Improving the convergence properties of the moving-mesh code AREPO

TL;DR

This paper enhances the moving-mesh code AREPO by implementing simple modifications that improve its convergence order from first to second, significantly benefiting simulations requiring high angular momentum conservation.

Contribution

The authors propose and validate modifications to AREPO's time integration and gradient estimation, achieving second-order accuracy and better angular momentum conservation.

Findings

AREPO's original formulation is only first-order accurate for some problems.

Modified scheme achieves second-order accuracy under the L1 norm.

Improvements significantly impact binary star simulations, less so cosmological galaxy formation.

Abstract

Accurate numerical solutions of the equations of hydrodynamics play an ever more important role in many fields of astrophysics. In this work, we reinvestigate the accuracy of the moving-mesh code \textsc{Arepo} and show how its convergence order can be improved for general problems. In particular, we clarify that for certain problems \textsc{Arepo} only reaches first-order convergence for its original formulation. This can be rectified by simple modifications we propose to the time integration scheme and the spatial gradient estimates of the code, both improving the accuracy of the code. We demonstrate that the new implementation is indeed second-order accurate under the $L^1$ norm, and in particular substantially improves conservation of angular momentum. Interestingly, whereas these improvements can significantly change the results of smooth test problems, we also find that cosmological simulations of galaxy formation are unaffected, demonstrating that the numerical errors eliminated by the new formulation do not impact these simulations. In contrast, simulations of binary stars followed over a large number of orbital times are strongly affected, as here it is particularly crucial to avoid a long-term build up of errors in angular momentum conservation.