A comparison of approximate non-linear Riemann solvers for Relativistic MHD

TL;DR

This paper compares five approximate Riemann solvers for ideal relativistic magnetohydrodynamics, evaluating their accuracy and robustness across various benchmarks to identify the most effective methods for different magnetization regimes.

Contribution

It provides a systematic comparison of non-eigenvector-based Riemann solvers, highlighting the strengths and weaknesses of each in relativistic MHD simulations.

Findings

HLLD yields the most accurate results for weak/moderate magnetizations.

GFORCE is less dissipative and robust in strongly magnetized environments.

HLLEM is not cost-effective for improving accuracy.

Abstract

We compare a particular selection of approximate solutions of the Riemann problem in the context of ideal relativistic magnetohydrodynamics. In particular, we focus on Riemann solvers not requiring a full eigenvector structure. Such solvers recover the solution of the Riemann problem by solving a simplified or reduced set of jump conditions, whose level of complexity depends on the intermediate modes that are included. Five different approaches - namely the HLL, HLLC, HLLD, HLLEM and GFORCE schemes - are compared in terms of accuracy and robustness against one- and multi-dimensional standard numerical benchmarks. Our results demonstrate that - for weak or moderate magnetizations - the HLLD Riemann solver yields the most accurate results, followed by HLLC solver(s). The GFORCE approach provides a valid alternative to the HLL solver being less dissipative and equally robust for strongly magnetized environments. Finally, our tests show that the HLLEM Riemann solver is not cost-effective in improving the accuracy of the solution and reducing the numerical dissipation.