An entropy stable high-order discontinuous Galerkin spectral element method for the Baer-Nunziato two-phase flow model

Abstract

In this work we propose a high-order discretization of the Baer-Nunziato two-phase flow model (Baer and Nunziato (1986) [5]) with closures for interface velocity and pressure adapted to the treatment of discontinuous solutions, and stiffened gas equations of states. We use the discontinuous Galerkin spectral element method (DGSEM), based on collocation of quadrature and interpolation points (Kopriva and Gassner (2010) [47]). The DGSEM uses summation-by-parts (SBP) operators in the numerical quadrature for approximating the integrals over discretization elements (Carpenter et al. (2014) [10]; Gassner et al. (2016) [32]). Here, we build upon the framework provided in (Renac (2019) [55]) for nonconservative hyperbolic systems to modify the integration over cell elements using the SBP operators and replace the physical fluxes with entropy conservative fluctuation fluxes from Castro et al. (2013) [12], while we derive entropy stable numerical fluxes applied at interfaces. This allows to prove a semi-discrete inequality for the cell-averaged physical entropy, while keeping high-order accuracy. The design of the numerical fluxes also formally preserves the kinetic energy at the discrete level. High-order integration in time is performed using strong stability-preserving Runge-Kutta schemes and we propose conditions on the numerical parameters for the positivity of the cell-averaged void fraction and partial densities. The positivity of the cell-averaged solution is extended to nodal values by the use of an a posteriori limiter. The high-order accuracy, nonlinear stability, and robustness of the present scheme are assessed through several numerical experiments in one and two space dimensions.