A discontinuous Galerkin spectral element method for a nonconservative compressible multicomponent flow model

In this work, we propose an accurate, robust (the solution remains in the set of states), and stable discretization of a nonconservative model for the simulation of compressible multicomponent ﬂows with shocks and material interfaces. We consider the gamma-based model by Shyue [J. Comput. Phys., 142 (1998), 208–242] where each component follows a sti ﬀ ened gas equation of state (EOS). We here extend the framework proposed in Renac [J. Comput. Phys., 382 (2019), 1–26] and Coquel et al. [J. Comput. Phys. 431 (2021), 110135] for the discretization of hyperbolic systems, with both ﬂuxes and nonconservative products, to unstructured meshes with curved elements in multiple space dimensions. The framework relies on a high-order discontinuous Galerkin spectral element method (DGSEM) using collocation of quadrature and interpolation points as proposed by Gassner [SIAM J. Sci. Comput. , 35 (2013)] in the case of hyperbolic conservation laws. We modify the integrals over discretization elements where we replace the physical ﬂuxes and nonconservative products by two-point numerical ﬂuctuations. The contributions of this work are threefold. First, we analyze the semi-discrete DGSEM discretization of general hyperbolic systems with conservative and nonconservative terms and derive the conditions to obtain a scheme that is high-order accurate, free-stream preserving, and entropy stable when excluding material interfaces. Second, we design a three-point scheme with a HLLC solver for the gamma-based model that does not require a root-ﬁnding algorithm for the approximation of the nonconservative products. The scheme is proved to be robust and entropy stable for convex entropies, to preserve uniform proﬁles of pressure and velocity across material interfaces (material interface preservation), and to satisfy a discrete minimum principle on the speciﬁc entropy and maximum principles on the parameters of the EOS. Third, the HLLC solver is applied at interfaces in the DGSEM, while we consider two kinds of ﬂuctuations in the integrals over discretization elements: the former is entropy conservative (EC), while the latter preserves material interfaces (CP). Time integration is performed using high-order strong-stability preserving Runge-Kutta schemes. The fully discrete scheme is shown to preserve material interfaces with CP ﬂuctuations. Under a given condition on the time step, both EC and CP ﬂuctuations ensure that the cell-averaged solution remains in the set of states; satisfy a minimum principle on any convex entropy and maximum principles on the EOS parameters. These results have allowed us to use existing limiters in order to restore positivity, and discrete maximum principles of degrees-of-freedom within elements. Numerical experiments in one and two space dimensions on ﬂows with discontinuous solutions support the conclusions of our analysis and highlight stability, robustness and accuracy of the proposed DGSEM with either CP, or EC ﬂuctuations, while the scheme with CP ﬂuctuations is shown to o ﬀ er better resolution capabilities. ﬂuctuations. We ﬁrst extend this framework to multidimensional unstructured grids with curved elements and derive conditions under which the semi-discrete scheme is high-order accurate, free-stream preserving and entropy stable. We then design a robust, entropy stable, material interface preserving, and maximum principle preserving HLLC solver for the SG-gamma model that does not require a root-ﬁnding algorithm to evaluate the nonconservative product. The HLLC solver is then used as interface ﬂuctuations in the proposed DG scheme, while we consider either EC, or CP ﬂuctuations in the integrals over discretization elements. The DGSEM is shown to either satisfy a semi-discrete entropy inequality with EC ﬂuctuations (when excluding material interfaces), or to preserve material interfaces with CP ﬂuctuations. We then analyze the fully discrete numerical scheme with a forward Euler time discretization. We derive conditions on the time step to guarantee that the cell-averaged solution remains in the set of states and satisﬁes a minimum principle on the entropy and maximum principles on the EOS parameters with either EC, or CP ﬂuctuations. We use a posteriori scaling limiters [70, 71] to extend these properties to all DOFs within elements, while a strong-stability preserving Runge-Kutta scheme [57] is used for the high-order time integration and keeps properties of the ﬁrst-order in time scheme. High-order accurate numerical simulations of ﬂows in one and two space dimensions with discontinuous solutions and complex wave interactions conﬁrm robustness, stability and accuracy of the present scheme with either EC, or CP ﬂuctuations, and the scheme with CP ﬂuxes present better resolution capabilities.