn{x)\ .
Because of this, the problem of searching for min ||'Pn(^)|| can be
reduced to the well-known minimax problem for the polynomial Vn{x) in
question. By interchanging the variables by the rule
B2) :c = 0.b[{j,-j,)t + j,+j,]
the segment [7^,72] carries into the segment [—1, 1], so that V„{x) = Vni^),
t G [-I,l],and-P„@)=: 1.
With the detailed forms in mind, a revised statement of the problem
consists of finding a polynomial with minimal deviations from zero on the
segment [—1,1] such that max |'Pn@| is minimal under the additional
condition of normahzation Vn(fo) = 1, where the point t^ corresponds to
the point x = 0. From formula B2) it follows for x = Q that
B3) i^ = _2i±2i.
T2 - 7i

662 Methods for Solving Gi-id Equations
Thus, the well-known Chebyshev polynomial defined by
-p (t) ~ -^"^^^
where
B4) Tn{t) = cos {n arccos t) for \t\ < 1,
is just the solution of the original problem concerned, For |i| > 1 the
polynomial of interest is specified by the formula
B5) T„{t) = 0.b
[t + Vf' - 1)" + (t - Vf' - 1)"
> 1^1 >1
Since max |T„(i)| = 1, the relations occur:
B6) min max |-p„(a;)| = min max \Tnit)\ = .„ ,. x, = gn ■
1tj.}7i<-t<72 lTj;}~i<*<i Mn(to)|
In an attempt to find the unknown parameters Tj , Tj , . . . , r,, by the
approved rule saying that the zeroes of the sought polynomial Vnii) should
coincide with known zeroes of Chebyshev's polynomial such as
2-fe - 1 , ,
B7) i,. = cos TT, k = 1,2,... ,n ,
2n
we recall from calculus that the polynomial
Vnix) = A - TiX){l - T^X) •••A - T,^X)
has zeroes at the points x^. = 1/rj., k = 1,2,.,, ,n. By formula B2),
relating x and t, we deduce that
giving
n
k= 1,2,.,
(Ti +72 + G2 '-7i)h) '
Also, it will be sensible to introduce more compact notations
72 1 + 't 1 + V", 7i + 72

Two-layer iteration schemes
663
All this enables us to write down
B9)
Tk
^+Potk'
k= 1,2,
thereby completing the task of motivating the choice of the parameters
Tj, r2,. . . , r„. The expression for q^ — l/|T„(io)|, ig = —l/p^, is needed as
further developments occur. Since |i|o > 1, applying formula B5) to T„{t^)
yields
\Tn{to)\ =
1
Po
'1
Pi
1
+
1
Po
Pi
-1
By minor change.s the expre.ssions in various bracket.s are modified into
1
Po
+
pI
1
1 +
4e
1
Po
and, hence,
C0)
Po Po^^"V(^+^)'
l+e + 2v^^ 1 + v^^ 1
Po^
i-e
l + v^
<in
l+pf
Thus, for .scheme A4) with optimal set of the parameters t^,T2,
given by formula B9) the relation
C1)
Ayn~f\\<in\\Ay,~f\
i.s valid with q„ still subject to C0).
The number n = n{e) is so chosen as to .satisfy the condition
In
_2p^
1+P?^
< £.
For this, it suffices to require in the further derivation that p" < e/2 or,
what amounts to the same, that
C2)
n >
lnB£-)
In(l/Pj

664 Methods for Solving Grid Equations
The well-established expansion of the function
where 0 < a; < a;, is aimed at establishing the relations
ln|^>2.., lnl = lni+4>2v^
and, hence, inequality C2) holds true for
C3) n>n,{e), „„(£) = i^^l^ = i ,/^ In ^
2 v^ 2 V 7i £ ■
This estimate is more convenient in practical implementations than
estimate C2).
4. The simple iteration scheme. By formally setting n = 1 in formula B9)
the preceding is referred to as the simple iteration method
C4) Mi:U^ + A,,=./
To
with the parameter Tq incorporated:
C4') r„ = -^~ .
7i +72
Here t^ = cos | = 0, Tj = Tq and
2pj
C5) q, = "—^ = Po ■
The equation for the residual 1\ = Ay^. ~ f reduces to yj._|_j =: S't/^.,
5 = E~TqA. As far as the operator Ti = 5 is concerned, formulas B0) and
C5) together imply the estimate for the norm of the transition operator
\\S\\ = Po = \i^.
By making n iterations of the simple iteration method we find that
„ _ on II II ^ n II II
' n — '-' ' 0 ' I r 71 11 _ Fo I r 0 11 •

Two-layer iteration schemes 665
The condition p'^ < e is ensured if
ln(l/£)
n >
ln(l/Po)'
which is certainly true for
C6) «>"o(£), noie)
■2^
5. A model problem. Comparison of methods. Further comparison of
various iterative methods will be conducted by having recourse to the Dirichlet
problem associated with Poisson's equation in the square {0 < Xj < 1,
0 < x^ < 1} of the unit sides l^ = l^ = I and posed on a square grid cj^
with steps /Zj = h^ — h. As a special case of problem B) in Section 2, the
problem of interest is characterized by the grid equations
C7) hy = hiy + h2y = -f{x), xeLu,,, 2/l7^=0,
where
Aa2/= %,,.„ = ^^ , a =1,2.
The system of equations C7) can be recast in the operator form as
0
Ay = f, where Ay = —Ay in the space H = Q of all grid functions given
on the grid cj^j and vanishing on its boundary 7^^. An inner product and
associated norm in that space H are defined by
iy,i>)= J2 yix)v{x)h'^, \\y\\ = \/iij,y) ■
We spoke about the operator A a lot of time in Chapter 4: it is self-
adjoint, positive and possesses the eigenvalues
4 / . , TT jfe, /l . 2 1" (^2 ^ \
^k, k, = T^\ sm- -^— + sm -^— ,
that
C8)
k^ = l,2,...,N~l, a=l,2,
8 . ,Trh
7i =min A^.,i^ = — sur—
8 2 ""/^
72 = max X^.^ k._ = t^ cos —-

666 Methods for Solving Grid Equations
In what follows problem C7) will be treated as a model one in the
further comparison of various methods in a step-by-step fashion in line with
established priorities and answering real needs. We concentrate primaril}'
on the total number of the iterations required in the simple iteration method
C4)-C4') and the method with optimal set of Chebyshev's parameters A4),
B9).
Recall that it is fairly common to write the iteration number k over
the sought function y within the frameworks of iterative methods available
for difference equations. The same procedure works in the simple iteration
scheme (SIS) which has been designed for problem C7):
C9) 'V =y+ To{Ay+.f),
where
2 /r
7i + 72 4
Upon substituting the assigned value into C9) we derive the formula
fC h> fC K ^-^
4 4
I 22 '
by means of which the [k + l)th iteration is completed.
For the explicit scheme A4) with Chebyshev's parameters (SCP) the
calculations are performed by the formula
k + l _k T^ + i
y I'l 82 Ki 12 """
/l2
/k k k k a'^ \
[yii-l,i2 +%'i+l,i2 +%i,J2-l +^ii,i2 + l "^J/jji^j
where
+ "'"S' + l/ji i-j
'''k + l
h'' 1
^+Poh 'i ^+ Poh
Since the volumes of computations in determining y in these methods
differ slightly, the main criterion in such matters is the total number of the
iterations. By formula C8) we find for /i <C 1 that
7, .-,% h TT^ /}2
,^ = -- = tg — « -^ .
For example, the reasonable accuracy e = 2e~''^ Ki lO^'' is attained for
^^(e) f« ^ in the case of SCP and for no(s) ~ rr in the case of SIS,
making it possible to fill in the following table regardless of the initial
approximation, that is, for any element y^ of the space H.

Two-layer iteration schemes
667
h
1/10
1/50
1/100
SIS
200
5000
20000
SCP
32
160
320
From here it seems clear that much more iterations are needed in SIS than
in SCP which is, generally speaking, preferable.
However, some progress may be achieved in reducing the total number
of operations by making the well-founded choice of the initial
approximation.
6. On computational stability of iterative methods. Until recent years the
iterative method with optimal set of Chebyshev's parameters was of little
use in numerical solution of grid equations. This can be explained by real
facts that various sequences turn out to be nonequivalent in computational
procedures.
At the initial stage such of such an analysis of algorithms it is
usually supposed that a computing process is ideal, that is, computations are
carried out with an infinite number of significant digits. But any computer
makes calculations with a finite speed and a finite number of digits. Not all
numbers are accessible to computers, there are computer null and computer
infinity. For instance, abnormal termination occurs when computer infinity
arises during the course of execution. A computing process may become
unstable, thus causing difficulties. In such cases rounding errors may
accumulate to a considerable extent so that the algorithm will be useless in
practical applications
For example, for doing so with the set of Chebyshev's parameters tf.
in increasing order
D1)
t.
2k-\
2n
k
1,2,,
or in reverse order
D2) tk
2k- 1
■ cos TT ,
2n
K i , .i , . . . , ^^ ,
abnormal terminations may occur for sufficiently small <J in connection with
growing intermediate values yj. for k < n. Such a danger is caused by a
nonmonotone character of the approximation j/j, to u, since the norm of the

668
Methods for Solving Grid Equations
operator Sk — E — Tj.A associated with the transition from the [k — l)th
iteration to the kih iteration may be greater than 1 for negative values of
Before proceeding to further discussions, let us give the details of a
simple example.
Example The system of equations to be solved is
v[xi_y)-2v{Xi) + v{Xij^^)^Q, x^
1,@) = 1, t;(l) = 0,
relating to the problem
w" = 0 , 0 < a; < 1 , w@) =
for which the exact solutions are known:
u{x) = 1 — a;, ij{Xi)
In that case the ingredients become
= ih, l<i <N
h = l/N ,
1
«A)=0:
Ay
-Vs
7i
4 . T TT /}.
-—■ sm'
/i2 2
TT h
72
cos
r,wh
'"■' " /i2 2 ' " /i2 """ 2
For convenience in analysis and clarity, we take into consideration 19
equations (N=20) and set e = 10"''. The analytical estimate C3) gives ng{e) =
63.2, so that n = 64 and the parameters
' 1 ! '2 ! ■
, T„ for n = 64 are
specified by formulas B9) and D1). The final results of computational
procedures are presented below in Table 3.
Table 3
Table 4
53
54
55
56
57
58
59
60
61
62
63
Ai
0.12
1.5
27
6.3-102
1.9
7.2
3.7
2.6
2.5
3,3
5.0
10^
10^
10^
109
10"
1013
1015
k
1
2
3
4
5
6
7
8
9
10
11
Aj:
39.6
2.6-103
1.6
8,2
3,7
1,2
3.3
7,0
1.2
1.7
1,9
10^
10"^
10^
101°
10"
1012
1014
1015
1016

Two-layer iteration schemes 669
The iteration number k is recorded along the first column and the
quantity of interest
Afc = WVk - Vk-iWc = „?i^5, IVki^i) - yk-i{Xi)\
is placed along the second one, From here it is easily seen that the iteration
process is divergent, thus causing abnormal termination for ^=64.
By obvious rearranging of the parameters Tj, in reverse order in
conformity with formula. D2) the inherent instability of this process is more
significant due to the fact that abnormal termination occurs very fastly for
k=12 (see Table 4).
In this regard, rounding errors can be treated as possible perturbations
of the right-hand side of equation A) at every step, The iteration scheme
A4) with parameters B9), D1) or D2) becomes unstable with respect to
the right-hand side by exactly the same reasoning as before; the norm of
the operator Sk = E — Tf.A for the transition from the [k — l)th iteration
to the ^'th iteration may exceed 1 for negative values of ij., since
\\Sk\\ = -. —TT-r for ifc<0,
\\Sk\\>\ for p,{\ + 2\t,\)>\.
By the same token,
11^^-11 ^ 1 ,, , < 1 for ij.>0,
^ -r Po T-k
The preceding formulas need certain clarification. Since Sk = S^., we obtain
||S'i: II = sup \(SkX,x)\ .
II ,-; 11 = 1
By virtue of the relations JiE < A < j^E we find that
{TkJ,-l)E<T,A-E<{T,j,-l)E,
which are followed by
1 + Po ^fc 1 + Po ^*

670 Methods for Solving Grid Equations
upon substituting
'^k ^ 1 , ° , . T-o7i = 1-^0 and ToJ^ = I + p^ .
i + Po '■k
From such reasoning it seems clear that || Sk \\ = '''k72 ~ 1 for ^k < 0 and
II ■S'j; II = 1 — r^.7j for tj. > 0. If ij. < 0 and k ^ kg, where kg is the minimal
number for which tj.^ < 0 and pg{l + 2\ti;J) > 1, then
||5fc+i||>||5', ||>1.
This provides enough reason to conclude that
n
n II 5; II > II 5^-011""'" >1
and rounding errors in specifying j/j. will grow with increasing k from kg
to n.
Let^ = ^ <C l,sothat/>o = 1-2^ + 0(^2) and ij.^ = cos ^4t^ ^^ < 0-
72
All this enables us to deduce that
Under the assumptions \tj. \ > 0.5 and <J < 0.01, we might have
l|S,.,|| = 3(l-60 + O«")>.3 . 0.9 = 2.7,
n
j=ki
5; II > 2.7"
As a matter of fact, formulas C0)-C1) expres.s the stability of scheme
A4) with parameters B9) with respect to the initial data. Those ideas
supported by the preceding calculations provide proper guidelines for the
further stability analysis of computational procedures with respect to the
right-hand side as well as with respect to the initial data in passing from
t/g to j/j, for any k = 1,2,... ,n. From the general stability theory outlined
in Chapter 6, Section 1 we recall that stability with respect to the right-
hand side is a corollary to the uniform stability with respect to the initial
data, meaning stability in the process of moving from any ?/,■ to any j/j, with
k > j > 0.

Two-layer iteration schemes 671
7. Re-ordering of iteration parameters. With regard to scheme A4) one
interesting problem arises in connection with re-ordering of the iteration
parameters {tj.} so as to minimize as much as possible the influence of
rounding errors and to avoid large intermediate values dependent on n.
The main goal of subsequent considerations is to constitute a "stable
collection" M„ of parameters Tj , r,, .. . , r„ for which scheme A4) becomes
stable and then show the way this result is used in practical
implementations. We improve our chances of ordering the set
M„- \^-cos/3i, /?. = —6'„(i), i=l,2,...,n|
if a sequence of odd integers will be available such as
On = {^n(l), ^nB), ■ ■ ■ , ^n(«)} , 1 < ^n@ < 2n - 1 ,
for i = 1,2,... ,n.
With the aid of its members the parameters {r^} are calculated by the
formula
D3) r, = —^ , a, = -cos\^0„{k)\, k=l,2,...,n.
l + PoO-k '■2n J
Thus, the further composition of the set consisting of n numbers 6'„ may
be of help in achieving these aims.
The traditional tool for carrying out this work is step-by-step
transitions from the sets 61^ to the sets 62^ and from the sets 62^ to the sets
^2m+i ■ '^he intention in this direction is to use the formulas
D4) 02™Bi - 1) = e^{i), 02mBO = 4m - d,^{2i - 1) ,
i = 1,2,... ,m,
or, what amounts to the same,
D5) d2^{2i-l) = d^{i), 02mBi) = 4m + 2-02mBi-l),
i = 1,2,... ,m
for the first operation in passing from the sets 6',„ to the sets 6'2„, with
placing the extra member 6'2,„_|_|B??? -|- 1) for the second one in passing
from the sets 62^ to the sets 02m+i '■
D6) ^2m+l@ = ^2mW, * = 1, 2, . . . , 2m ,
^2m+iBm + l) = 2m+l.

