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On energy conservation of the simplifiedTakahashi-Imada method

Published online by Cambridge University Press:  08 July 2009

Ernst Hairer ,
Robert I. McLachlan  and
Robert D. Skeel
Ernst Hairer
Affiliation:
Section de Mathématiques, Université de Genève, 1211 Genève 24, Switzerland. Ernst.Hairer@unige.ch
Robert I. McLachlan
Affiliation:
Institute of Fundamental Sciences, Massey University, Palmerston North, New Zealand. R.McLachlan@massey.ac.nz
Robert D. Skeel
Affiliation:
Department of Computer Science, Purdue University, West Lafayette, Indiana, 47907-2107, USA. skeel@cs.purdue.edu
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Abstract

In long-time numerical integration of Hamiltonian systems,and especially in molecular dynamics simulation,it is important that the energy is well conserved. For symplecticintegrators applied with sufficiently small step size, thisis guaranteed by the existence of a modifiedHamiltonian that is exactly conserved up to exponentially smallterms. This article is concerned with the simplifiedTakahashi-Imada method, which is a modificationof the Störmer-Verlet method that is as easy to implement buthas improved accuracy. This integrator is symmetric andvolume-preserving, but no longer symplectic. We study itslong-time energy conservation and give theoreticalarguments, supported by numerical experiments, whichshow the possibility of a drift in the energy (linear or like a random walk).With respect to energy conservation, this article provides empiricaland theoretical data concerning the importance of using a symplecticintegrator.

Keywords

Symmetric and symplectic integratorsgeometricnumerical integrationmodified differential equationenergy conservationHénon-Heiles problemN-body problem in molecular dynamics.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 43 , Issue 4: Special issue on Numerical ODEs today , July 2009 , pp. 631 - 644
Copyright
© EDP Sciences, SMAI, 2009

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