A physical-constraints-preserving arbitrary Lagrangian-Eulerian discontinuous Galerkin scheme on adaptive quadrilateral meshes for compressible multi-material flows

Abstract

In this paper, a high-order physical-constraints-preserving arbitrary Lagrangian-Eulerian (ALE) discontinuous Galerkin (DG) scheme on adaptive quadrilateral meshes is proposed for compressible multi-material flows. Our scheme couples a conservative equation related to the volume fraction model with Euler equations for describing dynamics of fluid mixture. The mesh velocity in the ALE framework is obtained by using an adaptive mesh method which can automatically concentrate the mesh nodes near the regions involving large gradient values, and it can help the scheme greatly reduce the numerical dissipation near material interfaces. Using this adaptive mesh, the resolution of solution near some special regions such as material interfaces can be improved effectively by our scheme. With the appropriate time step condition and using a physical-constraints-preserving limiter, our scheme can ensure the positivity of density and pressure and the boundness of volume fraction, which further ensures the computational robustness and degree of confidence of simulations under large density or pressure ratios and so on. In general, our scheme can be applied into the simulations of compressible multi-material flows efficiently with the essentially non-oscillatory property and physical-constraints-preserving (bound-preserving and positivity-preserving) property, and its steps are more concise compared to some other methods such as the indirect ALE methods. Some examples are tested to demonstrate the accuracy, essentially non-oscillatory property and physical-constraints-preserving property of our scheme.