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We show at the PDE level that the monolithic parabolic regularization of theequations of ideal magnetohydrodynamics (MHD) is compatible with all thegeneralized entropies, fulfills the minimum entropy principle, and preservesthe positivity of density and internal energy. We then numerically investigatethis regularization for the MHD equations using continuous finite elements inspace and explicit strong stability preserving Runge-Kuta methods in time. Theartificial viscosity coefficient of the regularization term is constructed tobe proportional to the entropy residual of MHD. It is shown that the method hasa high order of accuracy for smooth problems and captures strong shocks anddiscontinuities accurately for non-smooth problems.

Monolithic parabolic regularization of the MHD equations and entropy principles