Constraint preserving discontinuous Galerkin method for ideal compressible MHD on 2-D Cartesian grids

TL;DR

This paper introduces a novel discontinuous Galerkin method for 2D ideal compressible MHD that preserves the divergence-free magnetic field property automatically, ensuring physical accuracy in simulations.

Contribution

The method uniquely combines Raviart-Thomas polynomials with DG schemes and develops divergence-free reconstruction to maintain magnetic field divergence-free condition.

Findings

Achieves up to fourth order accuracy in tests

Effectively maintains divergence-free magnetic fields

Performs well on a range of benchmark problems

Abstract

We propose a constraint preserving discontinuous Galerkin method for ideal compressible MHD in two dimensions and using Cartesian grids, which automatically maintains the global divergence-free property. The approximation of the magnetic field is achieved using Raviart-Thomas polynomials and the DG scheme is based on evolving certain moments of these polynomials which automatically guarantees divergence-free property. We also develop HLL-type multi-dimensional Riemann solvers to estimate the electric field at vertices which are consistent with the 1-D Riemann solvers. When limiters are used, the divergence-free property may be lost and it is recovered by a divergence-free reconstruction step. We show the performance of the method on a range of test cases up to fourth order of accuracy.