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Explicit, time reversible, adaptive step size control

Hairer, E and Söderlind, Gustaf LU (2005) In SIAM Journal on Scientific Computing 26(6). p.1838-1851
Abstract
Adaptive step size control is difficult to combine with geometric numerical integration. As classical step size control is based on "past" information only, time symmetry is destroyed and with it the qualitative properties of the method. In this paper we develop completely explicit, reversible, symmetry-preserving, adaptive step size selection algorithms for geometric numerical integrators such as the Stormer-Verlet method. A new step density controller is proposed and analyzed using backward error analysis and reversible perturbation theory. For integrable reversible systems we show that the resulting adaptive method nearly preserves all action variables and, in particular, the total energy for Hamiltonian systems. It has the same... (More)
Adaptive step size control is difficult to combine with geometric numerical integration. As classical step size control is based on "past" information only, time symmetry is destroyed and with it the qualitative properties of the method. In this paper we develop completely explicit, reversible, symmetry-preserving, adaptive step size selection algorithms for geometric numerical integrators such as the Stormer-Verlet method. A new step density controller is proposed and analyzed using backward error analysis and reversible perturbation theory. For integrable reversible systems we show that the resulting adaptive method nearly preserves all action variables and, in particular, the total energy for Hamiltonian systems. It has the same excellent long-term behavior as that obtained when constant steps are used. With variable steps, however, both accuracy and efficiency are greatly improved. (Less)
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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
backward error analysis, reversible, and reversible step size control, explicit, Hamiltonian systems, Stormer-Verlet method, symmetric methods, time reversible and, adaptive integration, geometric integration, perturbation theory
in
SIAM Journal on Scientific Computing
volume
26
issue
6
pages
1838 - 1851
publisher
SIAM Publications
external identifiers
  • wos:000231357700002
  • scopus:27844551675
ISSN
1064-8275
DOI
10.1137/040606995
language
English
LU publication?
yes
id
b23073b3-306e-45c7-ba55-884a7eac889e (old id 226528)
date added to LUP
2007-08-10 12:23:22
date last changed
2017-02-13 13:10:02
@article{b23073b3-306e-45c7-ba55-884a7eac889e,
  abstract     = {Adaptive step size control is difficult to combine with geometric numerical integration. As classical step size control is based on "past" information only, time symmetry is destroyed and with it the qualitative properties of the method. In this paper we develop completely explicit, reversible, symmetry-preserving, adaptive step size selection algorithms for geometric numerical integrators such as the Stormer-Verlet method. A new step density controller is proposed and analyzed using backward error analysis and reversible perturbation theory. For integrable reversible systems we show that the resulting adaptive method nearly preserves all action variables and, in particular, the total energy for Hamiltonian systems. It has the same excellent long-term behavior as that obtained when constant steps are used. With variable steps, however, both accuracy and efficiency are greatly improved.},
  author       = {Hairer, E and Söderlind, Gustaf},
  issn         = {1064-8275},
  keyword      = {backward error analysis,reversible,and reversible step size control,explicit,Hamiltonian systems,Stormer-Verlet method,symmetric methods,time reversible and,adaptive integration,geometric integration,perturbation theory},
  language     = {eng},
  number       = {6},
  pages        = {1838--1851},
  publisher    = {SIAM Publications},
  series       = {SIAM Journal on Scientific Computing},
  title        = {Explicit, time reversible, adaptive step size control},
  url          = {http://dx.doi.org/10.1137/040606995},
  volume       = {26},
  year         = {2005},
}