On energy conservation of the simplified Takahashi-Imada method

Ernst Hairer; Robert I. McLachlan; Robert D. Skeel

ESAIM: Mathematical Modelling and Numerical Analysis (2009)

  • Volume: 43, Issue: 4, page 631-644
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Hairer, Ernst, McLachlan, Robert I., and Skeel, Robert D.. "On energy conservation of the simplified Takahashi-Imada method." ESAIM: Mathematical Modelling and Numerical Analysis 43.4 (2009): 631-644. <http://eudml.org/doc/250589>.

@article{Hairer2009,
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 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. },
author = {Hairer, Ernst, McLachlan, Robert I., Skeel, Robert D.},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Symmetric and symplectic integrators; geometric numerical integration; modified differential equation; energy conservation; Hénon-Heiles problem; N-body problem in molecular dynamics.; symmetric and symplectic integrators; geometric numerical integration},
language = {eng},
month = {7},
number = {4},
pages = {631-644},
publisher = {EDP Sciences},
title = {On energy conservation of the simplified Takahashi-Imada method},
url = {http://eudml.org/doc/250589},
volume = {43},
year = {2009},
}

TY - JOUR
AU - Hairer, Ernst
AU - McLachlan, Robert I.
AU - Skeel, Robert D.
TI - On energy conservation of the simplified Takahashi-Imada method
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2009/7//
PB - EDP Sciences
VL - 43
IS - 4
SP - 631
EP - 644
AB - 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.
LA - eng
KW - Symmetric and symplectic integrators; geometric numerical integration; modified differential equation; energy conservation; Hénon-Heiles problem; N-body problem in molecular dynamics.; symmetric and symplectic integrators; geometric numerical integration
UR - http://eudml.org/doc/250589
ER -

References

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