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.

翻译：非线性守恒律（如理想磁流体动力学（MHD）方程组）可能随时间发展出奇异性。在此类情形下，粘性正则化是恢复解正则性的常用方法。本文提出一种用于正则化MHD方程组的新型粘性通量，该通量具有诸多优良性质。具体而言，我们证明了所提出的粘性通量能保持密度与内能的正性，满足最小熵原理，与所有广义熵相容，并具有伽利略不变性和旋转不变性。我们还给出了可守恒角动量的粘性通量变体形式。为使分析结果更适用于数值格式，这里并未假设磁场散度为零。通过连续有限元方法，我们展示了包含接触波和磁重联在内的若干数值实验。