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On energy conservation of the simplified Takahashi-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 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.

Keywords

Symmetric and symplectic integratorsgeometric numerical 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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References

Blanes, S., Casas, F. and Murua, A., On the numerical integration of ordinary differential equations by processed methods. SIAM J. Numer. Anal. 42 (2004) 531–552. CrossRef
J.C. Butcher, 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. CrossRef
Chartier, P., Faou, E. and Murua, A., An algebraic approach to invariant preserving integrators: the case of quadratic and Hamiltonian invariants. Numer. Math. 103 (2006) 575–590. CrossRef
Cottrell, D. and Tupper, P.F., Energy drift in molecular dynamics simulations. BIT 47 (2007) 507–523. CrossRef
Faou, E., Hairer, E. and Pham, T.-L., Energy conservation with non-symplectic methods: examples and counter-examples. BIT 44 (2004) 699–709. CrossRef
Hairer, E. and Lubich, C., Symmetric multistep methods over long times. Numer. Math. 97 (2004) 699–723. CrossRef
E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration, Structure-Preserving Algorithms for Ordinary Differential Equations, Springer Series in Computational Mathematics 31. Springer-Verlag, Berlin, 2nd Edition (2006).
McLachlan, R.I. and Perlmutter, M., Energy drift in reversible time integration. J. Phys. A 37 (2004) L593–L598. CrossRef
Omelyan, I.P., Extrapolated gradientlike algorithms for molecular dynamics and celestial mechanics simulations. Phys. Rev. E 74 (2006) 036703. CrossRef
Rowlands, G., A numerical algorithm for Hamiltonian systems. J. Comput. Phys. 97 (1991) 235–239. CrossRef
Skeel, R.D., Zhang, G. and Schlick, T., A family of symplectic integrators: stability, accuracy, and molecular dynamics applications. SIAM J. Sci. Comput. 18 (1997) 203–222. CrossRef
Skeel, R.D., What makes molecular dynamics work? SIAM J. Sci. Comput. 31 (2009) 1363–1378. CrossRef
D. Stoffer, On reversible and canonical integration methods. Technical Report SAM-Report No. 88-05, ETH-Zürich, Switzerland (1988).
Takahashi, M. and Imada, M., Monte Carlo calculation of quantum systems. II. Higher order correction. J. Phys. Soc. Jpn. 53 (1984) 3765–3769.
Tupper, P.F., Ergodicity and the numerical simulation of Hamiltonian systems. SIAM J. Appl. Dyn. Syst. 4 (2005) 563–587. CrossRef
J. Wisdom, M. Holman and J. Touma, 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.