The first paper of this series presents a discretely entropy stable discontinuous Galerkin (DG) method for the resistive magnetohydrodynamics (MHD) equations on three-dimensional curvilinear unstructured hexahedral meshes. Compared to other fluid dynamics systems such as the shallow water equations or the compressible Navier–Stokes equations, the resistive MHD equations need special considerations because of the divergence-free constraint on the magnetic field. For instance, it is well known that for the symmetrization of the ideal MHD system as well as the continuous entropy analysis a non-conservative term proportional to the divergence of the magnetic field, typically referred to as the Powell term, must be included. As a consequence, the mimicry of the continuous entropy analysis in the discrete sense demands a suitable DG approximation of the non-conservative terms in addition to the ideal MHD terms.

This paper focuses on the resistive MHD equations: Our first contribution is a proof that the resistive terms are symmetric and positive-definite when formulated in entropy space as gradients of the entropy variables, which enables us to show that the entropy inequality holds for the resistive MHD equations. This continuous analysis is the key for our DG discretization and guides the path for the construction of an approximation that discretely mimics the entropy inequality, typically termed entropy stability. Our second contribution is a detailed derivation and analysis of the discretization on three-dimensional curvilinear meshes. The discrete analysis relies on the summation-by-parts property, which is satisfied by the DG spectral element method (DGSEM) with Legendre–Gauss–Lobatto (LGL) nodes. Although the divergence-free constraint is included in the non-conservative terms, the resulting method has no particular treatment of the magnetic field divergence errors, which might pollute the solution quality. Our final contribution is the extension of the standard resistive MHD equations and our DG approximation with a divergence cleaning mechanism that is based on a generalized Lagrange multiplier (GLM).

As a conclusion to the first part of this series, we provide detailed numerical validations of our DGSEM method that underline our theoretical derivations. In addition, we show a numerical example where the entropy stable DGSEM demonstrates increased robustness compared to the standard DGSEM.

该系列的第一篇论文提出了离散熵稳定不连续Galerkin（DG）方法，用于求解三维曲线非结构六面体网格上的电阻磁流体动力学（MHD）方程。与其他流体动力学系统（例如浅水方程或可压缩Navier–Stokes方程）相比，电阻MHD方程由于对磁场的无发散约束而需要特别考虑。例如，众所周知，为了理想MHD系统的对称化以及连续熵分析，必须包括与磁场的发散成比例的非保守项，通常称为鲍威尔项。作为结果，

本文关注于电阻MHD方程：我们的第一个贡献是证明当在熵空间中用熵变量的梯度表示时，电阻项是对称的且是正定的，这使我们能够证明熵不等式对于电阻性成立。 MHD方程。连续分析是DG离散化的关键，并为构建近似模拟熵不等式（通常称为熵稳定性）的近似方法提供了指导。我们的第二个贡献是对三维曲线网格离散化的详细推导和分析。离散分析依赖于部分求和属性，该属性通过带有Legendre–Gauss–Lobatto（LGL）节点的DG光谱元素方法（DGSEM）满足。尽管无散度约束包括在非保守项中，但所得方法没有对磁场散度误差进行特殊处理，这可能会污染溶液质量。我们的最终贡献是扩展了标准电阻MHD方程，并利用基于广义拉格朗日乘数（GLM）的发散清除机制对DG进行了近似。

作为本系列第一部分的结论，我们提供了我们的DGSEM方法的详细数值验证，这些验证了我们的理论推论。此外，我们显示了一个数值示例，其中与标准DGSEM相比，熵稳定DGSEM表现出更高的鲁棒性。