A Quasi-Conservative Discontinuous Galerkin Method for Solving Five Equation Model of Compressible Two-Medium Flows

In this work, we develop a quasi-conservative discontinuous Galerkin method for the simulation of compressible gas-gas and gas-water two-medium flows by solving the five-equation transport model. This spatial discretization is a direct extension of the quasi-conservative finite volume discretization to the discontinuous Galerkin framework, thus, preserves uniform velocity and pressure fields at an isolated material interface. Furthermore, for discontinuities with a large pressure ratio, low density, and a dramatic change of material property where nonphysical values may occur, a strategy for imposing the bound-preserving limiting for volume fraction and a positivity-preserving limiting for density of each fluid and internal energy is developed and analyzed based on the quasi-conservative DG(p₁) discretization. Typical test cases for both one- and two-dimensional problems are provided to demonstrate the performance of the proposed method.