Chichester: Wiley (ISBN 0-471-96758-0/hbk). xiv, 425 p. £ 65.00; $ 115.00 (2003).
This book is a modernisation and expansion of the previous 1987 volume by the same author (1987; Zbl 0616.65072) on the same topic of numerical methods for initial value problems of ordinary differential equations (IVODEs). It is not just only a revision with some additional pages added. In contrary, the present book keeps only a part of the earlier edition more or less unchanged but presents much of the material now in a changed spirit with a considerable amount of different subjects added. As a result, this book should widen the range of readers who might find it interesting and useful, especially engineers and other scientists with profound mathematical background. In this respect it is also helpful that exercises are introduced at the end of each section.
The book covers now all the material that is modern standard in this field: explicit and implicit Runge-Kutta methods, implementable Runge-Kutta methods (singly implicit methods and generalizations), Taylor series, hybrid methods, linear multistep methods, one-leg methods, general linear methods. As one naturally expects from the present author all these methods are thoroughly studied with a high standard of mathematical elegance. But not only the theoretical point of view is covered. Many tables of specific methods, implementation aspects and numerical examples are included. Already the first chapter starts with 20 pages of examples for differential equations originating from different fields of applications followed by the needed fundamental results from the theory of differential and difference equations. In an easy to understand way the fundamental ideas and basic features of numerical methods for IVODEs are introduced using the Euler method as starting point.
It is clear that the expectation of the reader will be satified that the whole theory of Runge-Kutta methods (rooted trees, order conditions and barriers for explicit and implicit methods, the Runge-Kutta group), that is to a large extent due to the author’s own research, forms a chapter in the book. An other chapter contains the state of the art in general linear methods, also a speciality of the author. Much is said about stability and stability regions, besides many pictures for specific one- and multistep methods also with regard to theoretical respect, involving order stars where it is adequate.
This book belongs into any mathematical library. It is of high value for those involved in numericallly solving IVODEs or doing research in this field.