Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations

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Springer Science & Business Media, May 18, 2006 - Mathematics - 644 pages

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.

 

Contents

Examples and Numerical Experiments
1
Numerical Integrators
27
Order Conditions Trees and BSeries
51
The Butcher Group
64
3
71
4
80
Conservation of First Integrals and Methods on Manifolds
102
6
118
NonCanonical Hamiltonian Systems
237
StructurePreserving Implementation
303
Backward Error Analysis and Structure Preservation
337
Hamiltonian Perturbation Theory and Symplectic Integrators
389
Reversible Perturbation Theory and Symmetric Integrators
437
Dissipatively Perturbed Hamiltonian and Reversible Systems
455
Oscillatory Differential Equations with Constant High Frequencies
471
Oscillatory Differential Equations with Varying High Frequencies
531

8
124
Symmetric Integration and Reversibility
143
Symplectic Integration of Hamiltonian Systems
179
Dynamics of Multistep Methods
567
Bibliography
617
Index
637

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