672 Methods for Solving Grid Equations
When n = 2P, where p > 0 is an integer, 9„ is found by successively applying
formula D4) for m = 1,2,... , 2^"' with the starting point 9^ = {!}.
Before taking up the general case, we give below the final re.sult of
this procedure for n = 16 = 2''
9, = {1,3},
9, = {1,7,3,5},
9g = {1,15,7,9,3,13,5,11} ,
0jg = {1,31,15,17,7,25,9,23,3,29,13,19,5,27,11,21}.
When moving from 9^ to 6'2„, the member ^2mB'') ~ 4m — 9^{i) stands
on the right from the member 9^{i) by the approved rule D4).
Among other things, we may attempt an arbitrary positive integer
n > 0 in the form
n = 2*^' +2*= + h2*-'',
where i > 0 is an integer, k^ > are integers such that k^ > i,_|_| + 1,
i = 1,2,. . . ,t — 1, and k^ > 0, constitute a new sequence of odd integers
D7) n^. = X: 2*^'"'^ i = l,2,...,i!, n, = I ,
8 = 1
and then set '«(_|_i = 2?j+l, which formally corresponds to the value ij_|_j =
— 1. It follows from the foregoing that
4 ^ '
In terms of how "j+i may be affected, some of the possibilities of interest
are:
1) ?"■ = k; — kj_^_l > 2 , that is, n,'_|_i = 2''^^^ + 1 ,
2) Tj = kj — kjj^y < 2 , that is, r^ = 1, n^'_|_i = 2 n^ + 1 .
The algorithm of obtaining the ordered set 9„ is mostly ba.sed on the
approved rules for the transition from the sets 6'„ to the sets 6'„ . Having
formed the set 9„ , the differences r,- are sought in two ways:
A) if '',■ > 2, then the chain of the available sets
^2nj ^^ -11, -^ ■ ■ ■ ^ ^2'^'n,

Two-layer iteration schemes 673
does follow D4) in conformity with the values m = n-,2n •,...,
'2^'~^nj. Since "j + i = '^^'^j + 1, the set 9,-^ is described by
formula D6) under the condition 2m = 2''j «,■;
B) if ?'■ = 1, that is, n- = (wi+i — l)/2, then the set 6'2„ is given by
formulas D5) and the set d„ = 02^ ■, j is in line with formulas
D6). After that, we are moving from 6„ to 6„ , etc.
Starting from j — I with n^ — I and 6^ = 6'j = {!}, we proceed to
calculation,? by a numljer of different formula,s: if i^ = 0, then n =: n^ is
an odd number and the computation.? must terminate for j = t — i upon
receipt of formulas D6); if k^ > 0, then n is an even number such that
n — 2*'?'jj and n(_|_i = 2n + 1. Since i^, j = —1, we employ r.( > 2 and
formulas D4) in passing from 6'„ to 6'„ for the values
m = ?i(, 2n(, . . . , 2 '^'■Mj = i n .
We will elaborate on this later for several particular cases. In
conclusion it should be noted that formulas D4) must be used 2''-'"^ times as
opposed to a single application of formulas D5) with the accompanying
formula D6). Thus, the algorithm is completely described.
Example 1 Let n = 90, n = 2^ + 2^^ + 2^ + 2\ that is, k^ = 6, k^ - 4,
k^ = 3, ^4 = 1; Mj = \, n^ = 5, n^ — 11, n^ — 45; i\ = 2, r^ = 1, fg = 2.
The chain of describing sets is designated by the symbols
leading to 6'„ = 6^^. Here 6^^^ for m = 5 refers to the set specified by
formulas D5) in contrast to the others O^jn- The transition to the next
member of that chain can be done using formulas D4), D5) or D6) as
suggested before.
Example 2 Let n = 25, n = 2'^ + 2^ + 2°, that is, k^ = 4, k^ = 3, ^'3 = 0;
■rtj = Ij n^ = 3, n^ = 25; Tj — 1, r^ = 3. The sets
constitute a perfect chain for the purposes of the present section.
Summarizing, the preceding algorithm is showing a way of obtaining
the ordered set of n zeroes of Chebyshev's polynomial Tn(t) and a stable

674 Methods for Solving Grid Equations
collection in a certain sense of the parameters TjjTj,... , r,^ specified by
formula D3). In what follows the set so formed is denoted by
M: = LcoJ^e*Jt)\ i=l,2,...,n\.
Scheme A4) with new parameters
^KW], k=l,2,...,n,
■K
l + />o<T*' « \2n
possesses the property of computational stability. Assuming that the effect
of rounding errors is equivalent to possible perturbations of input data; the
initial approximation, the right-hand side and the operator A involved in
the iteration scheme A4). Under such an approach a numerical solution yj.
of problem A4) can be treated as the exact solution of the problem
Vk+i ~ Vk , J ~ 7 , 1- , nio
h Aj/fr = tk+i + '-^t+i, k = 0,1,2,... ,n.
'''k + l '''k + 1
In this context, it should be noted that the member lj^+i covers the error
of Vk-
'''k + l
When possible perturbations of the operator A are neglected for one
reason or other, the intermediate solutions ■(/,„ turn out to be bounded in
norm:
— ,m.ax ||/i||+ 7 max \\ui\\ ,
^i l<2<m ^ l<i<m I
where c,„ = 1 if m 7^ 2''; c„j = 0 if ni = 2^ and the error z„ = y^ — u of n
iterations satisfies the estimate
1 - g ~ 4
II ^n - w II < ?n II % - u II + ^ max II /; - / II + —^ max \\di^ \\ .
7l l<J<n i* Y4 i<!<n
The derivation of these estimates espressing computational stability of
iterative methods with optimal sets of Chebyshev's parameters {r^} is omitted
in the present book. In the sequel we involve only the collection {t^},
allowing a simpler writing of the ensuing formulcis without concern of symbols
The results of calculations for the example from Section 5 through
the use of the explicit scheme A4) with ordered sets of parameters {t^} are
presented in the Table 5.

Two-layer iteration schemes
675
Table 5
k
Afc
k
Afc
k
Afc
k
Afc
k
Afc
1
39,6
10
3,2
26
0,1
50
6,7. 10-3
61
1.1 ■ lO"''
4
4.7
16
0,2
32
0,04
56
2,2- lO"''
62
8.7-10-5
5
7.4
17
3.1
33
0,3
57
1,5' 10-3
6
3.2
18
1.5
34
0,14
58
7,2 ■lO-''
8
1.1
24
0,1
48
1,5'10-3
59
6,5'10-^
9
6,7
25
0,8
49
1,3' 10-2
60
2,1 ■ lO-''
In that case we are looking for a solution to the system of equations
X; = ih ,
i= i,2,... ,N -l
h = l/N, 'y@) = l> f(l) = 0,
under the following conditions: A^ — 1 = 19, £ = lO"'', n =; 64 = 2*^. The
chain in question
is constituted in agreement with formula D4) and it reveals a nonmonotone
character of convergence of the iterations. True, it is to be shown that
during the transition from the kth iteration {k = 4j) to the {k + l)th
iteration {k + I = Aj + 1) the error of approximation Aj. = ||j/j, — j/j;_i|||^
is being increased and, after this, is held down in passing to k = Aj + 2,
4i + 3,4i+4,

676 Methods for Solving Grid Equations
10.3 THE AITERNATIVE-TRIANGUIAR METHOD
1. Seidel method. As we have mentioned above, implicit schemes are rather
stable in comparison with explicit ones, Seidel method, being the simplest
implicit iterative one, is considered first. The object of investigation here
is the system of linear algebraic equations
A) Au^f
or
N
Z a,yu^=f,, i=l,2,...,N,
i = i
with nonzero diagonal elements a- ^ 0. When this is the case, the iterative
method ascribed to Seidel amounts to
i N
B) Yl % Vj + Yl "y Vj "^ J'i' "»■ ^ ° '
where y. is the kth iteration. Furthermore, .starting from i = I, the
k
y^, , ,
(k + l)th iteration is to be formed in line with
N
"ii ^i' + Yl "i; Vj = fi ' «u 7^ 0 .
;=2
giving t/j . At the next stage, knowing y^ and setting i — 2, we find y.^
from
N
«2i J/i + «22 ?/2 + 2_y ; Vj = h ' 2 t" U .
;=3
The matrix A rearranges itself as a sum
C) A = A-+A+ + D,
where A~ is a lower triangle matrix with zero elements on the main
diagonal: A~ = {a~.), ar. = a^ for j < i, ar. = 0 for j > i; A+ is an upper
triangle matrix with zero elements on the main diagonal: yl+ ~ (fl-)>
at. = ay for j > i, at. = 0 for j < i, and, finally, D is a diagonal matrix:
D = {ciiiSij)- Here 6^j stands, as usual, for Kronecker's delta: 6^; = 1 for
j = i and 6^; = 0 for j j^ i.

The alternative-triana:ular method 677
Within these notations, Seidel metliod can be written as follows:
D) [A-+ D)^'i/ + A+y :^ f , i/= (j/i,t/2,... ,;(/w)-
Alternative forms of such a two-layer scheme are
(A-+D)(T -l)+{A- + D+A+)l^f
and
E) {A-+D)Ct/ ^y)+Al^!.
Further identification with the preceding canonical form reveals
k + \ k
F) B '^ ~'^ +Ay =f, k = 0,l,... ,n-l, y&H,
'''k + l
with B = A~ + D and Tj, = 1. This scheme is certainly implicit. The
matrix B concerned is triangular and, hence, it is not symmetric, because
the operator i.s non-.self-adjoint: B j^ B*.
On the basis on the model problem
Ay--'fi, X ^uif^, y\^^^ = 0,
arising from Section 2, Seidel method acquires the form
k + l k + [ fc + 1 k k .,
SO that
fc + l k + \ k k r,
\ ) y ^
In such a setting it seems reasonable to begin operations at the node
k4-l k4-1
ij = Ij i, = 1 in terms of Icnown values t/, _j and y i-,_i and tlie right-
hand side of G) at the boundary nodes @, 1) and A,0). With knowledge
k-\-l k-\-l
of y at the node ij = 1, i^ = 1 the values y are yet to be determined
along the lower row by setting ij = 2, 3,... for ij = 1 with further transition
to the rows with ij = 2,3,.... During the course of Seidel method it is
k + \
possible to get the values y at all grid nodes.

678 Methods for Solving Grid Equations
On account of the basic tlieorem proved in Section 1 of the present
chapter Seidel method converges if the operator A is self-adjoint and
positive. IVlore specifically, the sufficient stability condition A1) for the
convergence of iterations in scheme C') with a non-self-adjoint operator B takes
the form
(8) 5o - ^ A > 0 .
Within the framework of Seidel method we thus have B = A~ -|- -D, r = 1
and
5o = I \{A- + D) + [A- + DY\ = i (A- + A+ + 2 D),
since yl+ = (^~)* and D* = D > 0 in light of the properties of the operator
A: A — A* > 'd. With this in mind, condition (8) becomes
5o-~A=5o-^A>0=^D>0,
thereby justifying that Seidel method converges with the rate of a geometric
progression.
2. The upper relaxation method. In order to accelerate the iteration
process in view, we are forced to revise Seidel method by inserting in E)
the iteration parameter lo so that
(9) (A- + -D)(T-^^)+l(/ = /.
This method falls within the category of relaxation methods and gives rise
to Seidel method in one particular case where uj — \. In the modern
literature the iteration process (9) with cj > 1 is known as the upper
relaxation method.
By comparing of (9) with F) it is easily seen that
1
A" -h - D , Tj; = 1
or, what amounts to the same,
B -uj A~ ^ D , rJ, = cj .
As far as a non-self-adjoint operator B is concerned, the workable procedure
reduces to inversion of a lower triangle matrix.

The alternative-triangular method 679
As shown above, Seidel method is quite apphcable for any operator
A = A* > 0. However, the extra restriction 0 < cj < 2 is necessary for
the convergence of the upper relaxation method. This is certainly true
under condition (8) with a known operator Bq. Along these lines, it is
straightforward to verify that B = ui A~ + Z), Tj, = cj and
5o = \ {[u A- + D) + {oj .4+ ^D))=\[uA^['l-u)D).
This supports the view that 5o > 0 for 0 < w < 2 and condition (8) is
fulfilled for cj < 2, since
D > 0 for cj < 2.
But the convergence rate depends on the parameter cj. However, there are
analytical estimates for lo and the convergence rate when the subsidiary
information on the spectral bounds of the operator D~^iyA~ +71+) is
available, but their determination is some problem in itself. Just for this reason
the parameter lo is so chosen as to minimize the total number of iterations.
In dealing with numerous similar problems this approach is more convenient
and effective.
By appeal once again to the model problem of interest it is plain to
show that the upper relaxation method is a perfect tool in such matters,
since the work and storage require
n > n„{e), n„{e) =o(^- In - j .
3. Imiplicit iteration schemes. Convergence of implicit iteration schemes
was the subject of investigation in Section 2 for the special case
A0) yk+±^jh ^ ^ y^ ^ J ^ y^ ^ ^ gj^gj^^
'''it+i
when the iterations j/;i, i j can be immediately found by the formula
Vk+i =yk- Tk+i{Ay^ -/).
As a matter of fact, the upper relaxation method and Seidel method
are nothing more than the implicit scheme F) with B ^ E incorporated.
Still using the usual framework of implicit iterative methods, the value t/j._,_i
is determined from the equation
A1) 5 y,+, = Fk, Ffc = 5 y, - r,,+, [A y, - f),

680 Methods for Solving Grid Equations
with a known right-hand side Fk. In both cases the matrix B falls within
the category of triangle matrices, since B = A~ + D and B = A~ -|- ~ D do
arise during the course of Seidel and upper relaxation methods, respectively.
Due to this property every new iteration j/j.,i can be obtained by making
a minimal number of iterations. For the model problem concerned, the
number of the necessary iteration.? is equivalent to the total number N of
the grid nodes.
Proper guidelines for the well-founded choice of the operator B are
provided by the following requirements:
• a minimal number of iterations;
• the operator B is economical, the work and .storage require 0GV).
The meaning of the latter property for difference second-order
equations of the elliptic type i.s that the equation 5j/fc_|_i = F^ with a
known right-hand side Fk must be solved in a minimal number of
operations.
The main result in Section 2 regarding the optimal .set of parameters
{tj,} can be generalized directly for implicit schemes with B ^ E a.s follows:
B y^+'~y'' +Ay, =/, ^■ = 0,l,...,n-l, yo 6//given,
'''fc-l-i
under the conditions
A2) 5 = 5* >0, A = A*>Q, 7iB <A<j^B, j, > 0,
whose key role is to specify the primary family F) of iterative methods
at the very beginning. But this is not the case for the upper relaxation
method and Seidel method both in connection with the property that the
operator 5 is non-self-adjoint: B ^ B*.
In mastering the difficulties involved, the intention is to use the
homogeneous equation related to the correction Wj. = B~^{Ay-i. — /):
A3) B ^^ ^-FAw^. =0, i = 0,1,... ,n-1,
'''k + l
XV, ^B-\Ay,~f)^H given.
The preceding scheme is equivalent to the explicit scheme
A4) i!k±LZj!*L + C'a:fc = 0, i = 0, 1, ,. , , n - 1 , Kq 6 i/ given,
'''k + l

The alternative-triangular method 681
where Xk = B^l'^w^ and C = B-^I'^AB-^I'\
Indeed, because the operator B is self-adjoint and positive: B = B* >
0, the square root B^'"^ does exist with the property
E'/2)* ^5i/2 >o.
By applying the operator 5"'/^ to equation A3) and inserting Wj. =
S^'/^Kj. we arrive at scheme A4). Arguing in reverse order we might
obtain the same result without any difficulties.
Lemma 1 Given operators
A = A*>0, B = B*>0, C = B'^'^AB-^'\
the operator inequalities
A5) 7,B<A<y,B, 7,E<C<y,E, 7^ > 0,
are equivalent.
Proof A reasonable form of the functional at hand is
7 = ((.4 - 7 5) y, y) = {A y, y) - 7 E y, y)
= {AB-'l\B'l\j),B-'l-\B'l-'y)) ~ ^[B'l\B'l^y)
= {Cx,x) -7(a;,a;)z= ((C - 'yE)x,x) ,
where x — B^''^y is an arbitrary element of the space H due to the
arbitrariness in the choice of the element y ^ H. Because of this, the equality
A6) I = {{A - 7 B) y, y) = {(C - y E) x, x)
implies that the operators yl — 7 5 and C ~ j E are of the same sign. For
the sake of simplicity we may assume A — j^B > 0 and then insert 7 = 7j in
A6), leaving us with the inequality / = ((C — j^E) x, x) > 0. This means
that C > 7ji?, etc. Thus, the assertion of the lemma is completely proved.
From what has been said above another conclusion can be drawn in
this direction: the possible applications of the implicit scheme F) in solving
the original equation An = f are equivalent to the numerical solution of
the auxiliary equation Cv = if through the use of the explicit scheme
A7) —— \-Cxj^ = (p, i = 0,l,...,n, Kq 6 i7 given,
'''k + l
if we accept C = B"'^!'^AB"^!"^ and (p = B~^l'-^f. Under this agreement
all the results obtained in Section 2 for a particular implicit scheme can be
covered by the following statement concerning explicit schemes.

682 Methods for Solving Grid Equations
Theorem 2 Let conditions A2) hold. Then an optimal set of parameters
{tj.} specified by formulas B7)-B9) exists and may be of assistance for
solving problem F), providing the validity of the estimate
A8) \\Ay„~f\\g., <gnll^%-/lli3-
with
1+?;■■' " 1 + ^
7?
Recall that Theorem 2 has been proved in Section 2 placing a
special emphasis on one particular case of explicit schemes with the identity
operator B = E involved. To make our exposition more transparent, the
implicit scheme transforms into the explicit scheme A7), Having stipulated
the conditions JiE < C < J2E, the estimate
B0) \\Cx,^-^\\<q„\\Cx,-^\\
is an immediate implication of estimate C1). The forthcoming substitutions
^ = 5-i/2j, B-^/'-'x,, = j/„ and C = B-^I'^AB-^I'^ may be useful when
providing current manipulations:
II C x„ -ipf = (C ,i;„ - f, Cx„ - v?)
= (^B-\Ay„~f),Ay„~f)
= \\Ay„~f\\l...
By inserting \\Ay„ — /||^-i in B0) in place of the norm ||C'x-„ — <^|| we
arrive at inequality A8), thereby completing the proof of the theorem.
In concluding this discussion it is worth noting that the type of the
original equation Au = f and the operator B have no influence on a
universal method of numbering the parameters Tj , , , , , r„ that can be obtained
through the use of the ordered set A4^ of zeroes of Chebyshev's polynomial
of degree n, whose description and composition were made in Section 2 of
the present chapter.

The alternative-triangular method 683
4. The alternative triangle method. Making a substantiated choice of the
operator B will be justified in more detail a little latter. Recall that any
product B of "economical" operators is also an "economical" operator. This
is certainly true for "triangle" operators Bi and B2 for which the operator
B = B1B2 would be "economical".
Any .such operator /?=/?*> 0 is representable by
B1) /Ji+/J2 = /?, R=R*>0, R\ = R2,
where Ri and /?2 are "triangle" operators. The associated matrix??. = (fj,)
is .symmetric, that is, r^ = r^j. It is obvious that the appropriate matrices
'TZi and 7?.2 are given by:
.„ . _. _ , '■,;■ for i <i,
rt = 0.5 Ti
' ^'^'' '^ [r,,- for j>i,
From here and the symmetry condition r^ = r.-j it follows that TZ^ = 7?.2.
The operator B built into scheme F) arranges itself as a product of
"triangle" operators:
B2) B = {E + LjRi){E + ioR2),
where cj > 0 is a numerical parameter. One succeeds in showing that B
is a self-adjoint positive operator; 5 = 5* > 0. Due to these properties
scheme F) with operator B2) will belong to the primary family of schemes
A2).
Indeed, it turns out that the operators
Bi=E + LjRi, B2 = E + ijR2
are adjoint and positive:
Bl =: {E + uiRiY = E + ui R2^ B2, Bi>E, ^2 > ^^ for cj > 0 ,
since Ri > 0 and R2 > 0;
{Ry, y) = (Ri y, y) + {R2 y,y) = 2 {R, y, y) = 2 {R2 y, y) > 0 .

