Investigation of the Discontinuous Galerkin Method for First-Order PDE Approaches to CFD

To simulate accurately and efficiently aerospace-type flows using solely first-order PDE models, new numerical methods may be required to overcome difficulties introduced by the model. We consider a Discontinuous Galerkin (DG) method that has been shown to possess an asymptotic preservation (AP) property for one-dimensional hyperbolic relaxation systems, in an effort to understand the source of this property and to determine if this property persists for problems in which we are interested. Comparison with traditional MUSCL schemes suggests that the source of the known AP property results from a coupling of the flux and source terms by the direct evolution of the solution slope. For a canonical model problem in a single spatial dimension, we use Von Neumann analysis to demonstrate that the limiting flux function of the DG scheme is of the Harten-Lax-Van Leer type. Further Von Neumann analysis of both a MUSCL scheme and the DG scheme show that near equilibrium, both methods have mesh size restrictions related to the resolution of physical dissipation. The DG scheme is less numerically dissipative than the HR scheme in this limit. Numerical simulations demonstrate these findings in one dimension. In one and two dimensions, our numerical results demonstrate that the convergence rate of the DG scheme eventually drops as the mesh is refined.