Gauss collocation methods for efficient structure preserving integration of post-Newtonian equations of motion

TL;DR

This paper introduces a highly efficient and accurate Gauss collocation method for long-term, structure-preserving numerical integration of post-Newtonian equations of motion, outperforming previous schemes in speed and accuracy.

Contribution

It transforms post-Newtonian equations into a symplectic form and applies Gauss Runge-Kutta schemes, providing a novel, efficient approach for their long-time integration.

Findings

Faster than previous structure-preserving splitting schemes

More accurate than standard explicit Runge-Kutta methods

Effective for long-term integration of post-Newtonian equations

Abstract

In this work, we present the hitherto most efficient and accurate method for the numerical integration of post-Newtonian equations of motion. We first transform the Poisson system as given by the post-Newtonian approximation to canonically symplectic form. Then we apply Gauss Runge-Kutta schemes to numerically integrate the resulting equations. This yields a convenient method for the structure preserving long-time integration of post-Newtonian equations of motion. In extensive numerical experiments, this approach turns out to be faster and more accurate i) than previously proposed structure preserving splitting schemes and ii) than standard explicit Runge-Kutta methods.