684 Methods for Solving Grid Equations
This imphes that
{By, v) = {B,B2 y,v) = (y, B*Blv) = {y, B^B^v),
meaning B = B*. By the same token,
{By, y) = EiB^ y, y) = (^2 ?/, ^2, j/) = || 5. j/ H^ > 0.
It seems clear from the equation
B3) {E + LoR,){E + LoR2)^-^^^^^^ + Ay^ = }, ^ = 0,1
, -i , . . . 1 I" 1
given yo e H ,
that the value y^^i remains as yet unknown and can be found from the
equation
{E + LjRi){E+^R2)yk+i = Fk
with the right-hand side Fk = By^. — Tj._|_j(ylj/j. — /).
As a matter of experience, by solving successively two equations
B4) {E + LoR,)y=F,, {E + lo R2)y,^i = y
with upper and lower triangle matrices we get a final result. So, the title and
origin of the alternative-triangle method (ATM) owe a debt to scheme
B3) with operator B2) that can add interest and help in understanding of
how one could progress in this direction. Before going further, we need to
know the values of parameters j^ and j^.
Lemma 2 Let operators R, Ri and R2 be in hne with the conditions
/J = /Ji + R2, R* = Ri and
B5) R=R*, R>6E, 6>0,
A
B6) RiR2<-jR, A>0.
Then the estimates are vahd
B7) 7iB <R<7^B

The alternative-triangular method 685
with
0 S 0 1
where an operator B is specified by formula B2), The ratio
c
72(^)
attains its maximal value for
2
B9) id = Lj„ = , ,
^ ^ -yiA'
giving in that case
^ ^ ^^TTT^' ''=A' ^^ = 2A + 7^)' ^^ = v^-
Proof Inequalities B5) and B6) together imply that
{Ry,y) > Hy,y),
{R1R2y,y) = (/?2y,R\y) = || /?2j/1|' <^iRy,y),
whence it follows that 6 < A and j; < 1.
Before giving further motivations, we may attempt operator B2) in
the form
C1) B = E + Lj{Ri+R2)+Lj^ RiR2 = E + uiR + ui^ R1R2.
With the relations R > 6E or E < jR in view, we obtain for the operator
B the upper estimate
1 CJ^A /I Cj2A\ 1
B <-R + LjR+ -^ R=^-+Lj+ -^j R= -^R,
0
yielding the relation R > 1 ^B. Formula C1) transforms into
B = E-Lj{Ri +R2)+Lj'-' RiR2+2lj{Ri + R2)
= {E - io Ri) (E ~ io R2) + 2lj R ,

686 Methods for Solving Grid Equations
when performing current manipulations with the operators B and R:
{Btj,y) ={iE^LoR,){E~Lo i?2) y, y) +2Lo{Ry, y)
= {{E ^ Lj R2) y,{E ~ Lj R2) y) + 2lj {Ry,y)
= \\{E^ioR2)yf + 2io{Ry,y)
>2Lo{Ry,y)^^{Ry,y).
0
From such reasoning it seems clear that {Ry,y) <12 {By,y)-
The next step is to find a maximal value of the ratio
by observing that
4 _,, I^ljHA/4
(l+a;6 + (a;2^A)/4y
From calculus it is known that the maximal value of ^{lj) is attained for
2 . °
cj = cjn = , since ij"(cj) < 0 for cj = cjg, Upon substituting lj^ —
5A
2 ^/rj/6 into B8) we derive formulas C0).
Theorem 2 Let under the conditions of Lemma. 2 the inequalities
C2) Cj/?< A < Cj/?, Cj > 0,
hold simultaneously for operators A = A* > 0 and R = R* > 0, TAen for
ATM given by formula B3) with optimal set of Chebyshev's parameters
C3) T = ~— , Tg = ■ , Po - —-— , <J = — ,
^ l+PoO-fc 7l+72 1+^ 72
where
-y-j _ Cj 7 1 , I2 — ^2> 2) > \
2(l + v^)' ' 477]
'? = ^ • -^t e A^: ,

The alternative-triangular method 687
estimate A8) is true and it suffices to perform n iterations for the validity
of the inequality
ll^y„-/|lB-i <£ 11^%-/lis-',
where
[^ lnB/£)
C4) n>n,{e), n.ie)
Ci 2^2^'
Proof In complete agreement with the preceding theorem with known
coefficients 7j and j^, arising from the operator inequalities j^B < A < j^B,
we deduce from B7) and C2) that
0 0
A > Cj /i; > Cj 7 15 = 7i 5 , meaning 7i = Cj 7 j ,
0 , 0
A < CjR < c.^') ^B = f.yB , meaning 73 = r'.^ 7 , .
The conditionality parameter for lj — lj^ is equal to
72 C3 C2 1 + y?]
Theorem 2 and the results of Section 2 apply equally well to such a setting.
As stated in Section 2, the condition <?„ < £ is ensured by n > In ~/B\/<J).
c.
Substituting here <J = 71/72 and taking into account that <J < —^-Jrj,
we derive the sufficient condition C4), which can easily be verified in one
particular case of interest;
A = R = R1+R2, R*2 = Ri, Cj = Cj = 1 ,
The total number of iterations required in this connection is no less than
lnB/£)
n > Uoie), Uoie)
2^2^
5. ATM for the difference Dirichlet problem. The trace of those ideas can
clearly be seen in tackling the difference Dirichlet problem associated with
Poisson's equation in the rectangle G = {0 < X'„ < /„, a = 1, 2};
C5) Aj/= j/^^^.^ + j/g^^^ =-/(k), xElo^, y\^^^fi{x).

688
Methods for Solving Grid Equations
Here the composition of the grid is
'^ft = {^i = {hf^i,hf^2), i^a = 0, 1,2,, ,, ,Na, NJi^ = /„, a= 1,2}
where 7,j is its boundary. In that case
Ay= -Ay , y^ H = n ,
R = A, Riy = ysJh^+ys:Jh2, R-zy = -y^Jh^-y^^/h^,
and instead of C5) we obtain
C5') Ay = ^, yen,
where the right-hand side ip differs from the function / only at the near-
boundary nodes. Simple algebra gives
C6)
/l oTr/i, 1 97r/}o\
{RiR2y,y) = \\R2yf
y^^y^
t-lfi''
,.2 , ..2
<NJ, + ^jD+y:j,l
[^^+ll){Ry,y),
implying that
C7)
4 4
Here we should take into account that
\2 / /„2 , „2\ ,,,2 , ;,2n
and
(aj6j+a,,6J^ <(< + «;) F^+6,^)
(Ry, y) = (j/5;^, y^ J ^ + (y^,.^, j/j.J ^^ > || j/,,^ f + \\ y^.^ ||'

The alternative-triangular method
689
under the inner product structure
Wi N2-I
(w,f]i = X] X] "ni^^iii^i /»2
!l=l l2 = i
The inner product {u, v]2 can be introduced in a similar way.
Knowing 6 and A, it is not difficult to determine the values of 1], 7j,
72 and ^ and, after this, to estimate the number of the iterations required
in such matters.
Available data processing is a special starting procedure in the further
comparison of three methods in line with established priorities by having
recourse to the model problem C5) that we have set up on a square grid
//j = h^ = h in the square G with the unit sides (/j = l^ = 1), Plain
calculations show that
TT h
__ 2y^
TT h
V = sm" —- , C= / ' ?» 2 7?] = 2 sin —- ,
2 1 + ^ ^ 2
For small values ^ ^ 1 we might have
^2 y^Ri V^Ri l.nVh,
2.9
0.29 2
for
2e-
10
10-
The first criterion among others is the number of the iterations in the
following three schemes;
h
1/10
1/50
1/100
SIS
200
5000
20000
SCP
32
160
320
ATIVl
9
21
29
We present below the algorithm of finding the (k + l)th iteration from
the equation
C8)
k+l , ,*.■+! i:
]y = {E + LJa Ri) (E + i-Jo R2) y =F,
F =By - rj,.+i (Ay - ^) = By + r^+i (A ^ + /) ,

690 Methods for Solving Grid Equations
with the supplementary conditions
k+l k k k k r ^
y = 2/ ~ 0 on 7;j, V = y lor x 6 uij^, v = l^i- for x E 7,^ .
A solution V of problem C8) on coj^ +Jh can be obtained through the use
J, , , k + l k + l k + l rr.^ r
01 such a procedure: v ~ tJ on LJf^ and v = p on 7,^, ihe recovery 01
k 4-1
y requires successive solution of the problems
k k-\-l
C9) [E + Lo^Ri)y = F , {E + Lj^R2) y =y, x^lj,^,
wh'
ere
D0) Ri y
hi + hi
i\fi 1
hi ' hi)^ \hi^''-^ ^ hi^"'-^
D1) R,y = -yii±pL^yi,±LpL
In preparation for this, we have at our disposal
y = y{hh^,i^h^), 2/,.j±i = y({i^±l)h^,i^hr^), yi,_^±^ = y{i^h^,{i^^±l)h^).
The operator E-\-lJqRi is specified on a three-point pattern (fj h^, i-,h^),
((jj — 1) /ij, fj h^j, («'i /ij, ((I'g " 1) /*2)i while the operator E + cJq_R2 ^ on
a three-point pattern (i^h^^i^h^), [{i^ -\- I) h^,!^ h^), (*i/jj, ((J2 + 1) ^2) ■
Upon substituting the above expressions for Riy and R2y into C9) we
establish the recurrence relations which will be needed in the sequel;
it
('') ^= i + x,+x, ' <=''
D3) T^^^^^^^V^^^i^, TU = 0^
 

The alternative-triangular method 691
giving the values y and y at the centre (ij h^, i.^ h.^) of the suitably chosen
pattern.
Further development with y on the grid LJf^ is due to a special choice of
the near-boundary node fj = 1, i^ = 1 at the left lower corner of the domain
of interest so that two other nodes ((fj — 1) /ij, i^ l^) and (ij h^, ((ij — 1) h^)
should belong to the boundary on which the values j/jj_i and Vi^-i are
already known, Formula D2) gives the value y with further contingency
either along rows or along columns.
The directions of row account are from the left to the right: for fixed
i^ = 1 we are moving from i^ — 2,3,.,. to i^ = Ni — I and, after this,
for fixed i^ = 2 - from ij = 1, 2 , . , to ij = A^i — 1. The directions of
column account are bottom up. The value y depends on y ,- , j and
k4-l
y i -1-1, therefore account starts at the right upper corner of the domain:
jj = A^i — 1 and /j = Ny — I, so that two adjacent nodes should belong to
the boundary on which the values y ,;^_|_| and y ^^j^i are already known.
Further calculations are performed either along rows (from right to left) or
along columns (from top to bottom).
All of the calculations are best conducted by recurrence formulas,
whose algorithm is certainly stable and is called "through execution"
algorithm.
Within its framework it is necessary to perform 4 operations of
addition and 6 operations of multiplication at every node of the grid for
determination of y with knowledge of ^ . In giving ^10 operations of addition
and 10 operations of multiplication should be performed. Summarazing,
it is required to carry out 14 operations of addition and 16 operations of
multiplication m passing from y to ij .
In trying to make these numbers small enough, it is highly recom-
k
mended to save two numerical sequences instead of y, This can be done
using the algorithm
D4) iE + Lo,Ri)Lo = ^k, ^1.,,. =0,
{E + U^R'ijLJ-Ul, LO |^^_ = 0,
fc-l-i * i-l-l
y = J/ + rj,_^i cj ,
where $jt = At; -|- / and v\ = p, making it possible to perform 10
operations of addition and 10 operations of multiplication at one node for

692 Methods for Solving Grid Equations
k + 1
determination of j/ . In so doing the passage from the kth iteration
to the [k + l)th iteration necessitates the storage not only for the value
J/(«j/ij, ij/ij), but also for the value w (ij/ij, ij/ia)'
6. A higher-accuracy scheme in a rectangle. In Chapter 4, Section 5, the
Dirichlet problem
D5) Au = -f{x), X 6 G, u = fi{x), .i- 6 T ,
was completely posed, for which the difference scheme of fourth-order
accuracy was composed in the following way;
D6) A'y=-'f{x), KGcjft, y = fi{x), x E y,^ ,
where
A'y = Ay+ ' ^^ ' AiA-jy,
A = Ai-FA2, A„j/= j/j.^^.^, a =1,2,
h^ hi
^ = f+f^M.f+f^A,.f.
0
Let now H = Q be a space of all grid functions given on the grid lj/j
and vanishing on the boundary j/^. Having involved the operators Ay =
—A'y and Ry = —Ay specified for any y E H, we deduce instead of D6)
that Ay = tp, where (f ^ <p only at the near-boundary nodes. Applying the
estimates obtained in Chapter 4 for the operator A yields
2
^R<A<R,
•J
and reveals the coefficients to be Cj = | and c, = 1. The operator R
arranges itself as a sum
R=R^ + R^, iJiy=:_L+_L^ ij2j/ = --~- V^.
/ij n^ h^ h^
Being concerned with the same quantities 6, A and lj^ as in Section 5, we
.,0 0 0 0
might agree to consider 7i = ^1 x and 7, = 7,, where 1 ^,12 were specified
in Section 4:
08 0 S
^^ = 2(l + v^)' ^^"V^-

The alternative-triangular naethod 693
Knowing j^ and j^, it remains to find the iteration parameters {t^} with
a simple observation that formulcis D2)-D3) are still valid with another
member p ;
/.^x *-• *-■ *■ *-■ '*? + '*?» . *
D7) i^=FC8) + $. $ = -n^+i ^2 ^ ^ ^' ^ '
where i^og-i is given by formula C8). The total number of the iterations is
equal to
n>n,{e), n„[e) = \j~nl[e),
where "•*(£) is the total number of the iterations for the Dirichlet problem
associated with Poisson's equation C5), The total number of the iterations
required in ATM for a higher-order scheme is being increased in \/l.5 ?» 1,22
times as compared with a scheme of second-order accuracy.
In the two-dimensional case the iterative alternating direction method
or the direct decomposition method turns out to be more economical, but
for the multidimensional Dirichlet problem ATM is the most economical
one among other available methods. This advantage is stipulated by the
special structure of the operator A' (see Chapter 4, Section 5):
P P ^2
D8) A'y=J2^c n iE + xpAp), ^/? = jf - A„ j/= j/,^,,^ ,
a = l 13 = 1,pyia
a,ssuring the operator inequalities
/2\P-i
D9) h- R<A<R,
here
wnere
Ay = -~A'y, iZ j/=-Aj/= - ^ A„j/, y€H,
a = l
3\(p-i)/2
and n*{e) is the total number of the iterations required in connection with
solving the equation Ry = f, not depending, in fact, on the number of
observations p.

694 Methods for Solving Grid Equations
7. Difference scliemes for elliptic equations of general form. As we have
mentioned above, the applications of ATM in the numerical solution of an
operator equation of the first kind consist of several steps;
• determination of the operator R;
• design of special "triangle" operators Ri and R2;
• calculations of the equivalence constants Cj and c,^ and 8, A;
• selection of the number n = n[e) of the iterations and a set of
parameters {rj^}.
We begin our exposition with a discussion of examples that make it
possible to draw fairly accurate outlines of the possible theory regarding
these questions and with a listing of the basic results together with the
development desired for them. Common practice involves the Laplace
operator as the operator R in the case of difference elliptic operators A. The
present section is devoted to rather complicated difference problems of the
elliptic type. Here and below it is supposed that the domain of interest is
a p-dimensional parallelepiped G = {0 < a;„ < /„, a = 1, 2,... ,p] with
the boundary F (a rectangle for the case p = 2), on which the boundary
condition of the first kind is imposed:
In what follows it is required to find a continuous in G solution of the
Dirichlet problem
E0) Lu = -/(k) , x- 6 G , 'u|p = i.i{x),
with a second-order elliptic operator L involved.
With this aim, we introduce in the domain G the usual grid
'^h = {^i = ih'^1, «Vj2. ■ •. . «p'v) e G, i„ = 0, 1, 2,,,. , A^„,
h^Na = la, cv = 1,2,.,, ,p}
with the boundary 7;^, so that u)f^ = cj;, U7/,. As usual, an inner product in
0
the space H = Q oi all grid functions defined on w^j and vanishing on the
boundary 7;, is taken to be
(?/. f) = E y{x)v{x)h^h^ ■■ hp.
It seems worthwhile giving several particular cases.

The altei'iiative-ti'iangulai' naethod
695
a) An equEition of the elliptic type with mixed derivatives, fn that
ca.se we are agree to consider
E1) Lu= J2 ^«/?"' ^"^"" a7"(^""^^''^a^)
E2)
a,13=1
a = l aj3 = l ex—I
a
where £, = {£,i,£,2, • ■ ■ ,£,p) is an arbitrary vector and Cj > 0 and c^ > 0 are
constants,
fn dealing with difference operators on the grid cj^j at hand such as
E3) Ay= J2 ^cpy, Aapy = :^ (^a/? J/sJ,^^ + (^"a/?%-J,^.J >
a,p=:l
we may set up in conformity with problem E0) the difference Dirichlet
problem
E4)
Ay=-if{x), xEio,^, y = fi{x) , x E j,^ ,
and follow established practice: introduce in the space H = tl the operators
A and R acting in accordance with the rules
p
Ay = ~Ay, Ry = --Ay = ~2_,ys^x^>
where A is a difference {'2p+ l)-point Laplace operator, and accept instead
of E4) the governing equation Ay = ip, where (p ^ ip only at the near-
boundary nodes. Recall that
E5)
c^R< A< c^R,
where Cj and c, are constants arising from the ellipticity condition E2),
By analogy with Section 5 the operators Ri and R2 are specified by
the formulas
R^y = Y.ir' R'^y = -Y.
Ri + R'i = R .

