A general-purpose timestep criterion for simulations with gravity

TL;DR

This paper introduces a new adaptive timestep criterion based on the tidal tensor for gravitational simulations, improving accuracy and efficiency over traditional acceleration-based methods in various astrophysical models.

Contribution

The paper presents a novel tidal tensor-based timestep criterion that better respects physical principles and enhances simulation accuracy and efficiency.

Findings

The tidal criterion accurately estimates local dynamical timescales.

It provides smaller energy errors compared to acceleration criteria.

The overhead of computing the tidal tensor is often minimal.

Abstract

We describe a new adaptive timestep criterion for integrating gravitational motion, which uses the tidal tensor to estimate the local dynamical timescale and scales the timestep proportionally. This provides a better candidate for a truly general-purpose gravitational timestep criterion than the usual prescription derived from the gravitational acceleration, which does not respect the equivalence principle, breaks down when $\mathbf{a}=0$, and does not obey the same dimensional scaling as the true timescale of orbital motion. We implement the tidal timestep criterion in the simulation code GIZMO, and examine controlled tests of collisionless galaxy and star cluster models, as well as fully-dynamic galaxy merger and cosmological dark matter simulations. The tidal criterion estimates the dynamical time faithfully, and generally provides a more efficient timestepping scheme compared to an acceleration criterion. Specifically, the tidal criterion achieves order-of-magnitude smaller energy errors for the same number of force evaluations in potentials with inner profiles shallower than $\rho \propto r^{-1}$ (ie. where $\mathbf{a}\rightarrow 0$), such as star clusters and cored galaxies. For a given problem these advantages must be weighed against the additional overhead of computing the tidal tensor on-the-fly, but in many cases this overhead is small.