This work concerns the numerical approximation of a multicomponent compressible Euler system for a fluid mixture in multiple space dimensions on unstructured meshes with a high-order discontinuous Galerkin spectral element method (DGSEM). We first derive an entropy stable (ES) and robust (i.e., that preserves the positivity of the partial densities and internal energy) three-point finite volume scheme using relaxation-based approximate Riemann solvers from Bouchut (2004) [9] and Coquel and Perthame (1998) [22]. Then, we consider the DGSEM based on collocation of quadrature and interpolation points which relies on the framework introduced by Fisher and Carpenter (2013) [28] and Gassner (2013) [31]. We replace the physical fluxes in the integrals over discretization elements by entropy conservative numerical fluxes [Tadmor (1987) [71]], while ES numerical fluxes are used at element interfaces. We thus derive a two-point numerical flux satisfying the Tadmor's entropy conservation condition and use the numerical flux from the three-point scheme as ES flux. Time discretization is performed with a strong-stability preserving Runge-Kutta scheme. We then derive conditions on the numerical parameters to guaranty a semi-discrete entropy inequality as well as positivity of the cell average of the partial densities and internal energy of the fully discrete DGSEM at any approximation order. The later results allow to use existing limiters in order to restore positivity of nodal values within elements. The scheme also resolves exactly stationary material interfaces. Numerical experiments in one and two space dimensions on flows with discontinuous solutions support the conclusions of our analysis and highlight stability, robustness and high resolution of the scheme.

这项工作涉及使用高阶不连续伽辽金谱元法 (DGSEM) 对非结构化网格上多维空间流体混合物的多分量可压缩欧拉系统进行数值近似。我们首先使用 Bouchut (2004) [9] 和 Coquel 和珀萨姆 (1998) [22]。然后，我们考虑了基于正交和插值点搭配的 DGSEM，它依赖于 Fisher 和 Carpenter (2013) [28] 和 Gassner (2013) [31] 引入的框架。我们用熵保守数值通量替换离散化元素积分中的物理通量 [Tadmor (1987) [71]]，而 ES 数值通量用于单元界面。因此，我们推导出满足 Tadmor 熵守恒条件的两点数值通量，并将来自三点方案的数值通量用作 ES 通量。时间离散化采用强稳定性保持 Runge-Kutta 方案。然后我们推导出数值参数的条件，以保证半离散熵不等式以及完全离散 DGSEM 在任何近似阶数下的部分密度和内能的单元平均值的正性。后来的结果允许使用现有的限制器来恢复元素内节点值的正性。该方案还精确地解析静止的材料界面。