696
Methods for Solving Grid Equations
The constants 6 and A remain as yet unknown. Since 5 is the smallest
eigenvalue, it is not difficult to show that
p
S
J2 — ^i^'
TT h.
/l2
2L
By virtue of the relations
R2yr =
E
<t^t\\yrJ'<tl^Ry^y)
h
a= 1 «
plain calculations permit one to conclude that the constant A is equal to
P 4
y lY' With knowledge oi 8, A, Cj and c^ simple algebra gives u), n{e)
a=l a
and {t^}, which constitute what is needed in the applications of ATM, The
total number of the iterations
<^«' "*'^"*' = \/f^' ' = !■
is proportional to ./ —,
ft- Yd
In the case of a cubic grid (/ij = h^ = • • • = h = h) in a unit
p-dimensional cube we might have
4 p . r. IT h
0 = -— sm
Tj = sin'
A - ^"^^
/l2 2 ' /l2 ' ' 2
thereby justifying that the total number of iterations is independent of the
number of observations.
The iteration y is recovered from the equations
E7) Byy::^{E + LoRy)y=F, Bo'V ={E + ijR2)''V =y,
E8)
fc k k k h
F = By - Tj.^, [Ay - ^) = By + r^.+i (Ati + ^) .
A solution of problem E7) is given by v — y for x E coj^ and hy v = l^
for a; 6 7;j, so that v — y = y — y. The preceding difference operators
Bi and B'j can be written as
/ '' 1 \ '^ 1
E9) B,y ^ y + CO Riy = il +LJ Y. -j^ )y - "^ Y. /^^^"'"^'
F0)
/ '^ I \ '^ 1
a=l "^ a=:l "

The alternative-triangular niethocl 697
The calculations start from the corner fj = 1, i^ = 1, . ■ . j *p = 1, of the
parallelepiped of interest for which all of the adjacent nodes x'^^^"'' E -jf^
fall into the boundary ones at which the values j/'."i») ~ jj are already
known. With knowledge of y at that node we fix fj = 1, ,,,, i = 1
and change fj =; 1,2,,., ,A^i — 1, After that, continuing this process for
«j = 1, 2, , . . , TVi — 1 and for fixed i.^ = 2, etc. we find at all the nodes
X £ ijjf^ that
Along these lines, the starting nodes «„ = //„ — 1, a = 1,2,,.. ,p, suit us
k-\-Y
perfectly for determination of y on the grid cj^j by the rule
k+i /^ v~^ 1 *;+ir+i -1 -A /-, cjA\-i
y =(-E/^ y^+^»^ + yj(i + —) .
^ a=l « ^
b) A system of elliptic equations. Let u — (m^m^,,.. ,m'"°) be a
vector and let a block p x p-matrix k = (^^'?) with blocks of size rrig x rrig
will be so chosen as to obtain some suitable matrix k^a = {k""i) of size
nig X irig for later use. The Dirichlet problem for a system of equations is
first considered in the parallelepiped G:
F1)
where
F2)
Lu' =
: ~/^ XeG, 11'=^", X-er, 5 =
a,/J=l 771=1 " "
:1,2,..
. ,mo ,
In such a setting the condition of ellipticity becomes
p rng p ma p nig
F3) qEE(cr< E E ^";^wce^"<=.EE(cf.
a = l s = l a,/? = l 5,771 = 1 a=l 5 = 1
where £^ = (^^i-J^,-- ,(,2^°), ct = 1,2,,.. ,/;>, are arbitrary vectors and
Cj > 0, c,^ > 0 are constants. A reasonable form of the difference operator
is
P ma
F4) Aj/^- E HK'py"'
a,p=l m=\

698 Methods for Solving Grid Equations
with
F5) ATp ir = \ [(^:;y^),^^^ + {k:;c,),J ,
making it possible to set up the difference Diriclilet problem
F6) A J/' = ~f' , a; 6 cjft , j/' = ^r" , x- 6 7fe >
in the space H = ^ of all grid functions under the inner product structure
mo
(y,v) = 2^(J/^^;■'), (?/,t/')= l_^ tf{x)v'{x)h,h,-.-h^,
y = (J/^J/^,..,J/^..,,J/'"°), v={v\v\...,v\...,v"'^),
y' eH , v' & H , s=l,2,... ,m„.
Being concerned with the operator Ay'^ = ~Ay^ and the regularizer Ry" =
— Ay" = ~Yla=iyT- r ^"^ '^^^^ space H, where A is a Bp + l)-point
difference Laplace operator, we rely in the further derivation on the Green
formula and condition F3), whose combination gives the operator
inequalities C2), Having involved the same operator R as was done in problem
E3)-E4), we obtain the constant ui and the operator B in terms of known
members 6 and A, Just for this reason the same algorithm as in a) is
workable for determination of the (i-|-l)th iteration for either of the components
*-' + !
y" and so it is omitted here,
c) A system of equations in elasticity theory. The system of Lame's
equations arose from the stationary elasticity theory;
F7) Lu = fiAu+{\ + fi) grad div u = —f (k)
with vectors u = (w"', m^, , ,. , ^i'') and f = {f^,P,,,-,f^) and Lame's
constants A > 0 and /.i > 0, We may attempt the preceding system in the
form
a=l a p = l P «
Further comparison of this with F2) allows ug to deduce that
F9) k"^p = 1^16,p 6^„^ + (A + ^0 .5,„ 6p^ ,

The alternative-triangular method 699
where 6^j is, as usual, Kroiiecker's delta.
We learn from Chapter 9, Section 2 that the constants Cj and Cj
involved in inequalities F3) are equal to
Cj=p, C2 = \ + 2 fi.
With these, the difference Dirichlet problem associated with F8) is
described by
x'6 7;,, s— 1,2,.,, ,p.
G0) yV tf :
where
= ~r,
A'y' --
a; 6 cj,j , y" = fi" ,
p
= i^iYl ^^y' + (^ + t^)
p
a=l p=l
Apsv''
Ac,y = Vs^x^ . Kp,y = \ {y^^^^ + j/^^jJ ,
The above framework necessitates specifying for any y" ^ H the operator
Ay" = ~A"y" and the regularizer -Rj/*' = ~~ ^^0=1 ^-aV^ thereby clarifying
that the same operator R is adopted for these purposes as before.
By Green's formula it is straightforward to verify the inequalities
c^R < A < c^R with constants Cj = /./, and c^ = A + 2/y incorporated.
By exactly the same reasoning as in the preceding examples ATM requires
7io(£) iterations, where
G1) "oU') = \/-"^-^<
and n*{e) is the total number of the necessary iterations in solving the
Laplace equation,
8. ATM for solving grid elliptic equations in an arbitrary complex domain.
Two lines of research in subsequent considerations of such equations are
evident in available publications in this area over recent years. No much
is known in the case of an arbitrary complex domain. The first one is
connected with nonequidistant grids, while the second one deals with some
modifications of available methods. Even a constant step in each direction
does not allow to overcome the difficulties in the near-boundary zones as a
result of emerging non-uniform grids.
It turns out that the convergence of the aforementioned methods
become worse on account of widely varying bounds of the spectra of difference
operators.

700 Methods for Solving Grid Equations
Current explanations need certain clarification by having recourse to
one simplest example devoted to the governing equation
u" = ~f{x), 0 < a: < 1 , M@) = w(l) = 0,
and associated three-point scheme
G2) Ay = ~f(.r,), x^lo,, j/@) = 0 , j/(l) = 0 ,
which is constructed on the grid
'^h = {•'^0 = 0. ^'i = h^,x^ = h^ +h,x^^ = 1} ,
so that at two inner points
h V h h, J ' "' /),2
since y^ = y^ = 0. For clarity only, we take h^ < h and 2h -\-h^ =1.
The next step is to find the eigenvalues of the operator A. By
definition, Ky -\- \y = Q or, what amounts to the same,
''''-"^ '^^ + Xy, = o, ::%±A + A,, = o,
h \ h h^ ' K^
Having completed the elimination of j/ji we derive the quadratic equation
related to p = Xli?:
G3) ti^C--{i + U)i.i + 2 + t = Q, t = hjh,
with the roots
_ l + St±y/{l+My-'~AtBTT) __ /J(i,2)
/^(i,2)~ j^ . \uv~-j^
and the characteristic relation between A/i-j = A^^j^ = 6 and A(-2) = A^j^j, =
A:
__ S _ 2t{2 + t)
'' ~ A ~ A + ■itf -2t{2 + t) + {l + U) v/(l + 3iJ ~ 4i B -K ij ■
Whence it follows that for t = I, that is, on an equidistant grid r; = 1/3,
while rj K, 2t = 2hy/h for small ratios i = /ij//i <C 1; making worse the

The altei-iiative-ti'iaiagulai- naethod 701
conditionality of the system G2) of equations along with decreasing the
ratio /}j//i.
It seems clear that in solving the system G2) the number of the
iterations within the framework of the explicit scheme with optimal set of
Chebyshev's parameters or of the simple iteration scheme is proportional
to 1/s/fj or I/t], thus causing an enormous growth as h^ -^ 0.
The generalized problem on eigenvalues of the same operator A is of
the form
where D is a diagonal matrix such that Dy = d{x)y, d{x) > 0. In the case
of interest
and the problem of determining A amounts to
Instead of G3) we must solve the cjuadratic equation
(ij d^ t jj^ - [A + t) f4 + 2i fij] // + 2 + i = 0 ,
For the choice dj = l/t and dj = 1 we deduce from the foregoing that
rj = 1/B-|-i) and 1] ?» 0.5 when i( <C 1, that is, rj remains finite as h^ -^ 0.
From such reasoning it seems clear that the scheme
G4) ^Mi:ii^=Aj/, + /
'''fc+i
ofl^ers more advantages in comparison with the explicit scheme.
A similar situation exists during the course of ATM for solving elliptic
equations on nonequidistant grids or in arbitrary complex domains, giving
rise to obvious modifications of ATM with the intervention of the operator
D = D* > 0 built into the structure of the operator B. Making a
substantiated choice of the operator D is stipulated by economy reasoning for every
iteration as well as by a minimal number of iterations. Let D = D* > 0
be an arbitrary operator and the operator A = A* > 0 from the equation
Au = / be a sum of mutually adjoint operators Ai and A'j:
G5) A = A*>0, A = Ai+A2, A*, = A2,

702 Methods for Solviiig Grid Equations
by means of which the factorized operator B is selected by the rule
G6) B = {D + LjAi)D~\D + ioA2),
where cj > 0 is the iteration parameter. The operator B so defined is
.self-adjoint and positive. Once supplemented with the conditions
G7) A>6D or {Ax,x) > 6{Dx,x), 6>0,
G8) AiD-^A2<jA or {D'-^A2X,A2x)<—{Ax,x), A>0,
which are valid for all x 6 FI, Theorem 2 and formulas B7)-C0) continue
to hold with the operator A standing in place of the operator R, so there
is no need to rewrite them once again in a common setting.
Of special interest is the well-founded choice of the operator D under
the agreement that the appropriate matrix D is diagonal:
G9) Dy = d{x)y, d{x) > 0.
Because of this form, the member d(x) is so chosen as to maximize the
ratio 7] = 6/A in the modified alternative-triangular method (MATM) with
reasonable efficiency.
Adopting those ideas, some progress has been achieved by means of
MATM in tackling the Dirichlet problem in an arbitrary complex domain
G with the boundary F for an elliptic equation with variable coefficients:
(«°) ^" = ^('.wS;) + ^(*'W|;) = -^W'
X = (x^jX^) (i G, u{x) = fi{x), a; 6 F ,
^a(^) > Ci > 0, a = 1,2,
The usual assumptions are made here saying that the boundary F is smooth
enough and the intersection of the domain G and a .straight line passing
through any point x 6 G and in parallel to the axes Ox^, a = 1,2, consists
of a unique interval. The latter should not confine generality.
The design of a difference scheme for problem (80) is mostly based on
a nonequidistant grid cj^j (generally speaking, nonequidistant everywhere,
not only near the boundary F). When drawing up the family of straight

The alternative-triangular method 703
lines x^ = a;(^'"), i„ = 0, ±1, ±2,... , a = 1, 2, the points x^ = {xy'jX^'^')
constitute the basic pattern /?2 in the plane (x^/x^). Any such point x^
belonging to G is called an inner node of the grid, the total collection of
which is denoted by uif^, that is, uif^ = {x^ 6 GnR'i}-
The intersection of the domain G and any straight line passing through
a point X £ uij^ and in parallel to the axis Ox^ consists of the interval
Aa{Xi). The endpoints of this interval are called boundary nodes in the
X'„-direction. A set of all boundary nodes in the a;„-direction is designated
by 7^. The boundary jf^ of the grid cj^^ is just the union j/^ = j^ U j^j
making it poasible to write down W;, = cj;, U 7;,.
Furthermore, let uj^^[xp), jS =; 3 — a, a = 1,2, be a set of uodal
points from the interval A„; let uj~^{xp) be a set containing ui^ and the
right endpoint of the interval A^; let Lb^{xp) be a set containing uj^{xp)
and both endpoints of the interval A^. For an inner node x 6 uj^{xp),
we denote by a;(+-''») and x'^~^°'^ adjacent nodes belonging to ij^{Xp). If
j.(+i<») (z y^^ii jjjay happen that this node does not coincide with the node
2,(«o,+i) Qf ^}jg gj.jd j^^ hand. The accepted view is connected with steps
h^{x) of the grid as possible spacings between the nodal points x 6 uJJ^ and
the grid nodes x'^^^°'^ 6 0/^.
What is more, at all inner nodes of the grid cj;j the mean steps are
taken to be
S„(^J = 0.5(x-(^-^+i)-(^(:--i)).
When the pattern of interast happens to be nonuniform in either of the
directions x- , that is.
^ a'
^.(ic) _ ,; h ?■ — n +1 +2
the mean steps fe„ = /i„ become constant and H^ at a near-boundary zone
differ from h^:
if h+<h^, then x'(+i"N 7., 4+'^ V ^^+'^ = (*« + 1)/»« ■
With these ingredients, a reasonable difference scheme is
(81) Ay = -ifiix) , X 6 cj,j , yix) = fi(x), x 6 7,, ,

704 Methods for Solving Grid Equations
where
A=Ai+A
2
T I \ ^ ( 4-y'-'^^"^ - y y — y'^ '^^)\
(±U)^j,(,a±U)), a+=a„(x-(+'^))
J/
or Or
y,. = Y^y'^''''-yy
Here the coefficients a„ and <yJ'(a;) are .so chosen as to provide on a uniform
grid a local approximation of order 2. By analogy with Chapter 4, Section
3 it is plain to justify the uniform convergence of scheme (81) with the rate
0(|/zp).
We proceed to more a detailed exploration of MATM for solving the
system (81) of difference equations just established. For this, the sum
A J/ = Ai J/ + A2 J/ involves the members
(-) A..|:[g....ji-(|-|).
governing what can happen: y^. = -—y if a;'-"'"^ 6 7;^ and y^ = -—y if
Other ideas are connected with operators Ai and A2 such that Ao,y =
0 0
—Kay, a = 1, 2, for any j/ 6 Q = H< where il is the set of all grid functions
vanishing on the boundary 7;^. By the same token, A = Ai-\- A2 = —A.
0
Under the inner product .structure in the space H = U
(j/.'c) = E y{x)v{x)h, h^
the operators Ai and A2 are mutually adjoint to each other: {Aiu,v) =
{y, A'jv). Because of this fact, the operator A = Ai + A-j is self-adjoint.

The altei-native-ti'iaiagular method
705
Cumbersome calculations for a proper choice of d(x} will be excluded
from further consideration. It seems worthwhile giving only the final results
for the later use
(84) d(x) = J2
r^ V^-a/i+v^
2h^
h+ h-
1
/^ + Vv '
.(")r
(85) ^Jxp)= max v\°\x), i>Jxp)= max v!^°'){x)..
/?= 3-a
a= 1,2.
The functions v\ (x) and v^ (a;), a = 1,2, are declared to be solutions of
the relevant problems
(86)
(««^"i^!), =-~p["\ ^^'.ecj.'^)
(-^ or
(87)
K4^:),^ = -/>^"*,
.(;„ 6cj„ j:
.W|^^=0
o(")
2fi„
al-'YJ^
h+ h-
These will be given special investigation by means of the elimination method
with 0A) operations required at every grid node. With the intervention
of four new functions v\° (x), v^"' {x) it is not difficult to develop the grid
functions fai^e) ^'^'^ '/'(»(•''/?) of one variable and then specify d{x) by
formula (84). Under such a choice we obtain
E=1, A = 4 max
a = l,2
max
\Jva{^p) + 'Pa{^p,
leaving us with the iteration parameters uig and {tj.} and the iteration
scheme
it + l k
Tl.
k + 1

706
Methods for Solving Gi'id Equations
h-\-l
For determination of y we must solve the equation
{D + ljAi)D-\D + ljA2)''V =F
with the right part p = B y +t,._^_i (A v +(p) (v = y on uj,^ and v = /.« on 7,,).
As usual, this amounts to successive solution of the following equations:
{D + ioAi)y^ F
y\ = 0.
k + [
{D + ioA2) y = Dy= d{x)y,
k + l
In this regard, the more detailed forms of the members may be useful in
subsequent constructions:
(89)
where
(90)
2/ =
it+l
y =
K :
1
'
1
J:
^ c.a+ fc+i(+i<^)
jy-l a a
dy
^d-
The same procedure is workable here as was done in Section 5 of the present
chapter.
9. The Dirichlet problem for Poisson'.s equation in an arbitrary complex
domain. The algorithm of MATM is demonstrated by appeal to the Dirichlet
problem associated with Poisson'.s equation
d It d u
Au = —-x + TT-x =-f(x), xeG, u = pi{x), x^r.
ox^ ox^
111 working on a square pattern
'^A = {^i = (*l'i) «2^) £ ^i «a = 0,±l,..,, a =1,2}
.^(^.+l) ^ ^(i.) ^ h, i„ = 0,±l,..., a = 1,2,

The alternative-triangular method
707
where h^ = h and h^ differ from h only at the near-boundary nodes, the
intention is to use scheme (81) with the members a^{x) = 1 and H^ = /i,
so that
In an attempt to adapt here MATM, the functions Vi{x) and v^ix) are
declared to be solutions of one and the same equation
y\.„ V = V
a " — "£-„,t;o
"Pix), x^eLj^.{xp), v\ =0, /?=3-a, a =1,2,
but with different right-hand sides
p^{x) =
hh+
f'2i'^}
2h
1 1
Denote by x^ = lp,{xp) and x'„ = L^{xp) the endpoints of an interval
Aa and by h~ and /j+ irregular steps at the left and the right ends of this
interval; in so doing
1 /■u(+-'^) — V V — v'^ ^"^
AaV = <
/i V h h'T
a
h ^ h h
a^a = 'a + K
, la + h^ <X^< Lc- h +
K h\ h+
= La-h+.
The functions v["\x) and v;" {x) can be found in explicit form, while the
functions ^^{•'^b) ^^'^ V'aC'^'/j) ^^e expressed by
^d^p) = ^
1
1 /La.(x'«)-/„(X>)X2
)-'
V'a('^>) = 2' ,^ = 3-a, a =1,2,
giving A due to (88) and evaluating the number of the iterations:
nB/£)
(91
">"o(£). "o(£) =
3.4 VVo
where L is the diamieter of the domiain G of interest.

