On the long time energy conservation by high order geometric integrators


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.


C.I. Gheorghiu
A.C. Muresan


N-body problem; outer Solar system; composed geometric integrator; round-off errors; compensated summation; long time; energy conservation;


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C.I. Gheorghiu, A.C. Mureşan, On the long time energy conservation by high order geometric integrators, PADEU, 19 (2007), 213-219.



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Publications of the Astronomy Department of the Eötvös University (PADEU), vol. 19 (2007), pp. 213-219.

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Cluj University Press
(Presa Universitara Clujeana)

Conference name

International Conference on Actual Problems in Celestial Mechanics and Dynamical Astronomy,
Cluj-Napoca, Romania, May 25-27, 2006

B. Erdi and F. Szenkovits

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[1] Butcher, J. C., I. Austral. math. Soc. 3, 1963, 185
[2] Channell, P. J., Scovel, C., Nonlinearity 3, 1990, 231
[3] Gheorghiu, C. I., Muresan, A. C., RoAJ, 2006 (submitted)
[4] Hairer, E., Lubich, C., Wanner, G., Acta Numerica, 2003, 1
[5] Hairer, E., Lubich, C., Wanner, G., Geometric Numerical Integration-Structure Preserving algorithms for Ordinary Differential Systems, 2nd Ed. Apringer Verlag, Berlin Heidelberg, 2006.
[6] Hairer, E., Hairer, M., http://www.inige.ch/math/folks/hairer
[7] Henrici, P., Discrete Variable Methods for Ordinary Differential Equations,  John-Wiley & Sons, Inc., New York, London, 1962.
[8] Higham, N. J., SIAM J. Sci. Comput. 14, 1993, 783.
[9] Kirkgraber, U., Numer. Math. 48, 1986, 85
[10] Quinn, T., Tremaine, S., Duncan, M., Astron. J., 101, 1991, 2287
[11] Zhong, G., Marsden, J., Phys. Lett. A 133, 1988, 134


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