This monograph originated from a course given at the State Universities of Krasnoyarsk and Novosibirsk; an early version appeared as 'Methods of Computational Mathematics' by the first author and was published by Nauka Publishing House, 1977 (in Russian).
At the beginning of the century, Richardson introduced a completely new technique, based on extrapolation, for increasing the accuracy of numerical solutions of linear problems. This technique is now known as Richardson's method. For a long time this method has been used rather heuristically, without a proper theoretical basis.
The objective of the present monograph is to develop some general approaches to using the Richardson method for a wide range of problems in mathematical physics, and to formulate requirements which ensure the efficacy of its application. The work presents both the theoretical foundations and practical aspects of the method and provides the first coherent, general description of Richardson's method.
The exposition of the material is clear and the only prerequisites required of the reader are a sound course in calculus, some acquaintance with functional analysis, and an introductory course in numerical analysis.
The treatment of Richardson's method is confined to global extrapolation of numerical solutions obtained on the various grids; local extrapolation methods are not discussed. Furthermore, most of the considerations concentrate on linear problems for which a rather general analysis can be given. Apart from the (linear) Richardson extrapolation method, the authors discuss briefly rational extrapolation, exponential extrapolation, and the E-algorithm of Wynn.
Chapter 1 starts with a simple example illustrating the main features of linear extrapolation and explaining the reasons why global extrapolation is to be preferred to local extrapolation. General expansion theorems for linear problems are given, which are used throughout the book.
Chapter 2 treats the general, nonlinear scalar differential equation of first order and systems of linear first-order differential equations. A separate section is devoted to equations with nonsmooth solutions.
Chapter 3 deals with two-point boundary-value problems. The power of Richardson extrapolation is demonstrated by showing that discontinuous