In this paper, we present an entropy-stable (ES) discretization using a nodal discontinuous Galerkin (DG) method for the ideal multi-ion magneto-hydrodynamics (MHD) equations. We start by performing a continuous entropy analysis of the ideal multi-ion MHD system, described by, e.g., [Toth (2010) Multi-Ion Magnetohydrodynamics] \cite{Toth2010}, which describes the motion of multi-ion plasmas with independent momentum and energy equations for each ion species. Following the continuous entropy analysis, we propose an algebraic manipulation to the multi-ion MHD system, such that entropy consistency can be transferred from the continuous analysis to its discrete approximation. Moreover, we augment the system of equations with a generalized Lagrange multiplier (GLM) technique to have an additional cleaning mechanism of the magnetic field divergence error. We first derive robust entropy-conservative (EC) fluxes for the alternative formulation of the multi-ion GLM-MHD system that satisfy a Tadmor-type condition and are consistent with existing EC fluxes for single-fluid GLM-MHD equations. Using these numerical two-point fluxes, we construct high-order EC and ES DG discretizations of the ideal multi-ion MHD system using collocated Legendre--Gauss--Lobatto summation-by-parts (SBP) operators. The resulting nodal DG schemes satisfy the second-law of thermodynamics at the semi-discrete level, while maintaining high-order convergence and local node-wise conservation properties. We demonstrate the high-order convergence, and the EC and ES properties of our scheme with numerical validation experiments. Moreover, we demonstrate the importance of the GLM divergence technique and the ES discretization to improve the robustness properties of a DG discretization of the multi-ion MHD system by solving a challenging magnetized Kelvin-Helmholtz instability problem that exhibits MHD turbulence.

翻译：在本文中，我们针对理想多离子磁流体动力学（MHD）方程组，提出了一种采用节点型间断伽辽金（DG）方法的熵稳定（ES）离散格式。首先，我们对理想多离子MHD系统进行连续熵分析，该系统的描述参见例如[Toth (2010) Multi-Ion Magnetohydrodynamics] \cite{Toth2010}，其中每种离子物种具有独立的动量方程和能量方程，用于描述多离子等离子体的运动。在连续熵分析的基础上，我们提出对多离子MHD系统进行代数变换，使得熵相容性能够从连续分析传递到其离散近似中。此外，我们采用广义拉格朗日乘子（GLM）技术对方程组进行增广，以增加对磁场散度误差的额外清理机制。我们首先针对多离子GLM-MHD系统的替代公式，推导出满足Tadmor型条件且与单流体GLM-MHD方程中现有熵守恒（EC）通量一致的鲁棒熵守恒通量。利用这些数值两点通量，我们采用配点型Legendre-Gauss-Lobatto求和-积分（SBP）算子，构建了理想多离子MHD系统的高阶EC和ES DG离散格式。所得的节点型DG方案在半离散层面满足热力学第二定律，同时保持高阶收敛性和局部逐点守恒性质。我们通过数值验证实验证明了该格式的高阶收敛性以及EC和ES性质。此外，通过求解一个具有MHD湍流特征、具有挑战性的磁化开尔文-亥姆霍兹不稳定性问题，我们展示了GLM散度技术和ES离散格式对于增强多离子MHD系统DG离散格式鲁棒性的重要性。