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], 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：本文针对理想多离子磁流体动力学(MHD)方程组，提出一种基于节点间断伽辽金(DG)方法的熵稳定(ES)离散格式。我们首先对Toth(2010)[多离子磁流体动力学]中描述的理想多离子MHD系统进行连续熵分析，该系统通过为每种离子组分建立独立的动量和能量方程来描述多离子等离子体运动。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，使得连续分析中的熵一致性能够传递到其离散近似中。此外，我们采用广义拉格朗日乘子(GLM)技术扩充原始方程组，以增强对磁场散度误差的清除机制。首先为多离子GLM-MHD系统的替代公式推导满足Tadmor型条件且与单流体GLM-MHD方程现有EC通量一致的鲁棒熵守恒(EC)通量。利用这些数值两点通量，通过采用配置型Legendre-Gauss-Lobatto求和-by-部分(SBP)算子，构建理想多离子MHD系统的高阶EC与ES DG离散格式。所得到的节点DG格式在半离散层面满足热力学第二定律，同时保持高阶收敛性与局部节点守恒特性。我们通过数值验证实验证明了格式的高阶收敛性、EC与ES性质。此外，通过求解展现MHD湍流特性的磁化开尔文-亥姆霍兹不稳定性难题，验证了GLM散度处理技术与ES离散格式对提升多离子MHD系统DG离散格式鲁棒性的重要作用。