ADER discontinuous Galerkin schemes for general-relativistic ideal magnetohydrodynamics

TL;DR

This paper introduces a new high-order discontinuous Galerkin method for solving general-relativistic ideal magnetohydrodynamics equations, combining robustness at shocks with high accuracy in smooth regions, and demonstrating superior efficiency over finite-volume schemes.

Contribution

The paper develops a novel high-order DG scheme with an a-posteriori subcell limiter for relativistic MHD, enhancing robustness and accuracy in curved spacetimes.

Findings

The new DG scheme effectively handles shocks and discontinuities.

The method achieves high accuracy in smooth flow regions.

DG schemes outperform finite-volume schemes in efficiency.

Abstract

We present a new class of high-order accurate numerical algorithms for solving the equations of general-relativistic ideal magnetohydrodynamics in curved spacetimes. In this paper we assume the background spacetime to be given and static, i.e., we make use of the Cowling approximation. The governing partial differential equations are solved via a new family of fully-discrete and arbitrary high-order accurate path-conservative discontinuous Galerkin (DG) finite-element methods combined with adaptive mesh refinement and time accurate local timestepping. In order to deal with shock waves and other discontinuities, the highorder DG schemes are supplemented with a novel a-posteriori subcell finite-volume limiter, which makes the new algorithms as robust as classical second-order total-variation diminishing finite-volume methods at shocks and discontinuities, but also as accurate as unlimited high-order DG schemes in smooth regions of the flow. We show the advantages of this new approach by means of various classical two- and three-dimensional benchmark problems on fixed spacetimes. Finally, we present a performance and accuracy comparisons between Runge-Kutta DG schemes and ADER high-order finite-volume schemes, showing the higher efficiency of DG schemes.