Entropy stable, positive DGSEM with sharp resolution of material interfaces for a 4×4 two-phase flow system: a legacy from three-point schemes
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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.
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