A Discontinuous Galerkin Method for Conservation Laws Coupled with Algebraic-Type Nonlinear Constitutive Equations

Abstract

The discontinuous Galerkin (DG) finite element method has been popular as numerical techniques for solving the conservation laws. This method combines key features of the finite element and finite volume methods. In the present work, an explicit modal (cell-based) DG scheme is developed for solving the one-dimensional conservation laws in conjunction with Navier-Stokes-Fourier (NSF) constitutive laws and a nonlinear coupled constitutive relation (NCCR) in order to investigate the shock wave structures in thermal non-equilibrium. Moreover, a convergent iterative method for solving the nonlinear coupled algebraic constitutive relation is implemented into this DG scheme. The Maxwellian monatomic gas is selected for testing the shock structure at various Mach numbers. It is shown that the new scheme works well for all Mach numbers.