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On energy conservation of the simplified Takahashi--Imada method

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E. Hairer, R. I. McLachlan and R. D. Skeel

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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 St\"ormer--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.

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Key Words. Symmetric and symplectic integrators, geometric numerical integration, modified differential equation, energy conservation, Henon--Heiles problem, N-body problem in molecular dynamics.

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