This book covers numerical methods that preserve properties of Hamiltonian systems, reversible systems, differential equations on manifolds and problems with highly oscillatory solutions. It presents a

theory of symplectic and symmetric methods, which include various specially designed integrators, as well as discusses their construction and practical merits. The long-time behavior of the numerical solutions is studied using a backward error analysis combined with KAM theory.

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From the reviews of the second edition:

"This book is highly recommended for advanced courses in numerical methods for ordinary differential equations as well as a reference for researchers/developers in the field of geometric integration, differential equations in general and related subjects. It is a must for academic and industrial libraries." -- MATHEMATICAL REVIEWS

"The second revised edition of the monograph is a fine work organized in fifteen chapters, updated and extended. ... The material of the book is organized in sections which are ... self-contained, so that one can dip into the book to learn a particular topic ... . A person interested in geometrical numerical integration will find this book extremely useful." (Calin Ioan Gheorghiu, Zentralblatt MATH, Vol. 1094 (20), 2006)

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