Volume 43, Number 4, July-August 2009Special issue on Numerical ODEs today
|Page(s)||631 - 644|
|Published online||08 July 2009|
On energy conservation of the simplified Takahashi-Imada method
Section de Mathématiques,
Université de Genève, 1211 Genève 24, Switzerland.
2 Institute of Fundamental Sciences, Massey University, Palmerston North, New Zealand. R.McLachlan@massey.ac.nz
3 Department of Computer Science, Purdue University, West Lafayette, Indiana, 47907-2107, USA. email@example.com
In long-time numerical integration of Hamiltonian systems, and especially in molecular dynamics simulation, it is important that the energy is well conserved. For symplectic integrators applied with sufficiently small step size, this is guaranteed by the existence of a modified Hamiltonian that is exactly conserved up to exponentially small terms. This article is concerned with the simplified Takahashi-Imada method, which is a modification of the Störmer-Verlet method that is as easy to implement but has improved accuracy. This integrator is symmetric and volume-preserving, but no longer symplectic. We study its long-time energy conservation and give theoretical arguments, supported by numerical experiments, which show the possibility of a drift in the energy (linear or like a random walk). With respect to energy conservation, this article provides empirical and theoretical data concerning the importance of using a symplectic integrator.
Mathematics Subject Classification: 37M15 / 37N99 / 65L06 / 65P10
Key words: Symmetric and symplectic integrators / geometric numerical integration / modified differential equation / energy conservation / Hénon-Heiles problem / N-body problem in molecular dynamics.
© EDP Sciences, SMAI, 2009
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