An Energy Stable Local Discontinuous Galerkin Method for a Binary Compressible Flow

Abstract

This paper focuses on an energy-stable local discontinuous Galerkin (LDG) method for a binary compressible flow model. Since the densities and the momentum are highly coupled in the equations, and the test and basis functions in LDG discretizations have to be in the same finite element space, it is difficult to obtain stable LDG discretizations for the binary compressible flow model. To tackle this issue, we take the mass average velocity $\boldsymbol{v}$ and its square as auxiliary variables. These auxiliary variables are chosen in the stability analysis as the test functions for the momentum and density balance equations, respectively. Using the Crank-Nicolson (CN) time integration method, we can prove then the stability of the LDG-CN discretization. Computations are provided to demonstrate the accuracy, efficiency and capabilities of the numerical method.