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.

Classification: 37M15, 37N99, 65L06, 65P10

Keywords: 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: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique}, publisher = {EDP-Sciences}, volume = {43}, number = {4}, year = {2009}, pages = {631-644}, doi = {10.1051/m2an/2009019}, zbl = {1172.65067}, mrnumber = {2542868}, language = {en}, url = {http://www.numdam.org/item/M2AN_2009__43_4_631_0} }

Hairer, Ernst; McLachlan, Robert I.; Skeel, Robert D. On energy conservation of the simplified Takahashi-Imada method. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 43 (2009) no. 4, pp. 631-644. doi : 10.1051/m2an/2009019. http://www.numdam.org/item/M2AN_2009__43_4_631_0/

[1] On the numerical integration of ordinary differential equations by processed methods. SIAM J. Numer. Anal. 42 (2004) 531-552. | MR 2084225 | Zbl 1079.65075

, and ,[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 275679 | Zbl 0185.41301

,[3] An algebraic approach to invariant preserving integrators: the case of quadratic and Hamiltonian invariants. Numer. Math. 103 (2006) 575-590. | MR 2221062 | Zbl 1100.65115

, and ,[4] Energy drift in molecular dynamics simulations. BIT 47 (2007) 507-523. | MR 2338529 | Zbl 1130.82004

and ,[5] Energy conservation with non-symplectic methods: examples and counter-examples. BIT 44 (2004) 699-709. | MR 2211040 | Zbl 1082.65132

, and ,[6] Symmetric multistep methods over long times. Numer. Math. 97 (2004) 699-723. | MR 2127929 | Zbl 1060.65074

and ,[7] Geometric Numerical Integration, Structure-Preserving Algorithms for Ordinary Differential Equations, Springer Series in Computational Mathematics 31. Springer-Verlag, Berlin, 2nd Edition (2006). | MR 2221614 | Zbl 1094.65125

, and ,[8] Energy drift in reversible time integration. J. Phys. A 37 (2004) L593-L598. | MR 2100138 | Zbl 1064.37063

and ,[9] Extrapolated gradientlike algorithms for molecular dynamics and celestial mechanics simulations. Phys. Rev. E 74 (2006) 036703. | MR 2282154

,[10] A numerical algorithm for Hamiltonian systems. J. Comput. Phys. 97 (1991) 235-239. | MR 1134330 | Zbl 0749.65045

,[11] A family of symplectic integrators: stability, accuracy, and molecular dynamics applications. SIAM J. Sci. Comput. 18 (1997) 203-222. | MR 1433384 | Zbl 0868.65055

, and ,[12] What makes molecular dynamics work? SIAM J. Sci. Comput. 31 (2009) 1363-1378. | MR 2486834

,[13] On reversible and canonical integration methods. Technical Report SAM-Report No. 88-05, ETH-Zürich, Switzerland (1988).

,[14] Monte Carlo calculation of quantum systems. II. Higher order correction. J. Phys. Soc. Jpn. 53 (1984) 3765-3769.

and ,[15] Ergodicity and the numerical simulation of Hamiltonian systems. SIAM J. Appl. Dyn. Syst. 4 (2005) 563-587. | MR 2145198 | Zbl 1090.65139

,[16] 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 1406813 | Zbl 0871.65059

, and ,