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Entropy stable DGSEM for nonlinear hyperbolic systems in nonconservative form with application to two-phase flows. (English) Zbl 1451.76073
Summary: In this work, we consider the discretization of nonlinear hyperbolic systems in nonconservative form with the high-order discontinuous Galerkin spectral element method (DGSEM) based on collocation of quadrature and interpolation points [D. A. Kopriva and G. Gassner, J. Sci. Comput. 44, No. 2, 136–155 (2010; Zbl 1203.65199)]. We present a general framework for the design of such schemes that satisfy a semi-discrete entropy inequality for a given convex entropy function at any approximation order. The framework is closely related to the one introduced for conservation laws by T. C. Fisher and M. H. Carpenter [J. Comput. Phys. 252, 518–557 (2013; Zbl 1349.65293)] and G. J. Gassner [SIAM J. Sci. Comput. 35, No. 3, A1233–A1253 (2013; Zbl 1275.65065)] and relies on the modification of the integral over discretization elements where we replace the physical fluxes by entropy conservative numerical fluxes from [M. J. Castro et al., SIAM J. Numer. Anal. 51, No. 3, 1371–1391 (2013; Zbl 1317.65167)], while entropy stable numerical fluxes are used at element interfaces. Time discretization is performed with strong-stability preserving Runge-Kutta schemes. The present method is first validated on the Burgers and Euler equations in nonconservative form. We then use this framework for the discretization of two-phase flow systems in one space-dimension: a $$2 \times 2$$ system with a nonconservative product associated to a linearly-degenerate field for which the DGSEM fails to capture the physically relevant solution, and the isentropic Baer-Nunziato model. For the latter, we derive conditions on the numerical parameters of the discrete scheme to further keep positivity of the partial densities and a maximum principle on the void fractions. Numerical experiments support the conclusions of the present analysis and highlight stability and robustness of the present schemes.
##### MSC:
 76M10 Finite element methods applied to problems in fluid mechanics 76M22 Spectral methods applied to problems in fluid mechanics 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs 76T10 Liquid-gas two-phase flows, bubbly flows 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 35L45 Initial value problems for first-order hyperbolic systems
HLLC
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