This work concerns the numerical approximation of the multicomponent compressible Euler system for a mixture of immiscible fluids in multiple space dimensions and its contribution is twofold. We first derive an entropy stable, positive and accurate three-point finite volume scheme using relaxation-based approximate Riemann solvers from Bouchut [Nonlinear stability of finite volume methods for hyperbolic conservation laws and well-balanced schemes for sources, Frontiers in Mathematics, Birkhauser, 2004] and Coquel and Perthame [SIAM J. Numer. Anal., 35 (1998)]. Then, we extend these results to the high-order discontinuous Galerkin spectral element method (DGSEM) based on collocation of quadrature and interpolation points [Kopriva and Gassner, J. Sci. Comput., 44 (2010)]. The method relies on the framework introduced by Fisher and Carpenter [J. Comput. Phys., 252 (2013)] and Gassner [SIAM J. Sci. Comput., 35 (2013),] where we replace the physical fluxes by entropy conservative numerical fluxes [Tadmor, Math. Comput., 49 (1987)] in the integral over discretization cells, while entropy stable numerical fluxes are used at cell interfaces. Time discretization is performed with a strong-stability preserving Runge-Kutta scheme. We design two-point numerical fluxes satisfying the Tadmor's entropy conservation condition and use the numerical flux from the three-point scheme as entropy stable flux. We derive conditions on the numerical parameters to guaranty a semi-discrete entropy inequality and positivity of the fully discrete DGSEM scheme at any approximation order. The scheme is also accurate in the sense that the solution at interpolation points is exact for stationary contact waves. 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.

中文翻译：

---

熵稳定的正 DGSEM，具有清晰的材料界面分辨率，适用于 $4\times4$ 两相流系统：三点方案的遗产

这项工作涉及多分量可压缩欧拉系统在多个空间维度上的不混溶流体混合物的数值近似，其贡献是双重的。我们首先使用来自 Bouchut 的基于松弛的近似黎曼求解器推导出一个熵稳定、正和准确的三点有限体积方案 [Nonlinear stable of有限体积方法用于双曲守恒定律和源的良好平衡方案，Frontiers in Mathematics，Birkhauser， 2004] 和 Coquel 和 Perthame [SIAM J. Numer. 分析，35 (1998)]。然后，我们将这些结果扩展到基于正交和插值点的搭配的高阶不连续伽辽金谱元法 (DGSEM) [Kopriva and Gassner, J. Sci.。计算，44 (2010)]。该方法依赖于 Fisher 和 Carpenter 引入的框架 [J. 计算。Phys., 252 (2013)] 和 Gassner [SIAM J. Sci. Comput., 35 (2013),] 我们用熵保守数值通量 [Tadmor, Math. Comput., 49 (1987)] 在离散化单元上的积分中，而在单元界面使用熵稳定的数值通量。时间离散化采用强稳定性保持 Runge-Kutta 方案。我们设计了满足 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熵稳定通量。我们推导出数值参数的条件，以保证半离散熵不等式和完全离散 DGSEM 方案在任何近似阶数下的正性。该方案在插值点处的解对于固定接触波是精确的这个意义上也是准确的。