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Geometric numerical integration. Structure-preserving algorithms for ordinary differential equations. 2nd ed. (English) Zbl 1094.65125
Springer Series in Computational Mathematics 31. Berlin: Springer (ISBN 3-540-30663-3/hbk). xvii, 644 p. (2006).
[For the 1st edition (2002) see Zbl 0994.65135.]
The second revised edition of the monograph is a fine work organized in fifteen chapters, updated and extended. Thus, Chapters VII and XIII become respectively: “Non-canonical Hamiltonian systems” and “Oscillatory Differential Equations with Constant High Frequencies” and Ch. XIV is a new one, entitled: “Oscillatory differential equations with varying high frequencies”. The bibliography is also enriched with some old and new titles including those of the authors.
All in all the second edition is larger than the previous one with more than 130 pages. In the Preface to this new edition the authors provide a detailed list of major additions and changes. It contains 16 issues. As a general remark, fairly sophisticated aspects of numerical algorithms coexist with more applicative aspects such as long-time energy conservation, or round-off error analysis. Although the book has a genuine bias toward numerical methods to solve conservative (Hamiltonian) systems, readers far removed from this “thin” subset of the set of smooth dynamical systems given by differential equations, would fairly profit.
In fact the authors are concerned with two large and quite different classes of numerical schemes. The first class contains methods which preserve the structure of the flow, i.e., symmetric and symplectic methods. The second one is the class of methods which conserve some first integrals of the systems. One-step methods as well as multi step methods are considered.
The material of the book is organized in sections which are rather self-contained, so that one can dip into the book to learn a particular topic without having to read the rest of the book or even the rest of the chapter. A person interested in geometric numerical integration will find this book extremely useful. However, the theory of the numerical methods that preserve some particular properties of the flow of a dynamical system has come to maturity. The authors provide an exhaustive narrative of this story.

MSC:
65P10 Numerical methods for Hamiltonian systems including symplectic integrators
65Lxx Numerical methods for ordinary differential equations
34Cxx Qualitative theory for ordinary differential equations
37Cxx Smooth dynamical systems: general theory
37M15 Discretization methods and integrators (symplectic, variational, geometric, etc.) for dynamical systems
37Jxx Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
70Fxx Dynamics of a system of particles, including celestial mechanics
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
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