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Geometric numerical integration. Structure-preserving algorithms for ordinary differential equations. (English) Zbl 0994.65135
Springer Series in Computational Mathematics. 31. Berlin: Springer. xiii, 515 p. (2002).
The book is organized into fourteen chapters: I. Examples and numerical experiments; II. Numerical integrators; III. Order conditions, trees and B-series; IV. Conservation of first integrals and methods on manifolds; V. Symmetric integration and reversibility; VI. Symplectic integration of Hamiltonian systems; VII. Further topics in structure preservation; VIII. Structure-preserving implementation; IX. Backward error analysis and structure preservation; X. Hamiltonian perturbation theory and symplectic integrators; XI. Reversible perturbation theory and symmetric integrators; XII. Dissipatively perturbed Hamiltonian and reversible systems; XIII. Highly oscillatory differential equations; and XIV. Dynamics of multistep methods.
The philosophy of the work consists in considering a numerical method as a discrete dynamical system which approximates the flow of a differential equation. Specifically, the authors analyze algorithms which preserve the geometric properties of the flow and produce accurate long-time results.
Theory, applications and the implementation problems (dangers), for such structure preserving algorithms are interwoven through the book. There are enough applications to keep an applied mathematician, scientist, or engineer interested, and enough theory to lay a firm foundation for further work.
The writing style is concise, bright and lively – with even a touch of humor at times. However, the most important feature of the book consist in the authors’ valuable insight into both theoretical and practical aspects of structure-preserving algorithms for ordinary differential equations.

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