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Achieving Brouwer's law with implicit Runge-Kutta methods

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E. Hairer, R.I. McLachlan, and A. Razakarivony

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Abstract. In high accuracy long-time integration of differential equations, round-off errors may dominate truncation errors. This article studies the influence of round-off on the conservation of first integrals such as the total energy in Hamiltonian systems. For implicit Runge--Kutta methods, a standard implementation shows an unexpected propagation. We propose a modification that reduces the effect of round-off and shows a qualitative and quantitative improvement for an accurate integration over long times.

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Key Words. probabilistic error propagation, implicit Runge--Kutta methods, long-time integration, efficient implementation.

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