A positivity preserving and entropy stable nodal discontinuous Galerkin scheme for ideal MHD equations
For researchers solving MHD equations, this work provides a unified DG method addressing three major numerical challenges, though it is an incremental combination of existing techniques.
The authors developed a discontinuous Galerkin scheme for ideal MHD equations that simultaneously ensures positivity, entropy stability, and divergence-free conditions, combining advantages of previous modal and nodal approaches. Numerical experiments demonstrate accuracy and robustness.
Numerically solving magnetohydrodynamic (MHD) equations faces many challenges: avoiding divergence error, maintaining positivity, and satisfying entropy conditions. Among discontinuous Galerkin (DG) schemes, there has been a modal version that is locally divergence-free and positivity preserving and a nodal version that is entropy stable. In this work, we develop a DG scheme that combines the advantages of these two and solves all the three challenges. The key ingredients that bring these two schemes together are an HLL numerical flux with entropy stable signal speed estimates and a locally divergence-free projection. To handle problems with strong shocks, the essentially oscillation-free damping is applied. Various numerical experiments verify the accuracy and robustness of our method.