The subject of this book is numerical methods that preserve geometric properties of the flow of a differential equation: symplectic integrators for Hamiltonian systems, symmetric integrators for reversible systems, methods preserving first integrals and numerical methods on manifolds, including Lie group methods and integrators for constrained mechanical systems, and methods for problems with highly oscillatory solutions. A complete theory of symplectic and symmetric Runge-Kutta, composition, splitting, multistep and various specially designed integrators is presented, and their construction and practical merits are discussed. The long-time behaviour of the numerical solutions is studied using a backward error analysis (modified equations) combined with KAM theory and related perturbation theories. The book is illustrated by many figures, it treats applications from physics and astronomy and contains many numerical experiments and comparisons of different approaches.

This book deals with numerical methods that preserve properties of Hamiltonian systems, reversible systems, differential equations on manifolds and problems with highly oscillatory solutions. A complete self-contained theory of symplectic and symmetric methods, which include Runge-Kutta, composition, splitting, multistep and various specially designed integrators, is presented and their construction and practical merits are discussed. The long-time behaviour of the numerical solutions is studied using a backward error analysis (modified equations) combined with KAM theory. The book is illustrated by numerous figures, treats applications from physics and astronomy, and contains many numerical experiments and comparisons of different approaches.

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.- XIV. Dynamics of Multistep Methods.