An Energy Stable Local Discontinuous Galerkin Method for a Binary Compressible Flow (Advances in Applied Mathematics and Mechanics, 2025)
This paper focuses on an energy-stable local discontinuous Galerkin (LDG)method for a binary compressible flow model. Since the densities and the momentumare 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 themass average velocity $\boldsymbol{v}$ and its square as auxiliary variables. These auxiliary variablesare chosen in the stability analysis as the test functions for the momentum and density balance equations, respectively. Using the Crank-Nicolson (CN) time integrationmethod, we can prove then the stability of the LDG-CN discretization. Computationsare provided to demonstrate the accuracy, efficiency and capabilities of the numericalmethod.