Viscous regularization of the MHD equations

TL;DR

This work develops a physically motivated viscous regularization for the ideal MHD equations by blending a Guermond-Popov–type viscous flux for the hydrodynamic part with a resistive-like diffusion for the magnetic field. The GP-MHD formulation preserves density positivity, the minimum entropy principle, generalized entropy inequalities, and Galilean/rotational invariance, and a symmetry-structured variant GP^s achieves angular-momentum conservation. The authors also analyze divergence-cleaning strategies (GLM and related formulations) and their impact on these properties, including an energy-conserving GLM variant. Numerical experiments using continuous finite elements and residual-based viscosity demonstrate high-order accuracy on smooth solutions and robust capture of shocks and discontinuities, with reconnection rates in GEM comparable to resistive MHD benchmarks. Overall, the GP family provides a rigorous, thermodynamically consistent framework for artificial viscosity and resistivity modeling in MHD simulations, including an angular-momentum–conserving extension and compatibility with divergence-control mechanisms.

Abstract

Nonlinear conservation laws such as the system of ideal magnetohydrodynamics (MHD) equations may develop singularities over time. In these situations, viscous regularization is a common approach to regain regularity of the solution. In this paper, we present a new viscous flux to regularize the MHD equations which holds many attractive properties. In particular, we prove that the proposed viscous flux preserves positivity of density and internal energy, satisfies the minimum entropy principle, is consistent with all generalized entropies, and is Galilean and rotationally invariant. We also provide a variation of the viscous flux that conserves angular momentum. To make the analysis more useful for numerical schemes, the divergence of the magnetic field is not assumed to be zero. Using continuous finite elements, we show several numerical experiments including contact waves and magnetic reconnection.

Figures (10)

• Figure 1: The resistive MHD flux violates the maximum principle around the location of the contact line. First order viscosity, ${\mathbb P}_1$ and ${\mathbb P}_3$ elements. Left: GP flux; ${\mathbb P}_1$ and ${\mathbb P}_3$ solutions preserve bounds exactly. Right: resistive MHD flux; for visibility only the solutions with 601 DOFs are shown in the zoomed-in plots. Note that GP and GP$^s$ are same in 1D.
• Figure 1: A comparison of the GP flux and the resistive MHD flux using ${\mathbb P}_1$ elements for the Brio-Wu problem: the density profile. Left figure: first-order viscosity is used. Right figure: high-order viscosity using the RV method is used.
• Figure 1: Conservation of angular momentum $\int_{\Omega}{\boldsymbol m}\times{\boldsymbol x}\,\mathrm{d}{\boldsymbol x}$ by different viscous fluxes: resistive MHD, GP, GP$^s$, and monolithic Dao2022a. Artificial viscosity by the RV method. ${\mathbb P}_1$ elements. The comparison is separated into two figures because the scales of GP and monolithic lines are different. Right figure: the monolithic line is divided by 5000. The GP$^s$ and the resistive MHD fluxes conserve angular momentum to ${\mathcal{O}}(10^{-11})$.
• Figure 1: Density of the Orszag-Tang solution $t=0.5$. Zoomed-in region $[0.4,0.6]\times[0.4,0.6]$
• Figure 2: Investigation of discrete minimum entropy principle on the contact line problem. First order viscosity, ${\mathbb P}_1$ elements. The GP/GP$^s$ flux preserves $\min_{\Omega}(s_h)$ to machine precision. The resistive MHD flux produces unphysical behaviors in $\min_{\Omega}(s_h)$.

Theorems & Definitions (28)

• Theorem 3.1: Positivity of density
• Lemma 3.2
• Proof 1
• Lemma 3.3
• Proof 2
• Theorem 3.4: Minimum entropy principle
• Proof 3
• Theorem 3.5: Positivity of internal energy
• Proof 4
• Theorem 3.6: Properties of strictly convex generalized entropies Dao2022a