Geometric Numerical Integration

Numerical methods that preserve the properties of Hamiltonian systems, reversible systems, differential equations on manifolds, and problems with highly oscillatory solutions are the focus of this book. A complete, self-contained theory of symplectic and symmetric methods, including Runge-Kutta, composition, decomposition, multistep, and various specially developed integrators, is presented, along with discussions of their construction and practical advantages. The long-term behavior of the numerical solutions is examined using a backward error analysis (modified equations) in conjunction with KAM theory. The book is illustrated with many figures, covers applications from physics and astronomy, and includes numerous numerical experiments as well as comparisons of different approaches. The second edition has been significantly revised and expanded, with many improvements in presentation and additions on specific topics.