The second paper of this series presents two robust entropy stable shock-capturing methods for discontinuous Galerkin spectral element (DGSEM) discretizations of the compressible magneto-hydrodynamics (MHD) equations. Specifically, we use the resistive GLM-MHD equations, which include a divergence cleaning mechanism that is based on a generalized Lagrange multiplier (GLM). For the continuous entropy analysis to hold, and due to the divergence-free constraint on the magnetic field, the GLM-MHD system requires the use of non-conservative terms, which need special treatment.

Hennemann et al. (2020) [25] recently presented an entropy stable shock-capturing strategy for DGSEM discretizations of the Euler equations that blends the DGSEM scheme with a subcell first-order finite volume (FV) method. Our first contribution is the extension of the method of Hennemann et al. to systems with non-conservative terms, such as the GLM-MHD equations. In our approach, the advective and non-conservative terms of the equations are discretized with a hybrid FV/DGSEM scheme, whereas the visco-resistive terms are discretized only with the high-order DGSEM method. We prove that the extended method is semi-discretely entropy stable on three-dimensional unstructured curvilinear meshes. Our second contribution is the derivation and analysis of a second entropy stable shock-capturing method that provides enhanced resolution by using a subcell reconstruction procedure that is carefully built to ensure entropy stability.

We provide a numerical verification of the properties of the hybrid FV/DGSEM schemes on curvilinear meshes and show their robustness and accuracy with common benchmark cases, such as the Orszag-Tang vortex and the GEM (Geospace Environmental Modeling) reconnection challenge. Finally, we simulate a space physics application: the interaction of Jupiter's magnetic field with the plasma torus generated by the moon Io.

本系列的第二篇论文针对可压缩磁流体动力学 (MHD) 方程的不连续伽辽金谱元 (DGSEM) 离散化提出了两种稳健的熵稳定激波捕获方法。具体来说，我们使用电阻 GLM-MHD 方程，其中包括基于广义拉格朗日乘子 (GLM) 的发散清理机制。为了使连续熵分析成立，并且由于磁场的无散度约束，GLM-MHD 系统需要使用非保守项，这需要特殊处理。

亨尼曼等人。(2020) [25] 最近提出了一种用于欧拉方程的 DGSEM 离散化的熵稳定冲击捕获策略，该策略将 DGSEM 方案与子单元一阶有限体积 (FV) 方法相结合。我们的第一个贡献是对 Hennemann 等人的方法的扩展。到具有非保守项的系统，例如 GLM-MHD 方程。在我们的方法中，方程的平流项和非保守项使用混合 FV/DGSEM 方案离散化，而粘阻项仅使用高阶 DGSEM 方法离散化。我们证明了扩展方法在三维非结构化曲线网格上是半离散熵稳定的。

我们对曲线网格上的混合 FV/DGSEM 方案的属性进行了数值验证，并通过常见的基准案例展示了它们的稳健性和准确性，例如 Orszag-Tang 涡流和 GEM（地理空间环境建模）重新连接挑战。最后，我们模拟了一个空间物理应用：木星磁场与卫星 Io 产生的等离子体环面的相互作用。