Symplectic fourth-order maps for the collisional N-body problem

TL;DR

This paper develops symplectic, fourth-order integrators for the N-body problem that accurately handle close two-body interactions using a Kepler solver, with corrections for higher-order accuracy and extensions for selective interaction treatment.

Contribution

It introduces a novel fourth-order symplectic integrator employing Kepler solvers for pairwise interactions, including a generalized version for selective interaction treatment without backward steps.

Findings

Achieves fourth-order accuracy with minimal computational overhead.

Correctly handles close two-body interactions using a Kepler solver.

Extends to integrators that treat only selected interactions.

Abstract

We study analytically and experimentally certain symplectic and time-reversible N-body integrators which employ a Kepler solver for each pair-wise interaction, including the method of Hernandez & Bertschinger (2015). Owing to the Kepler solver, these methods treat close two-body interactions correctly, while close three-body encounters contribute to the truncation error at second order and above. The second-order errors can be corrected to obtain a fourth-order scheme with little computational overhead. We generalise this map to an integrator which employs a Kepler solver only for selected interactions and yet retains fourth-order accuracy without backward steps. In this case, however, two-body encounters not treated via a Kepler solver contribute to the truncation error.