Ernst Hairer, Gerhard Wanner, Christian Lubich (auth.)9783540306634, 9783540306665, 3540306633
Numerical methods that preserve properties of Hamiltonian systems, reversible systems, differential equations on manifolds and problems with highly oscillatory solutions are the subject of this book. 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 many figures, it treats applications from physics and astronomy and contains many numerical experiments and comparisons of different approaches. The second edition is substantially revised and enlarged, with many improvements in the presentation and additions concerning in particular non-canonical Hamiltonian systems, highly oscillatory mechanical systems, and the dynamics of multistep methods. |
Table of contents :
Front Matter….Pages i-xiii
Examples and Numerical Experiments….Pages 1-22
Numerical Integrators….Pages 23-46
Order Conditions, Trees and B-Series….Pages 47-92
Conservation of First Integrals and Methods on Manifolds….Pages 93-130
Symmetric Integration and Reversibility….Pages 131-166
Symplectic Integration of Hamiltonian Systems….Pages 167-208
Further Topics in Structure Preservation….Pages 209-254
Structure-Preserving Implementation….Pages 255-286
Backward Error Analysis and Structure Preservation….Pages 287-326
Hamiltonian Perturbation Theory and Symplectic Integrators….Pages 327-374
Reversible Perturbation Theory and Symmetric Integrators….Pages 375-390
Dissipatively Perturbed Hamiltonian and Reversible Systems….Pages 391-406
Highly Oscillatory Differential Equations….Pages 407-453
Dynamics of Multistep Methods….Pages 455-491
Back Matter….Pages 493-515 |
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