Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential EquationsNumerical 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. |
Contents
Examples and Numerical Experiments | 1 |
Numerical Integrators | 23 |
Order Conditions Trees and BSeries | 47 |
Conservation of First Integrals and Methods on Manifolds | 93 |
Symmetric Integration and Reversibility | 131 |
Symplectic Integration of Hamiltonian Systems | 167 |
Further Topics in Structure Preservation | 209 |
StructurePreserving Implementation | 255 |
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 |
| 493 | |
| 509 | |
Backward Error Analysis and Structure Preservation | 287 |
Other editions - View all
Common terms and phrases
Algorithm analytic applied approximation assume becomes behaviour bounded called close coefficients collocation completely composition computation condition consequence conservation consider constant construction coordinates corresponding defined Definition denote depend derivatives differential equation energy equivalent error estimates Euler method Example Exercise exists expansion explicit expression flow formula function given gives Hamiltonian system Hence holds identity implicit implies independent initial values insert integrable interval invariant iteration Lemma linear manifold matrix midpoint modified equation numerical solution obtain parameter partitioned perturbation picture Poisson polynomial positive preserves problem projection Proof properties proves relation replaced requires respect result reversible rule Runge-Kutta methods satisfies scheme Sect shows similar solved space splitting stable starting step size Stormer/Verlet sufficiently symmetric symmetric matrix symplectic symplectic Euler Table Theorem theory tion transformation trees variables vector written yields
Popular passages
Page 499 - Mehrmann. The Autonomous Linear Quadratic Control Problem, Theory and Numerical Solution. Lecture Notes in Control and Information Science, Vol.163, Springer- Verlag, Heidelberg, July 1991.