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Numerical Hamiltonian problems. (English) Zbl 0816.65042
Applied Mathematics and Mathematical Computation. 7. London: Chapman & Hall,. xii, 207 p. (1994).
During the past decade exciting developments in the numerical treatment of Hamiltonian systems took place and much new insight into this subject has been obtained. This book gives a first unified view on these developments. It is written by well-known specialists of this subject.
The first two chapters provide examples of Hamiltonian systems and explain the symplecticness of its flow. The next three chapters recall one-step methods for general ordinary differential equations (Runge-Kutta methods, partitioned Runge-Kutta methods, Nyström methods, their order conditions and implementation). The central part of the book consists of chapters 6 to 10. There the conditions for symplecticness of a given method are studied and their effect as simplifying assumptions for the order conditions is explained in detail. Theoretical and practical aspects of symplectic integration methods – backward error interpretation, conservation of energy, variable step-size implementation, long-term behaviour of the global error – are discussed and numerical experiments are given. The final chapters introduce the reader to a more advanced material (generating functions, Lie formalism, high-order methods) and briefly mention some extensions without going into details.
The text is well written and directed more towards an introductory level. It is of interest for numerical analysis as well as for scientists confronted with the integration of Hamiltonian systems.
Reviewer: E.Hairer (Genève)

65L05 Numerical methods for initial value problems involving ordinary differential equations
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65L50 Mesh generation, refinement, and adaptive methods for ordinary differential equations
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
70H05 Hamilton’s equations