3R3. Theory of Difference Schemes. Pure and Applied Mathematics Series, Vol 240. - AA Samarskii (Fac of Comput Math and Cybernetics, Moscow MV Lomonosov State Univ, Moscow, Russia). Marcel Dekker, New York. 2001. 761 pp. ISBN 0-8247-0468-1. $225.00.
Reviewed by VD Radulescu (Dept of Math, Univ of Craiova, 13, St AI Cuza, Craiova, 1100, Romania).
The study of difference schemes is one of the central subjects in Numerical Analysis. Because of the variety and importance of their applications, in particular to Applied Mathematics, difference schemes caused developments in various areas of mathematics.
This monograph is intended as an introduction to difference schemes at the advanced undergraduate and beginning graduate level. The author aimed at breaching the gap that too often exists between engineering and example-oriented textbooks on the one hand, and needlessly abstract mathematical formulations on the other.
The book is divided into ten chapters, followed by a list of symbols and some concluding remarks. After several necessary prerequisites exposed in the first two chapters, the author develops in Chapters 3–5 concrete difference schemes for equations of elliptic, parabolic, and hyperbolic types. Chapter 3 focuses on homogeneous difference schemes for ordinary differential equations, by means of which the author solves the canonical problem of the theory of difference schemes in which a primary family of difference schemes is specified and schemes of a desired quality should be selected within the primary family.
Chapter 4 is devoted to various difference approximations of second-order elliptic equations. The approximation technique for the Laplace operator and formulations of difference boundary conditions are described for regions of arbitrary shape.
Chapter 5 provides the general theory of difference schemes in which it seems reasonable to eliminate constraints on the structure and implicit form of difference operators. Such a theory treats difference schemes as operator equations and operator-difference equations, which are analogs of difference schemes for time-dependent equations of mathematical physics.
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. This chapter also includes many good examples illustrating the practical use of general stability theory with regard to particular schemes to assist the users in subsequent implementations. The author’s strategy is that the stability is the most pressing problem in any algorithm, since it is a necessary rather than a sufficient condition for accuracy.
The main purpose of the next three chapters is to show how the results of the general theory of difference schemes are aimed at starting principles for constructing difference schemes of a prescribed quality. The difference schemes for elliptic equations are viewed as operator equations of the first kind, while the difference schemes relating to analogs of nonstationary equations of mathematical physics are treated as difference equations with operator coefficients in an abstract space of any dimension.
In the last chapter, several economical schemes and iterative methods for multidimensional problems in mathematical physics are developed.
The book is well written and is strongly influenced by the well-known Russian school of Numerical Analysis. The point of view taken here, rigorous presentation without excessive formalism, is however nonstandard.
In conclusion, Samarskii’s book Theory of Difference Schemes is original, interesting, and transmits a message in a clear and efficient way. On the whole, this monograph is an excellent contribution to the literature of numerical methods and computer algorithms for solving mathematical-physics problems. The book qualifies to be a reference work that certainly would be a valuable addition to libraries of universities and research laboratories pursuing research in Applied Mathematics.