Efficient entropy-conserving/stable discontinuous Galerkin solution of the multicomponent compressible Euler equations

This paper presents the development of an efficient discontinuous Galerkin (dG) solver for the multicomponent compressible Euler equations. The method provides global entropy conservation/stability at the discrete level, contributing to the robustness of the computations, cf. [4, 19]. The unsteady term of the governing equations is formulated for the conservative variables, while the spatial discretization is assembled from the L2-projection of the entropy variables [4], [19] onto the dG function space, as suggested by Chan et al. [44] and Alberti et al. [35]. This approach requires numerical over-integration to ensure entropy conservation/stability, significantly degrading the computational performance. The Direct Enforcement of Entropy Balance (DEEB) proposed by Abgrall in [11] is implemented to avoid this. The DEEB consists of an explicit correction to the discretization to avoid unphysical entropy evolution. As high-order discretizations give rise to spurious oscillations at flow discontinuities, a directional shock-capturing term is added to the discretized equations. The performance of the solver is compared to alternative approaches, i.e., solving directly for the conservative or the entropy variables, by computing several one-dimensional cases. The convergence of the numerical solution is also tested using the method of manufactured solutions (MMS). The interactions of a shock wave with a circular and a square inhomogeneity are finally considered, assessing the accuracy of the solver for reproducing complex two-dimensional phenomena.