A Bound-Preserving and Positivity-Preserving Path-Conservative Discontinuous Galerkin Method for Solving Five-Equation Model of Compressible Two-Medium Flows

Abstract

We develop a high-order path-conservative discontinuous Galerkin method to simulate compressible two-medium flows by solving the five-equation transport model. The proposed path-conservative discontinuous Galerkin discretization is able to satisfy the equilibrium condition, which states that uniform pressure and velocity fields are preserved around an isolated material interface. In order to improve the robustness of the proposed method in simulating complex two-medium flows with large density and pressure ratios, a bound-preserving and positivity-preserving limiting strategy is developed for maintaining the bound of volume fraction and the positivity of partial densities and internal energy. A series of typical one- and two-dimensional cases are tested, which demonstrate that the proposed high-order path-conservative discontinuous Galerkin method is very capable of simulating compressible gas-gas and gas-water two-medium flows with large density and pressure ratios.