Continuous approximations for long-term numerical simulations of the Solar System

We present and analyse the performance of di(cid:27)erent combinations of four higher-order numerical integrators and up to nine interpolation schemes applied to the problem involving the Sun and four Gas-giants (outer planets), namely, Jupiter, Saturn, Uranus, and Neptune. The Hermite interpolation schemes obtained by one, two, and three time-step and interpolants for ODEX2 and ERKN integrators are considered in this paper. The interpolants are a special example of an interpolation scheme, which produce an approximation that is continuous across one step and across the complete interval of integration. The interpolants are quite expensive in comparison with the other interpolation schemes. Therefore, one of the objectives of this paper is to investigate the possibilities of replacing the interpolants of certain integrators by other interpolation schemes, perhaps at a cost of a little bit of accuracy. The experiments are performed to examine the error growth in the positions, velocities, and relative error in energy and angular momentum using di(cid:27)erent combinations of integrators and interpolation schemes over a long interval of integration, as long as 100 million years for the Jovian problem with local error tolerances ranging from 10 − 16 to 10 − 08 .