Time-symmetric integration in astrophysics

TL;DR

This paper investigates the errors introduced by time-reversible numerical integrators in astrophysics, analyzing their limitations and comparing them to symplectic methods, especially regarding energy conservation and adaptability to changing scales.

Contribution

It provides an analytical and numerical study of errors in time-reversible integrators, highlighting their differences from symplectic methods and their implications for astrophysical simulations.

Findings

Time-reversible integrators can cause secular energy errors.

Symplectic integrators are generally better for energy conservation.

Time-symmetry and time-reversibility are distinct properties.

Abstract

Calculating the long term solution of ordinary differential equations, such as those of the $N$-body problem, is central to understanding a wide range of dynamics in astrophysics, from galaxy formation to planetary chaos. Because generally no analytic solution exists to these equations, researchers rely on numerical methods which are prone to various errors. In an effort to mitigate these errors, powerful symplectic integrators have been employed. But symplectic integrators can be severely limited because they are not compatible with adaptive stepping and thus they have difficulty accommodating changing time and length scales. A promising alternative is time-reversible integration, which can handle adaptive time stepping, but the errors due to time-reversible integration in astrophysics are less understood. The goal of this work is to study analytically and numerically the errors caused by time-reversible integration, with and without adaptive stepping. We derive the modified differential equations of these integrators to perform the error analysis. As an example, we consider the trapezoidal rule, a reversible non-symplectic integrator, and show it gives secular energy error increase for a pendulum problem and for a H\'{e}non---Heiles orbit. We conclude that using reversible integration does not guarantee good energy conservation and that, when possible, use of symplectic integrators is favored. We also show that time-symmetry and time-reversibility are properties that are distinct for an integrator.