On a regularization of the magnetic gas dynamics system of equations
Bernard Ducomet, Alexander Zlotnik
TL;DR
The paper addresses regularizing the magnetic gas dynamics system of equations for a viscous compressible gas in the presence of a body force and heat source. It develops a regularization framework introducing a relaxation parameter $\tau>0$ and auxiliary fields $\mathbf w$ and $\widehat{\mathbf w}$, yielding a regularized MGD system with a non-symmetric viscous stress tensor $\Pi$, a regularized heat flux $\mathbf q$, and a Faraday equation that preserves $\nabla\cdot \mathbf B=0$. The authors present the regularized mass, momentum, total energy, and internal energy balances in a standard form, along with two versions of the entropy balance equation. They prove that the entropy production rate is nonnegative when $\tau$ is constant and $\tau Q/(4\rho\theta\varepsilon_\theta)\le 1$, thereby ensuring physical admissibility and assisting discretization. The results generalize prior quasi-gas-dynamics formulations to the magnetic setting with body forces and heat sources, providing a theoretically sound basis for kinetically consistent numerical methods and simulations of MGD phenomena.
Abstract
A brief derivation of a specific regularization for the magnetic gas dynamic system of equations is given in the case of general equations of gas state (in presence of a body force and a heat source). The entropy balance equation in two forms is also derived for the system. For a constant regularization parameter and under a standard condition on the heat source, we show that the entropy production rate is nonnegative.