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 fluid mixture in multiple space dimensions on unstructured meshes with high-order discontinuous Galerkin spectral element method (DGSEM). We first derive an entropy stable (ES) and robust (i.e., that preserves positivity partial densities internal energy) three-point finite volume scheme using relaxation-based approximate Riemann solvers from Bouchut [Nonlinear stability methods hyperbolic conservation laws well-balanced schemes sources, Birkhauser] Coquel Perthame [SINUM, 35, 1998]. Then, we consider DGSEM based collocation quadrature interpolation points which relies framework introduced by Fisher Carpenter [JCP, 252, 2013] Gassner [SISC, 2013]. replace physical fluxes integrals over discretization elements conservative [Tadmor, MCOM, 49, 1987], while ES are used at interfaces. thus two-point flux satisfying Tadmor's condition use as flux. Time is performed strong-stability preserving Runge-Kutta scheme. then conditions parameters to guaranty semi-discrete inequality well cell average energy fully discrete any order. The later results allow existing limiters order restore nodal values within elements. also resolves exactly stationary material Numerical experiments one two flows solutions support conclusions our analysis highlight stability, robustness high resolution