New families of symplectic splitting methods for numerical integration in dynamical astronomy
Abstract
We present new splitting methods designed for the numerical integration of nearintegrable Hamiltonian systems, and in particular for planetary Nbody problems, when one is interested in very accurate results over a large time span. We derive in a systematic way an independent set of necessary and sufficient conditions to be satisfied by the coefficients of splitting methods to achieve a prescribed order of accuracy. Splitting methods satisfying such (generalized) order conditions are appropriate in particular for the numerical simulation of the Solar System described in Jacobi coordinates. We show that, when using Poincaré Heliocentric coordinates, the same order of accuracy may be obtained by imposing an additional polynomial equation on the coefficients of the splitting method. We construct several splitting methods appropriate for each of the two sets of coordinates by solving the corresponding systems of polynomial equations and finding the optimal solutions. The experiments reported here indicate that the efficiency of our new schemes is clearly superior to previous integrators when high accuracy is required.
 Publication:

arXiv eprints
 Pub Date:
 August 2012
 arXiv:
 arXiv:1208.0689
 Bibcode:
 2012arXiv1208.0689B
 Keywords:

 Mathematics  Numerical Analysis;
 Astrophysics  Earth and Planetary Astrophysics;
 Physics  Computational Physics
 EPrint:
 24 pages, 2 figures. Revised version, accepted for publication in Applied Numerical Mathematics