论文标题
可压缩多组分欧拉方程的熵稳定，健壮和高阶DGSEM
Entropy stable, robust and high-order DGSEM for the compressible multicomponent Euler equations
论文作者
论文摘要
这项工作涉及使用高阶不连续的Galerkin Spectral Element方法（DGSEM）的多个空间尺寸的流体混合物的多组分压缩欧拉系统的数值近似。 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 [Nonlinear stability of finite volume methods for hyperbolic conservation laws and well-balanced schemes for sources, Birkhauser] and Coquel and Perthame [SINUM, 35, 1998].然后，我们考虑基于正交和插值点的搭配的DGSEM，该插入点依赖于Fisher和Carpenter [JCP，252，2013]和Gassner [SISC，35，2013]引入的框架。我们通过熵保守的数值通量代替积分中的物理通量[TADMOR，MCOM，49，1987]，而ES数值通量在元素接口处使用。因此，我们得出了满足tadmor熵保护条件的两点数值通量，并将三点方案中的数值通量用作ES通量。时间离散化是通过稳定的保留runge-kutta方案进行的。然后，我们在数值参数上得出条件，以确保在任何近似顺序下的分离密度和完全离散DGSEM的分区密度和内部能量的细胞平均值的阳性以及阳性。后来的结果允许使用现有限制器，以恢复元素中节点值的阳性。该方案还精确地解决了固定材料界面。在一个和两个空间维度的数值实验具有不连续解决方案的流量上的两个空间维度支持我们的分析的结论，并突出了该方案的稳定性，鲁棒性和高分辨率。
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 [Nonlinear stability of finite volume methods for hyperbolic conservation laws and well-balanced schemes for sources, Birkhauser] and Coquel and Perthame [SINUM, 35, 1998]. Then, we consider the DGSEM based on collocation of quadrature and interpolation points which relies on the framework introduced by Fisher and Carpenter [JCP, 252, 2013] and Gassner [SISC, 35, 2013]. We replace the physical fluxes in the integrals over discretization elements by entropy conservative numerical fluxes [Tadmor, MCOM, 49, 1987], 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.
