Entropy conservative and entropy stable solid wall boundary conditions for the resistive magnetohydrodynamic equations

TL;DR

This paper introduces a new method for imposing entropy conservative and entropy stable boundary conditions in resistive magnetohydrodynamics, ensuring stability and accuracy across various wall types using high-order discretizations.

Contribution

The paper develops a novel boundary condition enforcement technique that guarantees entropy stability for resistive MHD equations on unstructured grids, compatible with multiple numerical schemes.

Findings

Method achieves non-linear stability in 3D MHD flows.

Compatible with various high-order discretization methods.

Numerical tests confirm accuracy and robustness.

Abstract

We present a novel technique for imposing non-linear entropy conservative and entropy stable wall boundary conditions for the resistive magnetohydrodynamic equations in the presence of an adiabatic wall or a wall with a prescribed heat entropy flow, addressing three scenarios: electrically insulating walls, thin walls with finite conductivity, and perfectly conducting walls. The procedure relies on the formalism and mimetic properties of diagonal-norm, summation-by-parts, and simultaneous-approximation-term operators. Using the method of lines, a semi-discrete entropy estimate for the entire domain is obtained when the proposed numerical imposition of boundary conditions is coupled with an entropy-conservative or entropy-stable discrete interior operator. The resulting estimate mimics the global entropy estimate obtained at the continuous level. The boundary data at the wall are weakly imposed using a penalty flux approach and a simultaneous-approximation-term technique for both the conservative variables and the gradient of the entropy variables. Discontinuous spectral collocation operators (mass lumped nodal discontinuous Galerkin operators) on high-order unstructured grids are used to demonstrate the new procedure's accuracy, robustness, and efficacy for weakly enforcing boundary conditions. Numerical simulations confirm the non-linear stability of the proposed technique, with applications to three-dimensional flows. The procedure described is compatible with any diagonal-norm summation-by-parts spatial operator, including finite element, finite difference, finite volume, nodal and modal discontinuous Galerkin, and flux reconstruction schemes.