And Descriptions of Tables and Books

Abstract

This book is a translation of the original Russian edition which was published in 1973. The title is quite general, but the main content is numerical methods for initial or boundary value problems in Mathematical Physics. After an opening chapter with fundamentals about difference schemes, a chapter foUows dealing with Ritz and Galerkin's methods. The methods are illustrated on simple model equations, and various ways of choosing the subspaces for the finite element method are discussed. Chapter 3 contains a discussion of the most commonly used methods for solving linear systems of algebraic equations. There is a general description of the most interesting iterative methods, such as SOR, Chebyshev acceleration, conjugate gradients and various splitting methods. The Fast Fourier Transform with different applications is also discussed, but there is nothing about the realization of general direct methods commonly used for finite element problems. Chapter 4 deals with implicit methods for nonstationary problems, and there is a thorough discussion of splitting methods. The so-called component by component splitting methods, where only one component of the space operator is involved in each substep, receive special attention. They are considered by the author as the most important methods for applications. Inverse problems, often ül posed, are usually not treated in books of this type, but regarding the importance of such problems, the inclusion of Chapter 5 is a very good idea. Here the problem of finding the coefficients of an operator or the initial state, given the current state, is discussed. Fourier series methods and methods based on perturbation theory are presented. The discretization of the simplest problems of mathematical physics is discussed in Chapter 6. The Poisson, heat and wave equations are treated as well as the equation of motion, all in the simplest linear form. Only second order schemes are considered, but there is a section about increasing the accuracy by Richardson extrapolation. Chapter 7 is devoted to the transport equation of radiative transfer theory. Different geometries are discussed, and the splitting technique is again applied. Chapter 8 is a review of methods in numerical mathematics. The content is based on a talk given by the author at the International Congress of Mathematicians in Nice 1970. As the author says in the preface, he has concentrated on basic ideas, and that is a good approach. The methods are presented in such a way that they can be easily generaUzed to more complicated problems. However, since the author emphasizes implicit methods, it would have been natural to include a discussion about methods for solving nonlinear systems of algebraic equations. Other topics which are treated very briefly or not at aU, are expUcit difference schemes, higher order schemes and the choice of boundary conditions for approximations where this is not trivial. The book is written in a very clear and nice way. It is weU suited to give a good

Cite this paper

@inproceedings{Marchuk2010AndDO, title={And Descriptions of Tables and Books}, author={Guri I. Marchuk}, year={2010} }