zbMATH — the first resource for mathematics

The theory of difference schemes. (English) Zbl 0971.65076
Pure and Applied Mathematics, Marcel Dekker. 240. New York, NY: Marcel Dekker. 786 p. (2001).
This book is an extensive theoretical survey and a textbook on the subject by one of the leading figures of the Moscow school of Numerical Analysis and is presented from the point of view of this school. Besides facts largely known from the monographic literature it contains many concepts and methods less familiar to the ordinary Western reader interested in difference procedures. It also presents the philosophy of the author on the theory of the numerical solution of partial differential equations (PDEs) and discusses, besides classical topics, less known items such as “economical schemes” requiring a minimal number of operations, the “summarized (summed) approximation method” dealing with multidimensional problems, “homogeneous difference schemes” describing schemes which are independent of the concrete problem to be approximated and of the choice of the grid, etc.
The book is very understandably written, definitions and theorems are highlighted. Besides classical PDEs the author also considers more complicated problems such as that for the equations of nonstationary gas dynamics.
Here are the chapters of the book: 1. Preliminaries. 2. Basic concepts of the theory of difference schemes. 3. Homogeneous difference schemes. 4. Difference schemes for elliptic equations. 5. Difference schemes for time-dependent equations with constant coefficients. 6. Stability theory of difference schemes. 7. Homogeneous difference schemes for time-dependent equations of mathematical physics. 8. Difference methods for solving nonlinear equations of mathematical physics. 9. Economical difference schemes for multidimensional problems in mathematical physics. 10. Methods for solving grid equations.

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
35Jxx Elliptic equations and elliptic systems
35Kxx Parabolic equations and parabolic systems
35Qxx Partial differential equations of mathematical physics and other areas of application