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Finite difference methods (Part 1). Solution of equations in \(R^ n\) (Part 1). (English) Zbl 0689.65001
Handbook of Numerical Analysis 1. Amsterdam etc.: North-Holland (ISBN 0-444-70366-7). vii, 652 p. (1990).

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This is the first volume of Handbook of Numerical Analysis intended to cover the basic methods of numerical analysis under the headings: finite difference methods, solutions of equations in \(R^ n\), finite method, techniques of scientific computing, optimization, system theory, and the numerical solution of contemporary practical problems from applied mathematics.
The present volume covers the first part on Finite difference methods, namely, Finite difference methods for linear parabolic equations by V. Thomée [pp. 5-16) and Splitting and alternating direction methods by G. I. Marchuk (pp. 197-462), and the first part on Solutions of equations in \(R^ n\), namely, least squares methods by Å. Björck (pp. 465-652).
The article by Thomē consists of an Introduction and two large chapters (II and III) on the initial value problem (IVP) and the initial boundary value problem (IBVP). The article emphasizes concepts and basic priniples and their application in terms of model problems. Initially, proofs of principal results are given, but as technicalities are building up, the analysis becomes more sketchy. The article reflects that the theory for IVP’s (also for IBVP’s in one space variable) was completed in the 1960’s with few new results published after 1970. The Introduction presents standard material on the heat equation. Chapter II concerns the present state for the IVP for linear parabolic PDE’s or systems with smooth coefficients of both the equations and the approximating difference schemes. This includes a brief discussion of the Lax-Richtmyer theory, Fourier analysis methods, discussion of stability and accuracy for specific finite difference methods, multistep schemes, stability bounds based on ideas by John (1952) and later work by Widlund, and convergence rates, also as related to the regularity of data (using Besov spaces and data smoothing). Chapter III on IBVP’s for parabolic equations with space domains suitable as mesh domains covers the three main approaches, namely, energy methods (stability and convergence obtained from discrete analogs of energy arguments, also for curved boundaries in several space dimensions), monotonicity and maximum principle methods, and spectral methods (spectra relating to operators by Godunov and Ryabenki, methods by Kreiss, quarter-plane problems), and some further special methods.
In the second article, by Marchuk, “splitting” is to be understood in the usual general sense as the decomposition into simpler problems in order to allow for effective use of parallel computation. “The author tried to analyze all basic splitting algorithms, naturally focussing special attention on the Soviet school... which has achieved important results... As for the rest of the world’s experience, it was in our opinion reflected sufficiently while considering the algorithms as well as in the detailed bibliography...” The majority of the about 300 papers listed are in Russian. The chapters are grouped into three large parts, Part 1 on splitting and alternating direction methods themselves, Part 2 on methods for investigating their convergence, and Part 3 on applications (to heat conduction, hyperbolic PDE’s, integro-differential transport equations, Navier-Stokes equations, meteorology oceanology). Part 1 includes componentwise splitting, splitting with operator factorization, etc., Part 2 convergence studies by Fourier methods, use of a-priori estimates, splitting by Fourier methods, use of a-priori estimates, splitting and decomposition for variational problems.
Least Squares Methods by Björck emphasizes numerical aspects and gives throrough discussion, with some overlap with the book by G. Dahlquist and Å. Björck “Numerical Methods” (1972; Zbl 0272.65002) and various more recent relevant results added (about 2/3 of the 300 references given appeared in 1975 or later). The article includes a simple explanation of the basics, numerical methods for the general linear problem (normal equations, QR decomposition by several methods, iterative improvements, weighted problems), methods for sparse problems (storage, sequential orthogonalization, block-structured problems, iteration), modifications (rank-one changes, etc.), constraint least squares, and nonlinear least squares (Gauss-Newton methods, Moré’s trust region method, constraint problems, etc.).
The three articles in this volume present thorough, readable and up-to- date accounts on the main ideas, results, and applications in their respective fields. They will certainly be a valuable help to both the expert and the beginner, and a valuable addition to the existing literature; in fact, a substantial portion of the material presented had so far not yet been surveyed in monographies.
Reviewer: E.Kreyszig

MSC:
65-00 General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to numerical analysis
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65N06 Finite difference methods for boundary value problems involving PDEs
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