Several fourth-order force gradient symplectic algorithms

Abstract

By adding force gradient operators to symmetric compositions, we build a set of explicit fourth-order force gradient symplectic algorithms, including those of Chin and coworkers, for a separable Hamiltonian system with quadratic kinetic energy T and potential energy V. They are extended to solve a gravitational n-body Hamiltonian system that can be split into a Keplerian part H0 and a perturbation part H1 in Jacobi coordinates. It is found that the accuracy of each gradient scheme is greatly superior to that of the standard fourth-order Forest–Ruth symplectic integrator in T + V-type Hamiltonian decomposition, but they are both almost equivalent in the mean longitude and the relative position for H0 + H1-type decomposition. At the same time, there are no typical differences between the numerical performances of these gradient algorithms, either in the splitting of T + V or in the splitting of H0 + H1. In particular, compared with the former decomposition, the latter can dramatically improve the numerical accuracy. Because this extension provides a fast and high-precision method to simulate various orbital motions of n-body problems, it is worth recommending for practical computation.