Embedded operator splitting methods for perturbed systems

Abstract

It is common in classical mechanics to encounter systems whose Hamiltonian $H$ is the sum of an often exactly integrable Hamiltonian $H_0$ and a small perturbation $\epsilon H_1$ with $\epsilon\ll1$. Such near-integrability can be exploited to construct particularly accurate operator splitting methods to solve the equations of motion of $H$. However, in many cases, for example in problems related to planetary motion, it is computationally expensive to obtain the exact solution to $H_0$. In this paper we present a new family of embedded operator splitting (EOS) methods which do not use the exact solution to $H_0$, but rather approximate it with yet another, embedded operator splitting method. Our new methods have all the desirable properties of classical methods which solve $H_0$ directly. But in addition they are very easy to implement and in some cases faster. When applied to the problem of planetary motion, our EOS methods have error scalings identical to that of the often used Wisdom-Holman method but do not require a Kepler solver, nor any coordinate transformations, or the allocation of memory. The only two problem specific functions that need to be implemented are the straight-forward kick and drift steps typically used in the standard second order leap-frog method.