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New Integration Methods for Perturbed ODEs Based on Symplectic Implicit Runge–Kutta Schemes with Application to Solar System Simulations


We propose a family of integrators, flow-composed implicit Runge–Kutta methods, for perturbations of nonlinear ordinary differential equations, consisting of the composition of flows of the unperturbed part alternated with one step of an implicit Runge–Kutta (IRK) method applied to a transformed system. The resulting integration schemes are symplectic when both the perturbation and the unperturbed part are Hamiltonian and the underlying IRK scheme is symplectic. In addition, they are symmetric in time (resp. have order of accuracy r) if the underlying IRK scheme is time-symmetric (resp. of order r). The proposed new methods admit mixed precision implementation that allows us to efficiently reduce the effect of round-off errors. We particularly focus on the potential application to long-term solar system simulations, with the equations of motion of the solar system rewritten as a Hamiltonian perturbation of a system of uncoupled Keplerian equations. We present some preliminary numerical experiments with a simple point mass Newtonian 10-body model of the solar system (with the sun, the eight planets, and Pluto) written in canonical heliocentric coordinates.

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M. Antoñana, J. Makazaga, and A. Murua have received funding from the Project of the Spanish Ministry of Economy and Competitiveness with Reference MTM2016-76329-R (AEI/FEDER, EU), from the Project MTM2013-46553-C3-2-P from Spanish Ministry of Economy and Trade, and as part of the Consolidated Research Group IT649-13 by the Basque Government.

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Correspondence to Mikel Antoñana.

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Antoñana, M., Makazaga, J. & Murua, A. New Integration Methods for Perturbed ODEs Based on Symplectic Implicit Runge–Kutta Schemes with Application to Solar System Simulations. J Sci Comput 76, 630–650 (2018).

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