708 Methods for Solving Grid Equations
This provides enough reason to conclude that in the case of any
complex domain the number of the iterations depends only on the main step h
of the grid cj,j regardless of near-boundary nodes.
Applying (91) to the model problem in a square of sides /^ yields
3.54 ^/h|l^
whence it follows that the number of the iterations in an arbitrary domain
G is being increased in 3.54/3.4 = 1.04 times, that is, in 4% in comparison
with those performed in a square, whose sides are equal to its diameter,
thus causing one-two iterations in practical implementations for e ■= lO"''
and h= 1/100.
Summarizing, the number of the iterations required during the course
of MATM in an arbitrary complex domain is close to the number of the
iterations performed for the same Dirichlet problem in a minimal rectangle
containing the domain G and numerical realizations confirm this statement.
10. On Solving difference equations for problems with variable coefficients.
In the preceding sections this trend of research was due to serious
developments of the Russian and western scientist.s. Specifically, the method for
solving difference equations approximating an elliptic equation with
variable coefficients in complex domains G of arbitrary shape and configuration
is available in Section 8 with placing special emphasis on real advantages
of MATM in the numerical solution of the difference Dirichlet problem for
Poisson's equation in Section 9.
One of the most important issues is concerned with a smaller number
of the iterations performed in the numerical solution of equations with
variable coefficients. It was shown in Section 7 that the number of the
iterations required during the course of ATM is proportional to ^Cj/cj,
where Cj and c^ are the smallest and the greatest values of coefficients,
respectively. The operator R in question can be put in correspondence
with the operator A with variable coefficients such that
c^R< A<c^_R.
One way of covering this is connected with further intervention of the dif-
0 0
ference Laplace operator y\.: /? = — A ■ A key role of R owes a debt to the
structure of the factorized operator B:
B = {E+ ioRi){E + Lj R2}, Ri+R2^R, R\ = R2 ■

The alternative—triangular method 709
But it may happen that the coefficients of the governing differential
equation are varying very fastly, but locally in a small region so that the spectral
bounds of the operator A change insignificantly.
In the case where G is a rectangle, it is possible to adopt B = R with
further reference to one of the direct methods available for determination
of y , thus causing the ratio (^ = Cj/cj and the independence of the total
number of the iterations upon h:
1 V =1 f
Recall that in the modification of ATM (MATM, see for more detail Section
8) there is no need for involving the operator R and so a proper choice of
parameters {r„} will be independent of constants Cj and c^ both. Here the
operator A arranges itself as a sum A — Ai -\- A2.
By having recourse to problem (81) in a unit square we will
illustrate the performance of MATM in the sequel. In preparation for this, we
introduce a square grid
^A = {^i = {iih,i,h), i^ = 0,l,... ,N , hN = 1, a =1,2]
and write down on it the equations
(«i VsX:, + ( J/,f J,., = -^ > a: e cjft , y|^^^ = 0 .
The coefficients a^{;v,) and a.2{.i:) are given by the formulas
a,{x)= 1 + Ko[{x, - O.bf + {x, - 0.5f] ,
a.,{x) = 1 + Ko [0,5 - (Kj - 0.5J _ ^^^ __ q^^^] ,
A'o = const > 0 .
A proper choice of the right-hand side f{x) is stipulated by the fact that
y{x) = Kj A — Kj) x^ A —X2) is the exact solution of the problem concerned.
Also, it will be sensible to introduce
Cj = min a^{x) = 1 , c^ = max a„(a;) = 1 -|- 0.5 A'o
and then take the operators Ri and R2 with the values

710
Methods for Solving Grid Equations
Adopting those ideas, the framework of ATM (see Section 7) involves the
operator B = {E + lj Ri) {E +lj R-j). By going through the matter
chronologically we rely on A =^ Ai + A2 with the members
^'>=j:rf>..+Si<-''
A2y
sHf-
y I +
The operator D arose in Section 8, by means of which it is plain to form
B = {D + ljAi)D'HD + ljA2).
No wishing to cover the computational procedures of MATM once
again, we fill in the table on the basis of numerical experiments for £ = 10"''.
Table 6
C2/C1
2
8
32
128
512
h=l/32
MATM
20
23
25
26
26
ATM
23
46
92
184
367
h=l/64
MATM
28
33
37
39
39
ATM
32
64
128
256
512
h=l/128
MATM
39
47
53
57
59
ATM
45
90
180
360
720
From here it is easily seen that MATM offers more advantages not only in
an arbitrary domain, but also in the case of variable coefficients,
10.4 ITERATIVE ALTERNATING DIRECTION METHODS
1. The alternating direction method for solving the difference Dirichlet
problem in a rectangle. So far we have considered the Dirichlet problem
for Poisson's equation in a rectangle G = @ < x'„ < /„, a = 1, 2):
A) Am = A1M+A2M =-/(.-c), xeujn, u\_^^=fi{x)
Ac,y = y,^^^^, a = 1,2,

Iterative altei-iiating direction methods 711
The numerical solution of problem A) by means of iteration schemes can
be done using the alternating direction scheme for the heat conduction
equation du/dt ~ Au + f{x), taking riow for j = 0,1,... the form
B) L_^L=Au/+^/' + A2j/'+/(x-),
xe^h, 2/^'+'^'!^, =A'(a;),
yi+l _ yj+1/2
y ^ = Ai 2/^+1/2 + A2 2/^- + ! + fix),
with any initial data y° = j/(a?, 0). As usual, j is the number of the iteration,
the intermediate rteration, rj^^ > 0 and t^Vi > 0 are the iteration
parameters which will be so chosen as to provide a minimal number of
iterations.
The transition from the ith iteration to the {j + l)th iteration can be
done by the elimination method along the rows as well as along the columns
for the following three-point equations:
y^ + '^'-TJl\A,y^+'r^ = F^
F' =y'+rjl\ A2tJ +TJl\ J.,
:y^■+'-rSA2 2/^■ + l=F^+l/^
(along the rows)
(along the columns)
Thus, the users must perform 0{l/(h^ h^)) arithmetic operations in
calculating one iteration or 0A) arithmetic operations at every node of the grrd
By analogy with the nonstationary heat conduction equation the
iterative process B) is referred to as the alternating direction method
(ADM), the convergence of which can be established on account of the

712 Methods for Solviiag Grid Equatioias
homogeneous equations for the error z^ + ^ = y-''^^ — w-
^;+i/2 _ .J
T^'^
^;+i
^; + i„^j + i/2
= Aiz^ + i/2+A2z^'+i, ,'c6c.,, ^^' + '1=0,
Here we accept, on the same grounds as before, z^'^^''^ = j/-'"'"^'^ — u.
In this view, it seems reasonable to turn to operator-difference schemes
and then follow the usual practice: the operators Aiy = —Aij/ and A^y =
0
—A2J/ are introduced for any y from the space H = U oi all grid functions
defined on the grid w^^ and vanishing on the boundary 7,j under the inner
product structure
{y,y) = Yl yix)v{x)h^h2
Wj-1 W2-I
J2 J2 y{hhi,i2h2}v{i^h^,i^h,)hji.
il — l 22 — 1
It is well known that the operators so defined possess some remarkable
properties:
Al = Aa, S^E <Aa <AaE, E„>0, a =1,2,
C) 4 , 2 7r/i„ , 4 2 tt/i
^ sm _- , A„ = ^ cos -
,5„ =-;- sin^-—^ , A„ =-P7 cos^-—^ , a = 1,2.
What is more, the relation A1A2 = A2A1 takes places in various rectangles
only.
2. The general framework of ADM. The object of subsequent discu.ssions
is an operator equation
D) Aii = f, A = Ai + A2,
where A : H 1^ H, H is a finite-dimensional Euclidean space with an inner
product (;(/,'«) and associated norm ||j/|| = \/{y,y)-

Iterative alternating direction methods 713
Also, we take for granted that
E) 1) A^Ai+A2, A\=A,, Al = A2,
F) 2) 6^E <A^<^^E, .5„>0, a =1,2,
3) the operators Ai and A2 are commuting: Ai A2 = A2 Ai ,
and develop the iterative alternating direction method with these members
in just the same way as was done before for scheme B):
G) yi+MlZlL + A^ y^^,i2 + A2 y^ = f ,
^^ + Ai j/^. + i/2 + A2 Vj + i = / ,
given iJo & H , j = 0, 1,2,. ..
For the error Zj_^i = t/_(_j — u, we must solve the homogeneous equations
{E + TJ!^\A^)z,^,,2={E-rj-l\A2)z,,
^0 = % - « e i7, j = 0,1,2,... .
Having completed the elimination of Zj_^_^,2, we find that
{E + T^l\ A^){E + TJl\ A2) z^^, = {E- rj% A,) [E - t^I\ A2) z-
by observing that
Zj + i = Sj + i z- , Sj+i = 5^._^i 5^._|_j ,
s\'^ = {E+tPAi)-HE-tPAi),
SP = {E + rP A2r (E - rl'^ A2) ,

714 Methods for Solving Grid Equations
since the operators Ai and A2 are commuting. As a final result we get
n
(8) Z^ = Tn Zg , ^n = !_[ Sj ,
;=i
where T„ is a self-adjoint operator {T* = T„) as a product of commutative
self-adjoint operators involved.
From (8) the estimate || z„ || < || T„ || ■ || Zg \\ immediately follows, in
which the quantity || TJ, || depends on the parameters rj and r- . Roughly
speaking, a proper choice of such parameters is stipulated by the minimum
condition for the norm \\Tn || in connection with a minimal number of the
necessary iterations. To be more specific, when making a substantiated
choice, we have at our disposal two collections of parameters r} , Tj ,
. , . ,r^'-) and rf , rj , . . . , r^^^ with a prescribed number n = n{e) that
are known to us in advance;
mm IIT;, II = g„ .
\ J • J S
3. A proper choice of iteration parameters by Jordan's rule. First of all,
observe that the spectra of the operators Ai and A2 are located, because
of F), on different segments S^ < \ (Aa) < A^ with 6^ 7^ 6^ and Aj 7^ A^.
One trick we have encountered is to replace Ai and A2 by the newly formed
operators A\ and Aj with coinciding bounds:
ilE<A'^<E, a = 1,2, ;;>0.
This can be done using the decompositions
(9) A, = {qE-rA\r\A\-pE), A2 = {q E + r A'^yHA', + p E) ,
where the numbers r, q, p are free to be chosen in any convenient way and
the new parameters taken to be
^ ' q-T^^)p q + T^^^p
With these, we arrive at
C, _ c(l) cB)
^'^{E + J^'^A'.r'iE-u^f^^^,
SP = {E + J'U'2)-\E-J'U'2).

Iterative alternating direction methods 715
The norm ||T„ || can be expressed through the eigenvalues of the operators
ylj and A'2 such as
"t. = Ai',^(A;) , /?*, = a1P(A'2) , i„ = l,2,,..,yV„, a =1,2.
As far as the operators ylj and ylj are commuting, other operators A\^
A2, A and T„ possess common systems of eigenfunctions. By utilizing this
fact we denote by Aj.(T'„) the eigenvalues of the operator T^ and take into
account that
\{S(^)) = A - J^^a)/{l+Lj^^'>a) , AEB)) = A _ c^(i)/?)/(l +c^B)^)
with missing subscripts j and k. All this enables us to find that
j^j 1 +cj) 'a 1 +cj) -'/?
with
0<V<ak,<l, 0<7]</?i^<l, ^»„ = 1,2,... ,yV„, a =1,2.
As known, the norm of the operator T,i is equal to the greatest
eigenvalue maxj; Aj.(T,i). Further replacements of cv^. and /?j.^ in A1) by
continuous variables a and /? lead to increased maximum of the right-hand side
of A1); meaning
A2) II 7; II < max
ae[v,l]
1 B)
n
V l-a,J'>/3
Also, we may accept uj^^^ = iJ-' '^ and a = /?, since the arguments a and
/? run over one and the same segment [7], 1] and their positions in formula
A2) are certainly symmetric. The problem statement here is to find the
parameters for which
A3) min max TT -
' jcsl ae[n,ll -'■-'■ 1
l~a;^.Q'
A solution of such a problem is already known and so it remains only to
write the final expressions for optimal parameters r. ' and r. in question.
Having stipulated the conditions a = p = 1] for A(yli) = S^, A(yl2) = S^

716 Methods for Solving Grid Equations
and a = /? = 1 for X{Ai) = Aj, X{A2) = Aj, the constants p, q, r, rj are
found by the formulas
A4) ,-1^ (A^^M(A2JZM
x-i _ (Ai -Ei) A2
p =
A5)
x + i' (A2 + .5i)Ai'
Ai - A2 + (Ai +A2)p l-p
^ — , q = r -\
2A1A2 ^ Ai
with X > t and p > 0.
At the next stage, with reasonable accuracy £ > 0 and knowledge of
the spectral bounds 6^, A„ of the operators Aa simple algebra gives ?], p, 5,
r by formulas A4)-A5). When providing a prescribed accuracy e > 0, that
is, II Vn ^ "" II S s II % ^ " 111 ii^ i® necessary to perform n = n{e) iterations
that can be most readily evaluated by the approximate formula
A6) nie)
In the new notations
14 4
—■ In- In-
^ 2/,^ 1 2\ 2j-l .
i = li2,.
16 V 2 ' / ' 2n
the quantities cj,- are specified by the formulas
A + 2^)A+^'^)
"^i 2^<^/2(l+^i-<^+^i+<^) '
In agreement with A0) the undetermined parameters become
^ 1 + CJ^. P ' ^ 1 — W^' p '
making it possible to solve problem G).
It should be noted that for particular values 81 = 82 =^ 5 and Aj =
A2 = A formulas A4)-A5) give x = ,J, p = r = 0, 9 = 1/A and ^ =
A — »])/(! + 7]), 1] = E/A. Via transforms (9)-A0), amounting for now to

Itei'ative alternating direction methods 717
Ai = AA'i, A2 = AA2, cjC'-) = At'^^^ and cj^^) = Ar^^), we might have
-^A) = -^B) — J- under the condition cj^'-) = cj(^^.
For the model problem C7) posed completely in Section 2 we obtain
giving in combination with A6)
1 27 4
nie) Ri0.2 1n-^— In-.
n e
Thus, for example, we find that n{e) ?» 6 for h = 1/10, n{e) k, 9 for
h = 1/50 and n{c) ?» 11 for h = 1/100 with reasonable accuracy e =
2e-iORilO-^
4. ADM for the case of noncommutative operators. Of special interest is
the equation of the form
ylH = (^1 +yl2)w = /
with noncommutative operators Ai and A^ still subject to conditions E)-
F):
Ac = Al>{}, S^E <A^<A^E, <5„>0, a = 1,2.
These properties have had a significant impact on modifications of
iterative methods and provide the possibility of applications of the two-
parameter iterative ADM:
A7) {E + c^j Ai) j/,.+i/2 = (^ - ^1^42) Vk +ioJ\
{E +Lj^A2)yk^i = {E-Lj^Ai)y^^ij2+^2f ,
given Vo^ H , i = 0,1,2,... ,
where a proper choice of parameters cjj and cjj will be substantiated a little
later for some reason or another.
Consistent with Zj._|_i = ■t/j._,_j — u and Zj^j^ij^ = yk+1/2 ~ " ^^ ^^^ idea
of setting up the homogeneous equations for the error such as
A8) {E + ^^Ai)z,^,,2 = {E-^,A2)z,,
[E + u^A2) Zfc+i -{E- u^_Ai) 2:^ + 1/2 ,
given Zg ^ H , k = 0, i,2,... .

718 Methods foi' Solving Grid Equations
Having involved v^. = (E + uj^A'2) z^. with furter elimination of ij._,_w3 from
A8), we derive the equation
A9) v,,_^_i = SiS2Vk, Vg^H, k = 0,1,2,... ,
whose transition operator 5 arranges itself as a product S = S[ S2 with the
multipliers
Si={E + cj,A,)-'{E-cj,A,),
B0)
S2 = {E + Lj,A2)-'{E-Lj,A2).
This is showing the gateway to subsequent considerations: since
ll-.+ill<l|.5'iS2||.||.,||,
it is necessary to evaluate the norm ||5'i,S'2|| and find miiiwj.w, H.S'i.S'sH
with the aid of the well-established relation.
A.s further developments occur, we need an auxiliary lemma.
Lemma Let an operator A : H 1-^ H be in line with the conditions
B1) A = A*>0, 6E<A<AE, 6>0.
Then the norm \\S{lj)\\ of the operator S{lj) = (E+ljA)~^[E—ljA) attains
for cj = cj„ = , the minimal value
min II S{uj) II = II ,S(u;J || = , where rj
^ I + ^/r|
, where 77 = -—
For the most part, the proof is connected with further treatment of
S{lj) as the transition operator of the two-layer scheme
B2) (E + cjA)'^^^±^~^ + Ay, = 0, k = 0,l,..., y, & H,
with the operator B = E + ljA involved. This supports the view that
j/fc+i =S{io)yi: , \\yk + i II < pWVk II ,
where p = p{uj) = \\ S{uj) \\ needs to be minimized.

