A general-purpose time-step criterion for simulations with gravity

Abstract

We describe a new adaptive time-step criterion for integrating gravitational motion, which uses the tidal tensor to estimate the local dynamical time-scale and scales the time-step proportionally. This provides a better candidate for a truly general-purpose gravitational time-step criterion than the usual prescription derived from the gravitational acceleration, which does not respect the equivalence principle, breaks down when a=0⁠, and does not obey the same dimensional scaling as the true time-scale of orbital motion. We implement the tidal time-step criterion in the simulation code GIZMO, and examine controlled tests of collisionless galaxy and star cluster models, as well as galaxy merger simulations. The tidal criterion estimates the dynamical time faithfully, and generally provides a more efficient time-stepping 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 ρ ∝ r⁻¹ (i.e. where a → 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.