Modified Hermite Integrators of Arbitrary Order

TL;DR

This paper introduces a family of high-order modified Hermite integrators that improve the accuracy of orbital simulations in systems like planetary systems and galactic centers, with minimal implementation effort.

Contribution

It derives a corrector expression for arbitrary order Hermite integrators that reduces periapsis errors and enhances performance in near-Keplerian orbit integrations.

Findings

Higher-order schemes outperform lower-order ones in accuracy.

The new correctors effectively minimize periapsis argument errors.

The integrators are suitable for parallel computing architectures.

Abstract

We present a family of modified Hermite integrators of arbitrary order possessing superior behaviour for the integration of Keplerian and near-Keplerian orbits. After recounting the derivation of Hermite N-body integrators of arbitrary order, we derive a corrector expression that minimises integrated errors in the argument of periapsis for any such integrator. In addition to providing an alternate derivation of the modified corrector for the 4th-order Hermite integrator, we focus on improved correctors for the 6th- and 8th-order integrators previously featured in the literature. We present a set of numerical examples and find that the higher-order schemes improve performance, even when considering their slightly higher cost in floating point operations. The algorithms presented herein hold promise for systems dominated by central potentials, such as planetary systems and the centres of galaxies. Existing Hermite integrators of any order can be modified to use the expressions presented here with minimal effort. Accordingly the schemes presented herein can be easily implemented on massively parallel architectures.