Itei'ative alternating direction methods 719
Along these lines, it is worth noting that we are still in the framework
of the general stability theory outlined in Chapter 6, Section 2 for difference
schemes like B2) asserting that a necessary and sufficient condition for the
estimate ||j/j;_li || ri < P llj/tlln to be valid for any 0 < /> < 1 and any operator
D = D* > Q commuting with A is
^^[E + LoA)<A< ^-±^[E+LoA).
2.L0 luj
This is acceptable if we agree to consider D = E oi D = A. In this regard,
it should be noted that the preceding is equivalent to the bilateral operator
estimate
-E < A< — E , where f - -^^ and p = -^ .
Putting these together with B1) we conclude that
f I 1
B3) - <S, — >A or (,uj < -- , so that (,^ < 1],
it being understood that the minimum of p will be attainable in the case
of the maximal value of ij in B3). This is certainly so with
whose use permits us to establish the chain of the relations
^ = i = v! = _i_=^
and arrive at
min p{lo) = p{uj„) = ——- .
1 + V»]
However, the same result can be obtained by a simple observation that
. 11 — cj A I
p = b{Lj) I = max -——- ,
E<A<A I 1 + CJ A I
where A = X{A) is an eigenvalue of the operator A. Plain calculations show
that
mm p(ui) = min max
w ' ' w i<A<A I 1 + CJ A I
■1 -cjE cj A - 1
= mm max
i-<u.<i \l+ui6 cjA+1
^^ °' 1 + y^

720 Methods for Solving Grid Equations
thereby completing the task of motivating the desired minimum.
Via transforms (9)-A0) available for the operators Ai and A2 and the
parameters cjj and ui^ we accept
— ^ )
q -oJiP q + UI2P
making it possible to reduce the operators specified by B0) to
S = §1 §2 ,
Si =iE + LjA\)-^{E-LjA[),
S2 = {E + LjA'2r^{E~LjA'2).
Their members satisfy the relations
rjE <A'^ <E, a =1,2, r] > 0 .
In the preceding reasoning the member 1] and the parameters p, q, r are
given by formulas A4)-A5). With the inequality ||,S'|| < H^^ || • \\S-2 \\ in
view, the lemma asserts that for cj = cj^ = s/rj/6
min||5i||=min||52||=i^
and a solution of problem A8) obeys the a priori estimate
B4) \\{E + ^,A2)zJ\<p^^\\[E + Lo,A2)zA, P = [\^^)''
with cjj and lj^ still subject to the relations
r^.^ r + q^o gt-^o -r ^
B5) uj, = , cj, = , Lj„ = —— .
In particular, i] = S/A for Si = 62 = S and Ai = A2 = A.
From B4) it follows that the condition p" < e of the termination of
iterations is ensured if
n>n(')(e), «(!)(£) = MlM , where ^ = ^^ •
2^ 1+7]

Iterative alternating direction methods 721
In order to understand the nature of this a little better, we refer to a
model problem posed in Section 2 all over again for which
8,=8^, = 8 = Asm'[^)/h'
.2/''r/l\ 1,2
Ai = A2 = A = 4cos^(—j//j^
^ I „TT h TT h
The main goal of subsequent considerations is the conrparison between
ADM of the type A7) with parameters B5) and the explicit method with
optimal set of Chebyshev's parameters
arising in Section 2 and requiring no less than n^°\e) iterations:
n>n(%), „@)(.) = MM, e = 7./7:
2
In that case j^ - 6.^ + 6^ = 2 6, j^ = A^ + A^ = 2 A and ^ = 6/A
Thus, we might have
n
"BA) „(ll,,^.. MIA
@)(,)^i!VViZ^ „AJ(,
27^ ' ' 4^
thereby justifying that ?j'-'^'(e) fn 2 7i''- \e).
When solving the model problem concerned, the transition from the
kth iteration to the (k + l)th iteration is performed either in 9 steps or in
26 steps: 5 operations of addition and 4 operations of multiplication during
the course of the explicit Chebyshev's method and 12 operations of addition
and 14 operations of multiplication in the case of ADM in connection with
the double elimination (first, along the rows and then along the columns).
This provides reason enough to conclude that in the case of noncommutative
operators the first method is rather economical than the second one. Both
11...
methods require 0{— In-) iterations in the process of solving the model
problem under consideration.

722 Methods for Solving Grid Equations
5. Factorized iteration schemes and ADM. The main idea behind this
approach is connected with the equivalence between ADM from the preceding
sections and the two-layer iteration scheme
B6) B ^^+^ ~^'' + Ayf, = f
T
with the factorized operator
B7) B = {E + Lo,Av){E + Lo,A2)
and the iteration parameter
B8) T = LO^+LO^.
To make sure of it, we rewrite A7) as
B9) 5iyi_^l/2 = C'2?4 -Fu-'i /, 52j/i. + l = Ci J/4 + i/2+^2/.
where B-^ = E + u)^Ai, B2 — E-l-w^A^, Ci = E—w^Ai and C'2 = E — uj-^A2.
The elimination of the iteration yk+1/2 can be done by successively applying
the operator C'l to the first equation and the operator B2 to the second one.
Combination of the resulting equations with further reference to the useful
relations BiCi = CiB^, ui^Ci + ui^Bi = Wj(£' ~ w^Ai) + w^{E + w^Ai) =
(wj +u)^)E gives
C0) Bi B2 j/fc+1 = C\ C'2 Vk + (^1+^2)/.
It seems worthwhile to reduce equation C0) to the canonical form B6) by
observing that B1B2 — C1C2 = (wj +lj^)(Ai -|-A2). The arguments in
reverse order are obvious.
Let us stress here that the applications of the above framework to
noncommutative operators Ai and A2 could result in wrong reasoning in
light of the property that operator B7) is non-self-adjoint and scheme B6)
does not fall within the category of two-layer iteration schemes lying in the
fundamentals of the general theory.
Before going further in more detail on this point, it is worth
mentioning that the factorized operator B7) is self-adjoint and positive: B = B* >
0. To decide for yourself whether the obtained results are acceptable for
commutative operators Ai and A2 and conditions E)-F), the first step is
to discover from C0) the structure of the transition operator of scheme B6)
such as
S = S'l 52, 5'a = 5~ C'a , a = 1, 2 .

Iterative alternating direction methods 723
We learn from Section 4 the values of parameters Wj and uj^ for which
the minimum p of the norm || S[oj) \\ is attained:
^1,^2 VI + y?;/ 1 +,J 1 + 7;
where 7^, w, and 0K are given by formula.s A4), A5) and B5). Knowing p,
0 0
u)] and ^2, it is simple to compute the constants of equivalence "; j and 7 ,,
for the operators B and A such that
C1) 1,B<A<1,,
which will be needed in subsequent discussions of the operator B from a
viewpoint of the possible general theory.
A necessary and sutEcient condition for the p-stabity of scheme B6)
is
IzJLb<a<'±P-b
T T
with assigned values of p and r = Wj + w,. The outcome of this is
C2)
l-p
C^l + ^2
l+P
2^
(i+e)(c^, +L0,)
2
Wj+W2 A+0(^1+^2)'
A keystone in the design of an operator of the type B7) is the possible
structure of an auxiliary operator R = R* > 0 being a sum of two operators
i?i and Ro such that
R = Ri + R'2 , i?i i?2 — R2 Ri ,
Ra^Rl, S^E <Ra <AaE , a =1,2,
by means of which the factorized operator in question reveals to be
C3) B = {E+u.'Ri)iE + ujR2).
0 0
This serves to motivate instead of C1) the relations 7 ^B < R < f 2^ with
constants 7j and 72 arising from C2).

724 Methods for Solving Grid Equations
When the operator R happens to be a regularizer for the operator A,
that is, Cji? < A < c^R, the constants of equivalence for the operators A
and B such that fiB < A < ■y^B become
The intention is to use such a factorized operator in the explicit
method with optimal set of Chebysliev's parameters:
5~^ti -.j^Ay^-j^ k -1,2,... ,n, Vo & f ,
which requires no less than ?Iq(c) iterationy:
In complete agreement with B2), rj is expressed through the parameters 6^,
6^, Aj and A2 of the operators Ri and R2 involved.
It is worth noting here that the same estima.te for ni^[e) was established
before for ATM with optimal set of Chebyshev's parameters, but other
formulas were used to specify rj in terms of E^ and A„. If R = —A, where
A is the difference Laplace operator, and the Dirichlet problem is posed on
a square grid in a unit square, then
„TT h
for both cases: ATM and the factorized scheme B6)-B8) relating to ADM.
Just for this reason these methods require the same number of iterations. As
a matter of experience, ATM is more economical and preferable, since one
iteration necessitates performing a smaller number of arithmetic operations.
Some consensus of opinion here is to accept B — R and then find j/j._|_| from
the equation
RVk+i = Fk , Fk ^ RVk- Tk + i (A Vk - f) .
by one of the available direct methods, say by the decomposition method
or Fourier fast transform method.
Summarizing, we are somewhat uncertain in which situations scheme
B6)-B8) would be more better than, for example, ATM.

Iterative alternating direction methods 725
Remark 1 From such reasoning it seems clear that ADM with variable
parameters is equivalent to the two-layer scheme B6) with iteration
parameter T = Tj. = Tj, + Tj, and factorized operator
5i=(£ + rl'Ui)(£ + rf A2).
We will not pursue analysis of this: the ideas needed to do so have been
covered.
Remark 2 If A is a sum of p > 2 pairwise commutative operators such
that
A= J2 A^, a; = ^„ > 0 , 6^E<Aa< A^E, 6^ > 0 ,
a = l
a = 1,2,..,,];, AaAfj = A^Aa , a,/? = 1, 2,... ,p ,
direct applications of ADM described by A7) is impossible in principle,
so there is a real need for constructing scheme B6) with the factorized
operator
B, = fl(E + Ti"^A,), r/"'>0.
a = l
In this regard, we are unaware of any exact solution to minimax problem
and the so-called cyclic set of the ensuing parameters may be of help in
achieving the final aims.
The main idea behind this a.pproach is connected with the equation
related to a new iteration j/j,^i:
'[liE + Ti"'^Ac,)y,+, = F,,
which reduces to successive solution of p equations
{E + Ti'^AOy('^ = F,, (£' + r^")A„) ?/") = :(/"-!>, a = '2,...,p,
with J/,;, I 1 =: j/*^^-* incorporated.
The difference Dirichlet problem for Poisson's equation in a p-dimen-
sional unit cube such as
AaV = -AaV , AaV = Vg^^.^^ ,

726 Methods for Solving Grid Equations
helps clarify what is done. As a matter of fact, searching for yf;^i amounts
to .successive elimination along the directions x^^x^^ ■ ■ ■ j ■'^« relating to
algorithms of the ADM-type. The availability of the cyclic set of
parameters gives us hopes to achieve a prescribed accuracy e by making n^fe) =
0(ln(l//i) ln(l/e)) iterations, where h — h^ = h^ = ■ ■ ■ = h„ is the grid
step. On the other hand, it is no difficult to achieve what we suggest by
means of ATM with the operators
p ^ P ^
a = l " a~l "
This applies equally well to the problem posed above and requires no less
than ng(e) iterations, where
rigle) K 7= = O —^ In -
meaning that the asymptotic behavior of ATM becomes worse in
comparison with ADM. However, when p = 3 and /; > 1/60, that is, the tota.l
number of the grid nodes < 2.16 ■ 10^, a smaller number of iterations is
performed during the course of ATM in contrast to ADM with the cyclic
set of parameters. With regard to the work and storage required, ATM
being rather economical (in 2-2.5 times) offers more advantages than ADM
on a.ny admissible grid no matter how it is chosen.
6. ADM for non-self-adjoint operators. The equation we must .solve is of
the form
Au = (Ai +A2)u = f,
where /li and A2 are non-self-adjoint positive definite operators subject to
the conditions
C4) Aa>S^E, A-^^>-—E, E„>0, A„ > 0, a =1,2.
As can readily be observed, the second condition is equivalent to the
inequality
C5) WA^yf <Ac,{Acy,y).
Indeed, by merely setting a; = /!«;(/, we are led to
0 < (A~^x, x) - -— {x, x) = (y, Aay) - -r- [A^y, Aay) ■

Iterative alternating direction methods 727
In trying to solve the equation Au = f the iterative ADM with
iteration parameter to is good enough for our purposes:
(£' + wAi)yi,+i/2 = {E -LjA2)yk +^7,
C6)
{E + uiA-2)yk + i = {E -coAi)yk + i/2 +w/.
By exactly the same reasoning as in Section 4 with regard to the error
^k+i = Vk+i - w we deduce for v,._^_^ = (£■ + CJA2) Zk + i that
Vk + i= SiS2Vk , Sa = {E + wAay^iE-uiAa), a =1,2,
Ih. + i II < 1151^2 II ■ ||t'.||<||.5i||- II 52 II .||.,||,
We are going to show that under conditions C4) the following estimates
hold true:
/Q7^ II C l|2 / t -^o 2(^„CJ
C7) ll^"ll<TT^' "«=r+c.25„A„' ^='''-
Indeed, from the obvious identities
II (E - LoA^) X f = \\{E + LoAa) X IP - 4 w (.4„x-, x),
II {E + LoAa)x IP = II X IP + Lo''\\ Acx \f + 2w {A^x, x),
which a.re put together with inequalities C4)-C5), it follows that
II {E + ujAa) X IP < (i/6„ + u;2 A„ + 2u>) [Aax, x),
(A„x,x)> 1" 9,,, \\iE+c^Aa)x\\\
||(i?-c.A„)x|p< 1:1^ ||(£ + c.A„)x|p.
By inserting here x = {E + uiAa)~^ we find that
l|5„j/|p<i^||j/|P

728 Methods for Solving Grid Equations
as required, By virtue of C7) it is plain to establish the relations
i + Xj 1+^2
Here the parameter w is so chosen as to satisfy the mininiiun condition for
the function _F'(w) by discovering that the functions i^i(w) = A —X])(l+Xj)
and F2{oj) = A — x^) A + x^) attain the minimal values, respectively, for
(jj = l/\/S^Ai and ui = Xj\JK^^2.. These points are are identical when
EjAi = i^^'i- III that case for uj — l/^yo^Ai = 1/^'6^1:^2 the inequality
^a=X^, a =1,2,
holds and the error 2„ = j/,, — tz admits the estimate
C9) \\{E + ujA2)zA<p''\\{E + ojA-2)z,\\.
If EjAi ^ <5,A2, then uj — l/v<5A, where ^ = mai{8^,8^) and A =
max(Ai, A2).
In comparison with the case of self-adjoint operators the number of the
iterations is being increased in two times. This is clearly seen, for example,
from C8) by setting b^ = S^_ = S and Aj = A2 = A:
It should be compared with
p
1 + nA^ '
emerging from estimate C9) with regard to self-adjoint operators Ai and
A2- One needs to exercise good judgement in deciding which to consider.

other iterative methods 729
10.5 OTHER ITERATIVE METHODS
1. Three-layer iteration schemes. So far we have considered two-layer
iteration schemes available for solving operator equations of the form Ait =
f with a self-adjoint operator A under the assumption that the spectral
bounds 7j and -y^ for the operator A are known in advance either in a space
H or in a space Hb, where B = B* > 0 is some stabilizator. Other iterative
methods find a wide range of applications in some or other aspects.
An excellent start in this direction is to describe three-layer (two-step)
iteration schemes in the general setting due to which it is required to solve
the equation
A) Au = f, A: H i~^ H,
with a self-adjoint positive definite operator A, whose spectral bounds are
already known:
B) A = A\ f,E <A<r,E, 7j>0.
The links between three iterations j/^__2, j^, and Vk+i are provided by
three-layer iteration schemes, by means of which j/j,,! can be expres.sed
through the values j/j.„i and j/^. The standard form of explicit schemes is
C) !4-M = (i + aM't/fc-aj/fc„i+ (l-Fa)ro/, A; =1,2,...,
Vi = Syg + Tgf , given y^ ^ H ,
where S = E — t^A is the transition operator for the two-layer simple
iteration scheme with the optimal parameter r^
D) -0 = ^, <^^ = P]. P. = \=r4~ ^=--
Ti +72 i + V^ 72
The two-layer simple iteration scheme permits us to find the first iteration
yi-
In an attempt to create scheme C), equation A) should be represented
in the so-called "preliminary" form
u = u- tAu + rf = Sir) u + rf , S{t) = E - tA,

730 Methods for Solving Grid Equations
and the parameter r should be so chosen as to minimize the norm ||5'||, We
now know from Section 2 that this aim can be achieved by setting t = t^,
so that the preceding becomes
E) u = S{Ta)u + To f ,
allowing alternative forms of writing:
{l + a)u={l + a)Su + {l + a)Tj,
u = (I + u) Su — a u + {I + a) r^f .
Having replaced (l + a)Su by (l + aMj/^ and au by aj/^_j, we try to adapt
the explicit scheme, the parameter a of which needs to be selected by the
approved rule in a minimal number of iterations. Unfortunately, more a
detailed exploration on this point and the convergence of scheme C) are
not available in the present book. A final result can be obtained through
such an analysis by utilizing the fact that the residual rj. = A Hf. — f satisfies
the homogeneous equations
F) r^^i = (l + aMr^-ar^_i, A; = 1,2,...,
ri = Sr^, I'a = Avo - f & H ,
no matter how the initial value r^ is chosen.
In a revised statement of the problem for a = p^ the estimate
G) \\Ay„-f\\<qJ\Ay,-f\\
is valid with
(8)
assuring
^n=/>'/(!+i+;p)
1 / 1 - pr
ln- + ln 1+- ^
In —
Pi
1
")
This is certainly so with
,„1+,„(! +Mil)

other iterative methods 731
Comparison of (8) with the resulting expression for q„ = —^—^, valid for
2/>r
1+pf
Chebyshev's scheme, shows that both schemes are of the same asymptotic
order as cj ^ 0:
n — n{e) — O ( —= In - j.
Here n is, as usual, the number of iterations. But in practical
implementations Chebyshev's scheme with known va.lues -j^ and j^ i^ preferable,
because the extra iterations and storage are necessary for later use of the
three-layer scheme concerned. What is more, the second scheme depends
more significantly on the errors in specifying j^ and j^ than the first one.
Remark The passage from explicit three-layer schemes to implicit ones
can be accomplished by the replacements of A by B^^A and / by B~^ f,
so that
yk+
1 = A + a) (£ - r„ B-'A) y, - aj/,_, + A + a) r„ fi'V
A0) 5j/,+i = A + a) E - T,A) y, - aBy,_, + {I + a) t, f ,
By, = B yg - Tg Aijg + Tg f ,k = 1,2,... , given y^ ^ H .
Observe that equation A0) emerged from the identity
5u = (H- a){Bu - TqAw) - cyBu+ A -\- ajr^f
with further indication of iteration number in the appropriate positions, if
any. formulas D) for r^ and a are still va.lid with known spectral bounds
7i and -^2
A1) liB <A<f^B, 7i>0, 5 = 5* >0,
for the operator A acting in a certain space Hb (not in the entire space
H), making it possible to establish for a solution of problem A0) instead
of G) the estimate
A2) ll^J/n-Zlls- <9nll^J/0-/|liJ-
with q^ still subject to (8). This is acceptable if operator G6) is involved
in the framework of ATM in Section 3.

