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

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Springer Science & Business Media, Mar 9, 2013 - Mathematics - 515 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.
 

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Contents

Numerical Integrators
23
6
46
Order Conditions Trees and BSeries
47
XI
55
Conservation of First Integrals and Methods on Manifolds
93
Symmetric Integration and Reversibility
131
V6 Exercises
164
Further Topics in Structure Preservation
209
Backward Error Analysis and Structure Preservation
287
Hamiltonian Perturbation Theory and Symplectic Integrators
327
Reversible Perturbation Theory and Symmetric Integrators
375
Dissipatively Perturbed Hamiltonian and Reversible Systems
391
Highly Oscillatory Differential Equations
407
Dynamics of Multistep Methods
455
Bibliography 493
492
Index
509

StructurePreserving Implementation
255

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