WHFast: A fast and unbiased implementation of a symplectic Wisdom-Holman integrator for long term gravitational simulations

TL;DR

WHFast is a highly efficient and accurate symplectic integrator for long-term planetary orbit simulations, improving energy conservation and computational speed over previous methods.

Contribution

The paper introduces WHFast, a novel implementation of the Wisdom-Holman integrator with enhanced Kepler-solver and stability, achieving superior accuracy and speed.

Findings

WHFast conserves energy better than previous integrators.

WHFast is significantly faster in simulations.

It accurately computes chaos indicators like LCN and MEGNO.

Abstract

We present WHFast, a fast and accurate implementation of a Wisdom-Holman symplectic integrator for long-term orbit integrations of planetary systems. WHFast is significantly faster and conserves energy better than all other Wisdom-Holman integrators tested. We achieve this by significantly improving the Kepler-solver and ensuring numerical stability of coordinate transformations to and from Jacobi coordinates. These refinements allow us to remove the linear secular trend in the energy error that is present in other implementations. For small enough timesteps we achieve Brouwer's law, i.e. the energy error is dominated by an unbiased random walk due to floating-point round-off errors. We implement symplectic correctors up to order eleven that significantly reduce the energy error. We also implement a symplectic tangent map for the variational equations. This allows us to efficiently calculate two widely used chaos indicators the Lyapunov characteristic number (LCN) and the Mean Exponential Growth factor of Nearby Orbits (MEGNO). WHFast is freely available as a flexible C package, as a shared library, and as an easy-to-use python module.