732 Methods for Solving Grid Equations
2. The minimal residual method. Until now the bounds 7j and 72 of the
operator A were known before giving special investigations. But it may
happen that these constants are either too rough or, generally speaking,
indeterminate in advance. In ma.stering the difficulties involved, variational
iterative methods sucli as the method of conjugate gradient, the minimal
residual method and the method of steepest descent can be employed in
the further elaboration on this subject.
We confine ourselves here to the minimal residual method and the
method of steepest descent relating to two-layer schemes. As usual, the
explicit scheme is considered first;
A3) ^'+'"^'+At/, = /, fc = 0,l,2,..., given ?/„ei^,
or
A3') Vk+i = Vk - Tk+i Tk, rk = Ay^- f ,
where r^, is the residual.
The only difference between the methods we have mentioned above
lies in the selection rules for the parameter Tf,,^. In the minimal residual
method the choice
A4) T,+, = Tjjrw . where r, ^Ay,-f,
is stipulated by the minimum condition for the norm || 7'^^| [| of the residual.
In this context, several Cjuestions are yet to be answered. The equation for
the residual
A5) ^k+i-r, ^^^^^Q^ k = Q,l,2,...,
'''k + l
implies that
A6) lk.+ill' = lh'Jl'-2r,+:(.4r,,r,) + r2^J|Ar,||2.
The right-hand side of relation A6) is the polynomial P2{T^.^f^) of degree
2 with respect to the parameter rj,^|. Equating the derivatives P2iTk+i)
to zero reveals Tf.^^ in complete agreement with A4). Because the second
derivative for that value of the parameter r^^j is positive, the quantity
II r^^j II of interest tiniis out to be minimal.

other iterative methods 733
Adopting those ideas to the case of a non-self-adjoint operator A, we
derive from the foregoing the a priori estimate
A7) II r,+, II < p, II r, II , meaning || A y„ - f \\ < p"^ || A y, - / || ,
where Po '= [^ — 0/A + 0' ■? ~ 7i/72 ^'^^ 7j, 7, are the accurate bounds
of the operator A — A* > Q.
Indeed, for the value r^^j assigned by A4) the right-hand side of A6)
is minimal for fixed r^ £ _ff. Due to this property it is being increased for
all other values and, in particular, for t = Tq. The meaning of this is that
we should have
Ik-fe+ill^ < ll''JI"-2T-o(Ari,,r^)-Fr^2[|ArJ|"
<\W-roAr,f<\\E-T,Af ■ \\r,\\\
lk-t+ill<ll^-T-oA|| ■ ||r,||,
On the other hand, we learn from Section 2 that ||-E — t^AW = p^ for
Tg = 2/Gj -1-72I thereby justifying estimate A7) and the convergence of
the minimal residual method with the same rate as occurred before for the
simple iteration method with the exact values 7j and 73.
During the course of MRM the same procedures A3') and A4) are
workable with increased volume of calculations in connection with formula
A4) for
Tk+i a.s compared with the simple iteration method.
The implicit minimal residual method can be designed in line
with established practice:
A8) Vk+i = Vk- T-k+^w^., w^ = B~\Ayk ~ f),
which is referred to as the minimal correction method, In that case
instead of the governing equation Au = / we are dealing with
A9) Cv = ^, v = B^I'^u, C = B-^I'^AB-^I'\ ^ = .S^^/^f.
Applying here the explicit minimal I'esidual method yields
^k + l = ^k ~ '''k + li^' ^k ~ f) J
where
{Ch.Cfk)

734 Methods for Solving Grid Equations
The forthcoming substitutions x^ = B^^^y^, C = B'^^^AB"^/^ and if =
B^^l"^f allow us to modify these into
h = B-'l\Ay,^f) = B-'l\,,
{Cf„f,) = {B~'l'AB~'r„B-"\,) = {Aw„w,),
{Ch;Ch) = {B~'f'Aw„B-'/'Aw,) = {B-'Aw,,Aw,),
leading to equation A8) with the correcting term lUf. and parameter Tj.^^
such that
B0) ^.. = 5-V,, r,,,= (^--^'^^^
iB-^Aw„Aw,)
Instead of A7) we eventually get one more useful estimate
B1) \\Ay,,~f\\s~. </>oll^J/o-/|ls--
3. The method of steepest descent. The explicit method of steepest descent
is given by the formulas
yk+i=yk-'^k+iiAyk - f), k = 0,l,2,..., given yo e H ,
where
B2) T,^,= ^'fl\, r,^Ay,-f, k = 0,l,2,...,
arising from the minimum condition for the norm of the error Zf. = yj. — u
in the space Ha, meaning
,min IUa-+iI1^i. \\-A\a = x/T^^-^) •
{^k+O
The error Zf. — yf. — u satisfies the equation
^k + \ = '^k ~ '^k + l A'^k ■
By interchanging the variables V). = A^I'^Zi. we are led to
B3) '^k+i = Vk-T^+iAvk.

other iterative methods 735
Having squared their norms, we find that
B4) II Vk+i IP = II Vk IP - 2 r,^i{vk,Av,) + r^^^ \\Av, |p,
giving
B5)
'k + l
\\Av,\r^
under the above condition min jlfi^+iir"*. The arguments of the preceding
{^yfe + ll
section serve to motivate the estimate
B6) II -^n II < Police II .
Thus, it remains to return to Z). = A^^I'^Vf^ by observing furthermore that
A^k = MVk -u) = Ayk - f = ri,,
{Av„v,) = \\Az,\\' = \\r,\\\
\\Av,f = {Ar,,r,).
All this enables us to transform formula B5) into formula B2) for later use,
Moreover, from inequality B6) it follows that
B7) \\yn-^\\A<Po\\yo-^\\A'
since
W^n W'^ = (I'n.l'n) = iAz„,AzJ = 11 2„ ||^ ,
No doubt, several conclusions can be drawn from such reasoning. First, the
method being employed above converges in the space Ha with the same rate
as the simple iteration method although it occurs in one of the subordinate
norms. Second, the minimal residual method converges in the space Ha^,
that is, in a more stronger norm.
By analogy with Section 2 the implicit method of steepest descent is
described by
B8) gyk+i-Vk ^j^y^^j^ fc = 0,l,2,..., given y, & H,
'''k + l

736 Methods for Solving Grid Equations
with parameters
B9) r,+i = -i^Al!^, ,,^^B-i,,^^ r, = Ay,-f.
{Awk,w,,)
Estimate B9) is still valid in that case under the following conditions;
fiB <A<j,B, Ti>0, B^B'yO.
4. On solving equations with non-self-adjoint operators. In dealing with
an equation
Au = f, A: H i~^ H ,
where A is a positive definite non-self-adjoint linear operator, the intention
is to use one of the stationary iterative methods associated with the two-
layer scheme, the parameter of which is constant:
^3Q^ yk±^ZJk+Ay,=f, k = Q,i,2,..., given y^ ^ H,
T
The homogeneous equation for the error Zj. =z y^, — u amounts to
z^^y=Sz^, S=E-tA, fc = 0,l,2,,,,, Zg e H,
permitting to esta.blish the convergence rate of iterations. At first glance,
II ^k+i II ^ II 'S' II ■ II '-k 11- ^"^s before, a proper choice of the parameter r is
stipulated by the condition niin^ ||'5'(t) ||.
In such a setting the assumptions are made on the lower bounds of
the operators A and A~ .
A>j,E or {Ay,y) >7,\\y\\\ 7i > 0,
C1) 1
A-'>~E or II Ay\\- < 7, {Ay, y), 7, > 0 .
A simple observation that the second condition for the case where
A = A* is equivalent to the condition A < j^^ may be of help in further
elaboration on this subject. Provided the condition 2 — r72 > 0 holds, we
are able to arrive at the chain of the relations
\\Sy\(' = \\y-TAy\\'' = \\y\\''-2TiAy,y) + T'^\\Ayf
< \\yW'-2T(Ay,y) + T''j,(Ay,y) = || j/||'- r B - r72) (A j/)
<||j/||2~rB-r7,O, ||j/||2 = (l-2r7, + r2^,7,)||j/||2.

other iterative methods
737
yielding
||5||2<l-2r7,+r^,7,.
Having stipulated the minimum condition for the preceding trinomial, the
parameter r is found to be r = I/72, so that || ,S|p < A ~ I1II2) o^'i what
amounts to the same.
C2)
S\\<^/l^i for r=l/7,, ^ = 7i/72 •
In an attempt to generalize the results obtained to the case of three
parameters 7j, 7,, 73 , the operator A arranges itself as a sum
C3)
.4 = Ao + Ai, Aa = i(A + A*), A,^\{A~A*).
Here Aq is a symmetric operator and Ai is a sketch-symmetric operator,
so that
Aq = Ao , A\~-A\, {A\x,x)~-[x,A\x)~'^,
meaning {Ax, x) = {AqX, x). Let the the members of the operator A be in
line with the conditions
C4)
7,E<Ao<7^E,
Ai II < 73 .
where 72 > 7i > 0 and 73 > 0 are known numbers.
An alternative form of the equation ^^ 1 j = {E — TA)Zf, for the error
^k + l ~ Vk + l - W IS
C5) z,^, = {E -T Ao-T Ai) z, ^ {0 E -tAo) z, + [{l -0) E -T Ai) z, ,
where the number 0 < (? < 1 is free to be chosen in any convenient way.
The main goal of further development is to minimize the norm
||5|| = |l^-r(Ao+ Ai)||
by observing that on account of the triangle inequality
C6)
Iki+i II <
E-
Af
2*
A-,
tAi
To this end, the numbers r and 6 are so chosen as to satisfy the minimum
condition that we have mentioned above. As far as 7j£' < Aq < 72^' for a
self-adjoint operator Aq, we might have
C7)
mm
r/e
E-
■Ao
= Po for
7i + 72

738 Methods for Solving Grid Equations
where
so that r = r^d. The second suramand on the right-hand side of C6) will
be the subject of special investigations. The outcome of this is
\\{\-e)y-TA,yf = {\-ef\\yf-2T{\-e) {A,y, y) + r^ || Ai y f
= {l^ef\\yf + r^\\A,yf
which assures us of the validity of the following inequalities for r = t^O:
\\^k+A\<\\s\\-\\zA, \\s\\<m,
C8)
m = ep, + ^{i-ey + e'^a^, a' = r^ ^l
In order to find the minimum of the function f{0), we calculate and analyze
its derivative
The equation /'(O) = 0 gives p^ \/a'^ + a^ ~ a — a^. It may be viewed as a
quadratic equation with respect to a:
{l-pl)a--2a'a + a''{a'-pl) = Q
with the first root
a + Po ^\- pI + a2
a = a .
We note in passing that the second root is unacceptable in connection with
the possible negativity for some value of the parameters a and p^. Also, it
will be sensible to introduce the notation h — Ts/ \/l\ t2 + tI > ^'^ that
2_ >^^ '2 _ 2 2 _ 4 7, 7, X^ A-Pq)
^3 - 1 _ ^2 ^1 ^2 . « - ^0 73 - (^^ ^ ^^^2 1 _ ^2 - 1 _ ^2
x2

other iterative methods 739
When providing current manipulations with
" i — x^ 1 — x"^ x^
we find that
a
_ a^(>^+PQ) _ >^(>^ + Pq)
1 + a
l + x/>o
1 + a 1 + x/>o
and, consequently, the expression for the function
fiO) = T—- Po + T—- \/a' + «' = ° ^ , •
1 + a 1 + a A + a) Po
Taking into account that
pI + a- - o^ = A + a) - A - pI + «2) = i + a - ~
l + x/>o 1 -Po _ Po
X
1 - X2 1 - x2 '^" 1 - X2 '
= Po
it is plain to show that
b\\ < tor r = Tn
l + x/>o l + ^^Po
The meaning of this is that a solution of problem C0) satisfies the
estimate
Wvn -m|| <p" IIj/o - '"II
with
C9) p= " + ^" , x^ , ^^ , r^r^"°(^-"'\
1 + ^Po \/7i72 + 7| l + >*Po

740 Methods for Solving Grid Equations
provided that conditions C4) hold. The number n of iterations for doing
so is no less than ng(^e):
" > '^o(^) > In - / In — .
P
What can happen upon replacing the describing explicit scheme by an
implicit one? Being concerned with one of the implicit .schemes
D0) B^'+'~'^' +Ay,^f, k = 0,l,2,..., y, e H ,
'''k+l
with a self-adjoint operator B = B* > 0, the intention is to employ the
explicit scheme
^k+i = ^k - T (Cxk - f), Xk = B^^'y^. ,
with the members
and rearrange conditions C4) with rega.rd to the operator C as revised
conditions for the operators ,4 and B:
D1) 7,B<Ao<j,B, {B''A,y,A,y)<'rl(By,y),
WMyWs-^ <li\\y\\B and ||j/„-'"lis </'"ll?/o-"lis
in contrast to the preceding estimate just established.
The minimal residual method can be employed for the equation Au =
f with a non-self-adjoint operator A, the convergence rate of which
coincides with that of scheme C0) for t = f.
For the explicit scheme A3) we have
Vk+i-Vk ^^y^^f^ fc= 0,1,2,..,, y,^H given,
'''k + l
where the parameter r^., ^ i.s chosen by the approved rule

other iterative methods 741
(for more detail see Section 2, formula A4)) governing the minimum
condition for II rj, I 1 Ip. No property of the self-adjointness of the operator A
applies here and below. In what follows minor changes will appear in
connection with the necessity of proving its convergence. In particular, f will
stand in place of t^.^^ on the right-hand side of identity A6) and will result
in the relations
lk,+i|P<||r,||2-2r(Ar„r,) + r2||Ar,|p
= \\r,-TAr,\\''<\\E-TA\f ■Wr.W'
<p'\\r,\\\
thereby assuring || r^^j II < p|| I'k \\ and justifying the estimate
IMy„-/||</>"||Aj/„-/||.
This is certainly true for the minimal residual method A3)-A4) under
conditions C4). Here y„ is a solution of problem A3) and p is specified by
formula C9). The fastest move method is useless in that case because the
opera.tor A is non-self-adjoint in such a setting.
Observe that under conditions C4) the estimate
II^J/„-/|l5-</>"l|Ayo-/IU-
is valid for the implicit minimal correction method with B = B* > 0
incorporated.
5. Gibrid (combined) methods. In mastering the difficulties involved in
solving difference elliptic equation,?, .some consensus of opinion i.s to bring
together direct and iterative methods in some or other aspect.s as well as
to combine iterative methods of various types (two-step methods). All the
tricks and turns will be clarified for the iteration scheme
D2) Bk^'+'~^' +Ay, = f, y,^H given,
'''k + l
yielding
D3) Vk+i =yk- Tk+i 'Wk .
where Wj. is the correction satisfying the equation BkWj. = r^. and Tj. =
Ay^ - / is the residual.

742 Methods for Solving Grid Equations
Let now R = R* > Q he a regularizer such that Cji? < A < Cji?,
Cj > 0. Knowing W/,, it is plain to find the {k + l)th iteration Vk+i in line
with D3), so there is some reason to be concerned about this.
The operator Bk can be specified in a number of different ways and,
in particular, in explicit form. One way of covering this is
D4) Bk = R,
D5) Bk = {E + w'f}^R,){E + wi'^R2),
where i?i, R'j are economical operators and coj. , u)\. are iteration
parameters. We dealt with the factorized operator D5) during the course of ATM
with Ri = R^, ui[^'^ = w'-p = w as well as of ADM with R^ = R„ > 0,
Of ;= 1,2, Ri i?2 =^ R2 Ri-
a) A direct method for determination of the correction. Let Bk = R
and one of the available direct methods will be employed for solving the
system of algebraic equations Riu = r^. We refer the users to Section
1 (items 2-3) in which such methods have been designed for elliptic grid
equations. Among them the decomposition method with j^ = Cj and 7, =
C2 is highly recommended for practical implementations. By the selection
rule for Chebyshev's parameters t^jT^,... , r„ we derive the estimate for
the number of the necessary iterations
n > n^is) ,
In this regard, it should be noted that for the difference elliptic
problems posed in Section 3, item 7, «o(^) i^ independent of the grid step. What
is more, when G is a rectangle and h^ = h^ = h, the work in doing this is
b) An iterative method for determination of the correction. The
starting point for subsequent considera.tions is the equation Rw^. = 7'^,, which can
be solved by means of some (internal) iterative method, whose use permits
us to find u)'-™-' = W)., where m is the number of the internal iteration.
Further reduction of a two-layer iteration scheme with the
accompanying self-adjoint operators to an explicit one by replacing the variable w
by R^l'^w leads to
i?i/2(^„(™) _ ,^) = T,„ iZi/2(t„(o) - w),

other iterative methods 743
where w is the exact solution to the equation Rw = r^ and T,„ is the
resolving operator subject to the relations
D6) T,l = T,n, \\n,\\<q<l.
Here the dependence q upon m is omitted. This should cause no confusion.
With the initial value w'^°-' = 0 in view, we find that
Upon substituting this expres.sion into the equation Rw = rj. we obtain the
equation BiUf, = r^ with the members
B = R'/\E-T,^)-'R'/\ w, = w(^-\
thereby completing the task of searching for the correction. By the same
token, j/^^i = j/fc -Tk+iw^.
Still using the framework of the general theory, it is straightforward
to verify that the operator B is self-adjoint [B ~ B*) with the aid of the
relation.s R = R* and T*-^ = T„j. We shall need yet the constants j^ and 7,
of the energetic equivalence of the operators B and A. Beca.u.se of D6), we
thu.s have
(l-g)^<^-r,„<(l + fi)^,
{l + qr'E<(E~T,„)-' <(l-<i)-'E,
{Bx,x) = {R'I\E - T,„y'R'^''x, x) = {{E - T^y'y, y) ,
where y = R^l'^x. From such reasoning it seems clear that
{l-q)B <R<{\ + q)E,
° °
giving 7i = 1 — (/ and 7, = I -\- q and implying that 7j = c^{l — q) and
I2 = Ci(l +<?).
For Chebyshev's scheme with such an operator we obtain
1 /c.(l + g) , -2
since
^^q {l-q)
' c. A + 9)

744 Methods for Solving Grid Equations
If iZ = -Ri + -R2, -R*, = -R« > 0 and Ri R2 = R2 R\ ADM from Section
4, items 2-3 may be employed, where the minimum condition for the work
in connection with solving the equation Au = / guides a proper choice of
the quantity q.
When the spectrum of i? happens to be unknown in advance, it seems
worthwhile adapting iterative methods of variational type for determination
of the correction. Unfortunately, in this book we have no possibility to say
niore about other coml)inations arising time and again in such matters.

