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.
Mots-clés : symmetric and symplectic integrators, geometric numerical integration, modified differential equation, energy conservation, Hénon-Heiles problem, $N$-body problem in molecular dynamics
@article{M2AN_2009__43_4_631_0,
author = {Hairer, Ernst and McLachlan, Robert I. and Skeel, Robert D.},
title = {On energy conservation of the simplified {Takahashi-Imada} method},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {631--644},
publisher = {EDP-Sciences},
volume = {43},
number = {4},
year = {2009},
doi = {10.1051/m2an/2009019},
mrnumber = {2542868},
zbl = {1172.65067},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2009019/}
}
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: Modélisation mathématique et analyse numérique PY - 2009 SP - 631 EP - 644 VL - 43 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2009019/ DO - 10.1051/m2an/2009019 LA - en ID - M2AN_2009__43_4_631_0 ER -
%0 Journal Article %A Hairer, Ernst %A McLachlan, Robert I. %A Skeel, Robert D. %T On energy conservation of the simplified Takahashi-Imada method %J ESAIM: Modélisation mathématique et analyse numérique %D 2009 %P 631-644 %V 43 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2009019/ %R 10.1051/m2an/2009019 %G en %F M2AN_2009__43_4_631_0
Hairer, Ernst; McLachlan, Robert I.; Skeel, Robert D. On energy conservation of the simplified Takahashi-Imada method. ESAIM: Modélisation mathématique et analyse numérique, Special issue on Numerical ODEs today, Tome 43 (2009) no. 4, pp. 631-644. doi : 10.1051/m2an/2009019. http://www.numdam.org/articles/10.1051/m2an/2009019/
[1] , and , On the numerical integration of ordinary differential equations by processed methods. SIAM J. Numer. Anal. 42 (2004) 531-552. | MR | Zbl
[2] , The effective order of Runge-Kutta methods, in Proceedings of Conference on the Numerical Solution of Differential Equations, J.L. Morris Ed., Lect. Notes Math. 109 (1969) 133-139. | MR | Zbl
[3] , and , An algebraic approach to invariant preserving integrators: the case of quadratic and Hamiltonian invariants. Numer. Math. 103 (2006) 575-590. | MR | Zbl
[4] and , Energy drift in molecular dynamics simulations. BIT 47 (2007) 507-523. | MR | Zbl
[5] , and , Energy conservation with non-symplectic methods: examples and counter-examples. BIT 44 (2004) 699-709. | MR | Zbl
[6] and , Symmetric multistep methods over long times. Numer. Math. 97 (2004) 699-723. | MR | Zbl
[7] , and , Geometric Numerical Integration, Structure-Preserving Algorithms for Ordinary Differential Equations, Springer Series in Computational Mathematics 31. Springer-Verlag, Berlin, 2nd Edition (2006). | MR | Zbl
[8] and , Energy drift in reversible time integration. J. Phys. A 37 (2004) L593-L598. | MR | Zbl
[9] , Extrapolated gradientlike algorithms for molecular dynamics and celestial mechanics simulations. Phys. Rev. E 74 (2006) 036703. | MR
[10] , A numerical algorithm for Hamiltonian systems. J. Comput. Phys. 97 (1991) 235-239. | MR | Zbl
[11] , and , A family of symplectic integrators: stability, accuracy, and molecular dynamics applications. SIAM J. Sci. Comput. 18 (1997) 203-222. | MR | Zbl
[12] , What makes molecular dynamics work? SIAM J. Sci. Comput. 31 (2009) 1363-1378. | MR
[13] , On reversible and canonical integration methods. Technical Report SAM-Report No. 88-05, ETH-Zürich, Switzerland (1988).
[14] and , Monte Carlo calculation of quantum systems. II. Higher order correction. J. Phys. Soc. Jpn. 53 (1984) 3765-3769.
[15] , Ergodicity and the numerical simulation of Hamiltonian systems. SIAM J. Appl. Dyn. Syst. 4 (2005) 563-587. | MR | Zbl
[16] , and , Symplectic correctors, in Integration Algorithms and Classical Mechanics, J.E. Marsden, G.W. Patrick and W.F. Shadwick Eds., Amer. Math. Soc., Providence R.I. (1996) 217-244. | MR | Zbl
Cité par Sources :
