Numerical convergence in self-gravitating disc simulations: initial conditions and edge effects

TL;DR

This paper investigates the numerical convergence of self-gravitating disc simulations, revealing that initial conditions and boundary effects influence the critical cooling time for fragmentation, with convergence achieved under specific setups.

Contribution

It demonstrates that boundary effects affect convergence in disc simulations and identifies conditions for obtaining consistent critical cooling times.

Findings

Convergence depends on initial conditions and boundary effects.

A critical cooling time scale of ~4 Ω^{-1} was identified.

Boundary effects can cause non-convergence in simulations.

Abstract

We study the numerical convergence of hydrodynamical simulations of self-gravitating accretion discs, in which a simple cooling law is balanced by shock heating. It is well-known that there exists a critical cooling time scale for which shock heating can no longer compensate for the energy losses, at which point the disc fragments. The numerical convergence of previous results of this critical cooling time scale was questioned recently using Smoothed Particle Hydrodynamics (SPH). We employ a two-dimensional grid-based code to study this problem, and find that for smooth initial conditions, fragmentation is possible for slower cooling as the resolution is increased, in agreement with recent SPH results. We show that this non-convergence is at least partly due to the creation of a special location in the disc, the boundary between the turbulent and the laminar region, when cooling towards a gravito-turbulent state. Converged results appear to be obtained in setups where no such sharp edges appear, and we then find a critical cooling time scale of ~ 4 $\Omega^{-1}$, where $\Omega$ is the local angular velocity.