Symbols
w,, = {Xj = ih, h > 0, i = 1,2,. .. , N — I, hN = /} - an equidistant
(uniform) grid on the interval @,/);
Wf^ = {x- ~ ih, h > 0, i = 0,1,.. . , N, hN = /} - an equidistant (uniform)
grid on the segment [0,/];
h - step of the grid ojf^;
X = Xj ~ a node of the grid uj^;
y = t/j = y{x^) - a function defined on the grid w^;
t/,. = (y,^i — yi)/h ~ the right difference deriva.tive at a point x^,
Ug — iVi ~ yi-i)lh ~ the left difference derivative at a point x{,
ysx — (.'A+i ~ 2jA' + yi-i)/h^ ~ the second difference derivative at a point
^h = {*i £ (Oj 0> 2 = 1, 2,... , iV — 1} - a non-equidistant (nonuniform)
grid on the interval @,/);
'^h — {^"i £ P' ^]j ' = 0,1,... , yV, x"o = 0, x'jy = /} - a non-equidistant
(nonuniform) grid on the segment [0,/];
hi — Xi — x-_^ - step of the grid w^,;
/), = 0,5(ft, + /!,+J;
Vt: = {Vi+i -yi)/hi+i,
ys = [yi -%-i)//ii;
Vi = (%+i -ydlh'^
745

746 Symbols
1 ( Vi+i - Vi Vi -Vi-i'
^^■* h- \ h- h-
A list of inner products and associated norms on a grid:
N-l
(y,v)= E Vi'^ih, \\y\\= \/iy, y)\
1=1
N
iy,v]=Y.yiVih, \\ij]] = s/iy, y];
2 = 1
IIylie ~ ^w \yi^i)\'^
ll^ll(_i)=^ E H,?/^^
ijj^ = {tj- = JT, r > 0, j = 0, 1,. .. } - a grid in time;
T ~ step of the grid u)^;
y = y^ = y(t;) - a function defined on the grid w^;
y = y^'+' = yitj+i), y = y^"' = yitj-i);
fit = {y - y)/T, yt = {y- y)/^, j/= = {{/ - y)l{2T);
;;; = x^ = ( a'l ,.... x^^''\ . .. , x^^"^ \ - a node of the p-dimensional
rectangular grid iv;,;
,,[i„) — U ■ .
'^ a ~ ' a '■a^
h^ - step of the grid w^j along the direction a;
y^yix,), y(±i«) = j/(x(±^''));
y.^ = {y^+'-^-y)/K.;
y,. = {y-y^-'''^)/K.;
Q - set of all grid functions given on the grid w^ and vanishing on its
boundary;
H - Hilbert space;
{y, v) - inner product of elements y, v £ H, and associated norm \\y\\ =
\/{y, y);
1^{A) - domain of definition of an operator A;
Ti(A) - ra.nge of values of an operator A;
E - identity operator;
A : H 1-^ H - operator A with the domain T>{A) = H and range Tl{A) C _ff;

Symbols 747
A* - adjoint operator;
A~^ - inverse operator;
S - transition operator from one layer to another;
T - resolving operator;
A > 0 - positive operator;
A > 0 - nonnegative operator;
A > SE, E > 0, - positive definite operator;
ll?/IU = Vi^V' ?/). y^H;
II ^ lU-i = \/(^~V. ^). ^ e i/.

Concluding Remarks
In concluding this book we give a brief commentary regarding the main
results forth in the foregoing. In order to make the book accessible not
only to specialists, but also to students and engineers, we give in Chapter 1
a complete account of definitions and notations and present a number of
relevant topics from other branches of mathematics. The detailed outline
of mathematical models leading to partial differential equations is available
in many textbooks and monographs on equations of mathematical physics.
In particular, we refer the reader to Courant and Hilbert A953, 1962), Go-
dunov A971), Morse and Feshbach A953), Tikhonov and Samarskii A963).
Current exposition follows the best legacies of the past: my first book with
my dear teacher - the late Academ. Tikhonov "Equations of
Mathematical Physics", throughout which the reader can find thorough, advanced-
undergraduate to graduate level treatments of problems leading to partial
differential equations: hyperbolic, parabolic, elliptic equations; wave
propagations in space, heat conduction in space, special functions, etc. with
emphasis on the mathematical formulation of problems, rigorous solutions,
physical interpretation of the results obtained.
The books by Gelfand A967), Samarskii and Nikolaev A989) cover in
full details the general theory of linear difference equations. Sometimes the
elimination method available for solving various systems of algebraic equa.-
tions is referred to, in the foreign literature, as Thomas' algorithm and this
749

750 Concluding Remarks
should cause no confusion. This method was independently proposed by
many authors in the early fifties. The bibliographical sources of Godunov
and Ryabenkii A964) and Richtmyer A957) help clarify its appearance a
little bit. Faddeev and Faddeeva A963) have treated the elimination method
as the well-established Gaussian elimination for systems of equations with
a tridiagonal matrix.
Most of the propositions in Chapters 2-4 are of independent value
although, for the present book they ave used only a,s part of the auxiliary
mathematical apparatus. Some of them were known earlier a,nd the rest
were discovered and proven in recent years in connection with the rapid
development of the theory of difference schemes.
Relevant results from functional analysis, operator theory and the
well-developed numerical methods are given at the very beginning of
Chapter 2. All proofs af the main statements as well as a detailed exposition of
the foundations of functiona.l analysis, operator theory and the theory of
difference schemes are outlined in the textbooks and monographs by Ames
A977), Mitchell and Griffits A980), Morton and Mayers A994), Ortega and
Poole A981), Samarskii A987), Samarskii and Gulin A989). The basic
concepts, notions and mathematical apparatus of the contemporary theory of
difference methods for solving mathematical physics problems completed in
late fiftieth owe a debt to Forsythe and Wasow A960), Richtmyer A957),
Richtmyer and Morton A967), Ryabenkii and Filippov A956). The
theorem on connection between approximation and stability of a difference
scheme and its convergence to a solution of the original problem appeared
in the works of Filippov A955), Lax a.nd Richtmyer A956).
The theory of homogeneous difference schemes for ordinary differential
equations of second order with variable coefficients and, in particular, with
discontinuous ones, was developed by A.N. Tikhonov and A.A. Samarskii
in the middle of 50"^ - in the early 60'*^. The first results in this area
were obtained by Tikhonov and Samarskii A956). Later a, complete and
systematic study was carried out by Tikhonov and Samarskii A961) along
these lines. Special investigations of convergence of difference schemes on
non-equidistant grids have been done by Tikhonov and Samarskii A961ab,
1962). The monograph of Samarskii, Lazarov and Makarov A987) includes
more a detailed exploration of the theory of explicit difference schemes and
schemes of any accuracy order for ordinary differential equations, while the
original works of Alekseevskii A984), Bagmut A969), Chao Show A963)
and Prikazchikov A965) concentrate on special cases in tackling singular-
perturbed problems, equations with singularity, some problem.s associated
with a four-order equation and the Sturm-Liouville problem, respectively.
The notion of the conservatism of a difference scheme played a crucial

Concluding Remarks 751
role in the further recognition and establishment of convergence of
homogeneous difference schemes with discontinuous coefficients (for more detail
see Tikhonov and SamarskiT A961ab)). It was shown therein that
conservatism is a necessary property for convergence of a homogeneous scheme in
the class of discontinuous coefficients. Further development of the
conservatism principle provided by Sa.marskiT and Popov A969, 1992) gave rise
to the notion of full conserva.tism.
Being concerned with homogeneous conservative difference schemes,
A.N. Tikhonov and A.A. Samarskii recommended the design of the
balance method being an integro-interpolation one and providing the
necessary framework for constructing difference schemes on arbitrary irregular
grids. SamarskiT, Koldoba, Poveshchenko, Tishkin and Favorskii A966)
followed these procedures in solving the globa,l problems of mathema.ticaJ
physics under such an approach, We note in passing that the methods based
on Marchuk's identity and well-established by Babuska et al. A966) and
Marchuk A975) are conceptually close to the balance method mentioned
above. A greater gain in accuracy of difference schemes on a sequence of
grids were the main idea behind this approach justified in full details by
Marchuk and Shajdurov A979).
Some modification of the describing monotone difference scheme for
"divergent" second-order equations was made by Golant A978) and Ka-
retkina A980). In Andreev and Savin A995) this scheme applies equally
well to some singular-perturbed problems. Vtirious classes of monotone
difference schemes for elliptic equations of second order were composed by
SamarskiT and Vabishchevich A995), Vabishchevicli A994) by mea.ns of the
regularization principle with concern of difference schemes.

References
Akhiezer, N. and Glazman, I. A966) The Theory of Linear Operators in
Hilbert Space. Nauka: Moscow (in Russian).
Alekseevskii, M. A984) Difference schemes of higher-order accuracy for
some singular-perturbed boundary-value problems. Differential Equations,
17, 1177-1183 (in Ras.sian).
Ame.s, W. A977) Numerical Methods for Partial Differential Equation.s.
Academic Press; New York-London.
Andreev, V. and Savin, I. A995) On the uniform convergence in a small
parameter of Samarskii' scheme and its modification. Zh. Vychisl. Mat.
i Mat. Fiz., 35, 739-752 (in Russian); English transl. in USSR Comput.
Mathem. and Mathem. Physics.
Babuska, I., Pager, M. and Vitasek, E. A966) Numerical Processes in
Differential Equations. Wiley: Chichester.
Bagmut, G. A969) Difference schemes of higher-order accuracy for an
ordinary differential equations with singularity. Zh. Vychisl, Mat. i Mat. Fiz.,
9, 221-226 (in Ru.ssian); Engli.sh transl. in USSR Comput. Mathem. and
Mathem. Physics.
Balakrishnan, A. A976) Applied Functional Analysis. Springer; New-York-
Berlin-Heidelberg.
753

754 References
Berezanskij, Yu. A968) Expansions in Eigenfunctions of ,Self-Adjoint
Operator. AMS: Providence, RI.
Bitsadze, A. A976) Equations of Mathematical Physics. Nauka: Moscow
(in Russian).
Cannon, J. A984) The one-dimensional heat equation. In: Encyclopedia of
Mathematics and Its Applications, v. 23, Addison-Wesley: Reading, MA.
Chao Show A963) Homogeneous difference schemes for a four order
equation with discontinuou.s coefficients. Zh. Vychi.sl, Mat. i Mat. Fiz., 5, 123-
132 (in Russian); Engli.sh transl. in USSR Comput. Mathem. and Mathem.
Physics.
Courant, R. and Hilbert, D. A953) Methods of Mathematical Physics.
Volume 1. Wiley: New York.
Courant, R. and Hilbert, D. A962) Methods of Mathematical Physics.
Volume 2. Wiley: New York.
Dezin, A. A980) General Questions of the Theory of Boundary-Value
Problems. Nauka: Moscow (in Russian).
Dunford, N. and Schwartz, J. A971a) Linear Operators (second edition).
Part I: General Theory. Wiley: New York.
Dunford, N. and Schwartz, J. A971b) Linear Operators (second edition).
Part II: .Spectral Theory, Self-Adjoint Operators in Hilbert Space. Wiley:
New York,
Faddeev, D, and Faddeeva, V, A963) Computational Methods of Linear
Algebra, Freeman&Co,: San Francisco,
Filippov, A, A955) On stability of difference equations, Dokl, Akad. Nauk
S'SSR, 100, 1045-1048 (in Russian); English transl. in Soviet Mathem. Dokl.
Forsythe, G, and Wa,sow, W. A960) Finite Difference Methods for
Differential Equations, Wiley: New York,
Gelfand, A, A967) Calculus of Finite Differences, Nauka: Moscow (in
Russian),
Godunov, S, A971) Equations of Mathematical Physics, Nauka: Moscow
(in Russian),
Godunov, S. and Ryabenkii, V. A964) Introduction to the Theory of
Difference Schemes, Interscience: New York,

References 755
Golant, E, A978) On conjugate families of difference schemes for equations
of parabolic type with lower members, Zh. Vychisl, Mat, i Mat, Fiz,, 18,
1162-1169 (in Russian); English transl, in USSR Comput, Mathem, and
Mathem, Physics,
Kantorovich, L, and Akilov, G, A977) Functional Analysis in Normed
Spaces, Nauka: Moscow (in Russian),
Karetkina, N, A980) Noncouditionally sta.ble difference schemes for
parabolic equations with the first derivatives, Zh, Vychisl, Mat, i Mat, Fiz,,
20, 236-240 (in Russian); English transl, in USSR Comput, Mathem, and
Mathem, Physics,
Ladyzhenskaj'a, 0, A973) Boundary Value Problems of Mathematical
Physics, Nauka: Moscow (in Russian),
La,x, P, and Richtmyer, R, A956) A servey of stability of linear finite
difference equation,?. Comm, Pure Appl. Mathem,, 9, 267-293.
Marchuk, G, A975) Methods of Computational Mathematics, Springer:
New York,
Marchuk, G and Shajdurov, V, A979) Gain in Accuracy of Difference
Schemes, Nauka: Moscow (in Russian),
Mitchell, A, and Griffits, D, A980) The Finite Difference Methods in Partial
Differential Equations. Wiley; New York,
Morozov, V, A984) Regularization Methods for Solving Improperly Posed
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Index
Approximation:
difference, 56
difference of order m, 56, 69, 73
exact, 74
summarized (summed), 691
A priori estimate, 97, 203, 420, 448
Cauchy-Bunyakovskii inequality,
101
Condition:
Courant, 376
first boundary, 8
norm concordance, 56
periodicity, 179
second boundary, 8
third boundary, 8
p-stability, 411
Difference:
left, 2
right, 2
Difference analog of the second
Green formula, 101
Difference derivative:
central, 57
left, 57
right, 57
Difference problem:
Dirichlet, 252
ill-posed, 127
well-posed, 96
Difference scheme, 74, 78, 126
additive, 598, 618
alternating direction, 547
best, 168, 169
conservative, 150
co-equivalent, 231
co-stable, 230
Crank-Nicolson, 303
disbalanced, 150
Du-Fort-Frankel, 324
Duglas-Rachford, 629
economical, 349, 619
explicit, 76, 302, 345, 358, 386
388
explicit iteration, 657
factorized, 565
forward, 302
fully conservative, 532
higher-accuracy, 207, 290
homogeneous, 146, 147
implicit, 76, 386, 388
implicit iteration, 657
locally one-dimensional, 600
monotone, 183
nonconservative, 150
759

760
Index
[Difference scheme]
Peaceman-Rachford, 548
predictor-corrector, 522
purely implicit, 302
Richardson, 323
"rhombus", 324
r-layer
stable, 132, 306, 429
asymptotically, 329
conditionally, 310
in every harmonic, 309
unconditionally, 310, 334
with respect to coefficients,
230
with increased accuracy, 347
symmetric, 303
three-layer, 86, 353, 445
truncated, 213
two-layer, 302, 386
with variable operators, 347
well-posed, 388
weighted, 301, 346, 441
Domain of operator, 42
Eigenvalue of an operator, 49
Eigenvector of an operator, 49
Energy:
identity, 318, 399, 430
inequality, 319
Energy of an operator, 44
Error of the difference
approximation, 57, 69, 159
Equation:
difference, 2
first-order linear, 4, 5
mth-order linear, 3
second-order linear, 4, 7
elliptic, 237
heat conduction, 300
hyperbolic, 84
operator, 140
Poisson, 237, 290
quasilinear, 300
Schrodinger, 349
stationary, 187
time-dependent, 299
transfer, 354
vibrations, 364
Formulae:
"difference differentiation" by
parts, 29
first Green, 32
second Green, 32
Function:
difference Green, 199
grid, 2, 51
pattern, 208
Grid, 2, 51, 246, 248
connected, 250, 259
equidistant, 51
non-equidistant, 52, 67
rectangular, 246
square, 246
Grid step, 51
Harmonic, 308
Input data, 87
Iterations, 653
Kronecker's delta, 201
Lattice, 2
Layer, 302
Lipschitz continuity of an operator,
420
Majorant, 16, 262
Maximum principle, 15, 20, 260
Method:
alternating direction, 711
alternative-triangle, 684
Bubnov-Galerkin, 221
counter elimination, 13
cyclic elimination, 37
decomposition, 645
energy inequality, 113, 316,
374
factorization, 39
integro-interpolation, 150, 215
left elimination, 13
matrix elimination, 651
minimal correction, 733
minimal residual, 733
Newton, 517
n-layer composite of period m,
618
one-step iterative, 653

Index
761
[Method]
right elimination, 10
Ritz, 221
Seidel, 676
separation of variables, 406
simple iteration, 664
steepest descent, 734
summator identities, 228
test function, 150
two-step iterative, 653
upper relaxation, 678
Model, 666
Neighborhood of a node, 256
Nodes, 2, 51, 245, 248
adjacent, 52
boundary, 53, 246, 249, 258,
703
inner, 53, 246, 249, 258, 703
irregular, 249, 250
near-boundary, 249, 250
strictly inner, 250
Norm of an operator, 42
Operator:
additive, 42
adjoint, 44
bounded, 42
commuting, 43
homogeneous, 42
inverse, 43
Laplace, 238, 280
linear, 42
resolving, 391
self-adjoint, 44
skew-symmetric, 425
transition, 386, 391
"triangular", 457
Operator-difference scheme, 384
Pattern, 56
"cross," 241
integral, 146
irregular, 67
Pattern functional, 146, 147
Primary family, 146, 156, 398
Problem:
boundary-value, 7, 75, 82, 108,
120, 136, 360
Cauchy, 7, 74, 75
eigenvalue, 106, 108, 278
periodic, 496
stable with respect to the
initial data and right-hand
side, 384
stationary, 507
Stephan, 523
well-posed, 96, 232
Range of an operator, 42
Regular behaviour, 329
Regularization, 454
Regularization principle, 575
Regularizator, 455
Square root of an operator, 45
Stabihty:
absolute, 389
asymptotic, 329
strong, 232
with respect to coefficients, 232
with respect to the initial data,
389, 403
with respect to the right-hand
side, 389, 411
Temperature wave, 514
Theorem:
comparison, 16, 262
difference embedding, 110, 112,
281
"Through execution" algorithm, 691