Positivity-preserving entropy filtering for the ideal magnetohydrodynamics equations

TL;DR

This paper introduces a positivity-preserving adaptive filtering method for ideal magnetohydrodynamics equations that ensures physical constraints are maintained while accurately resolving shocks and discontinuities on unstructured grids.

Contribution

It combines entropy filtering with divergence-free magnetic field enforcement and introduces an operator-splitting approach for non-conservative source terms, providing a robust high-order scheme.

Findings

Successfully resolves strong discontinuities and shocks

Maintains positivity and entropy constraints

Achieves high-order accuracy for smooth solutions

Abstract

In this work, we present a positivity-preserving adaptive filtering approach for discontinuous spectral element approximations of the ideal magnetohydrodynamics equations. This approach combines the entropy filtering method (Dzanic and Witherden, J. Comput. Phys., 468, 2022) for shock capturing in gas dynamics along with the eight-wave method for enforcing a divergence-free magnetic field. Due to the inclusion of non-conservative source terms, an operator-splitting approach is introduced to ensure that the positivity and entropy constraints remain satisfied by the discrete solution. Furthermore, a computationally efficient algorithm for solving the optimization process for this nonlinear filtering approach is presented. The resulting scheme can robustly resolve strong discontinuities on general unstructured grids without tunable parameters while recovering high-order accuracy for smooth solutions. The efficacy of the scheme is shown in numerical experiments on various problems including extremely magnetized blast waves and three-dimensional magnetohydrodynamic instabilities.