Entropy stable, robust and high-order DGSEM for the compressible multicomponent Euler equations

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.