Numerical experiments on the efficiency of second-order mixed-variable symplectic integrators for N-body problems |
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Authors: | Patrick Michel Giovanni H. Valsecchi |
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Affiliation: | (1) Dept. Cassini, Observatoire de la Côte d'Azur, URA CNRS 1362, 229, F-06304 Nice, France;(2) I.A.S.-Planetologia, viale dell' Università, 11, 00185 Roma, Italy |
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Abstract: | We discuss the efficiency of the so-called mixed-variable symplectic integrators for N-body problems. By performing numerical experiments, we first show that the evolution of the mean error in action-like variables is strongly dependent on the initial configuration of the system. Then we study the effect of changing the stepsize when dealing with problems including close encounters between a particle and a planet. Considering a previous study of the slow encounter between comet P/Oterma and Jupiter, we show that the overall orbital patterns can be reproduced, but this depends on the chosen value of the maximum integration stepsize. Moreover the Jacobi constant in a restricted three-body problem is not conserved anymore when the stepsize is changed frequently: over a 105 year time span, to keep a relative error in this integral of motion of the same order as that given by a Bulirsch-Stoer integrator requires a very small integration stepsize and much more computing time. However, an integration of a sample including 104 particles close to Neptune shows that the distributions of the variation of the elements over one orbital period of the particles obtained by the Bulirsch-Stoer integrator and the symplectic integrator up to a certain integration stepsize are rather similar. Therefore, mixed-variable symplectic integrators are efficient either for N-body problems which do not include close encounters or for statistical investigations on a big sample of particles. |
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Keywords: | numerical integrations symplectic integrators three-body problem |
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