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

• An efficient, entropy-stable high-order method for multicomponent flows is presented • The spatial residual is built via L 2 -projection of conservative to entropy variables. • The Direct Enforcement of Entropy Balance term is included to avoid over-integration • Performance is compared with other approaches in one-dimensional cases of increasing complexity • The entropy projection solver combines computational efficiency and entropy conservation/stability 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 L 2 -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.