Towards time symmetric N-body integration

TL;DR

This paper explores methods to improve the time symmetry of adaptive time-stepping schemes in N-body simulations, aiming to reduce artificial dissipation and energy errors for more accurate long-term integrations.

Contribution

It investigates approaches to make individual adaptive time steps more time-symmetric, enhancing the accuracy and stability of N-body simulations.

Findings

Significant improvements are achievable with minimal extra effort.

Reducing irreversibility to negligible levels requires more computational effort.

Exact time reversibility remains difficult with discretised individual step sizes.

Abstract

Computational efficiency demands discretised, hierarchically organised, and individually adaptive time-step sizes (known as the block-step scheme) for the time integration of N-body models. However, most existing N-body codes adapt individual step sizes in a way that violates time symmetry (and symplecticity), resulting in artificial secular dissipation (and often secular growth of energy errors). Using single-orbit integrations, I investigate various possibilities to reduce or eliminate irreversibility from the time stepping scheme. Significant improvements over the standard approach are possible at little extra effort. However, in order to reduce irreversible step-size changes to negligible amounts, such as suitable for long-term integrations of planetary systems, more computational effort is needed, while exact time reversibility appears elusive for discretised individual step sizes.