Forward Symplectic Integrators for Solving Gravitational Few-Body Problems

TL;DR

This paper introduces fourth order forward symplectic algorithms with positive time steps for efficient long-term integration of gravitational few-body problems, outperforming traditional methods in accuracy and step size.

Contribution

The paper presents a new class of fourth order symplectic integrators that require force gradients and are optimized for long-term gravitational simulations.

Findings

Achieve accuracy comparable to Runge-Kutta methods at larger step sizes.

Demonstrated efficiency in the circularly restricted three-body problem.

Require only positive time steps, enhancing stability and performance.

Abstract

We introduce a class of fourth order symplectic algorithms that are ideal for doing long time integration of gravitational few-body problems. These algorithms have only positive time steps, but require computing the force gradient in additional to the force. We demonstrate the efficiency of these Forward Symplectic Integrators by solving the circularly restricted three-body problem in the space-fixed frame where the force on the third body is explicitly time-dependent. These algorithms can achieve accuracy of Runge-Kutta, backward time step and corrector symplectic algorithms at step sizes five to ten times as large.