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On the long-time energy conservation by high order geometric integrators. (English) Zbl 1325.70005
Érdi, Bálint (ed.) et al., International conference on actual problems in celestial mechanics and dynamical astronomy, Cluj-Napoca, Romania, May 25–27, 2006. Cluj-Napoca: Presa Universitară Clujeană / Cluj University Press (ISBN 978-973-610-555-5/pbk). Publications of the Astronomy Department of the Eötvös University 19, 213-219 (2007).
Summary: The main aim of this paper is to investigate the long-time behavior of three high order geometric integrators, namely an implicit Runge-Kutta-Gauss method, the composed Störmer-Verlet method and a high order linear multistep method. All these three families of methods perform fairly accurate, at least qualitatively, when they are used in the integration of the outer Solar system. No spiral outwards or inwards are observed when their orders exceed six. With the long time energy conservation the situation change considerable. A significant improving in the computation of Hamiltonian is observed passing from order two to six but further, in contrast with the trajectories, almost nothing is gain by increasing the order of the method. A partial answer to this intriguing situation is furnished by the analysis of round off errors.
For the entire collection see [Zbl 1117.70002].
##### MSC:
 70-08 Computational methods for problems pertaining to mechanics of particles and systems 70F10 $$n$$-body problems 65P10 Numerical methods for Hamiltonian systems including symplectic integrators