DISPERSION-DISSIPATION ANALYSIS FOR ADVECTION PROBLEMS WITH NONCONSTANT COEFFICIENTS: APPLICATIONS TO DISCONTINUOUS GALERKIN FORMULATIONS

This paper presents an extended version of von Neumann stability analysis to study dispersion and dissipation errors in nonconstant coefficient advection equations. This approach is used to analyze the behavior of discontinuous Galerkin (DG) discretizations, including the influence of polynomial order, number of elements, and choice of quadrature points (Gauss or Gauss-Lobatto) on numerical errors. Additionally, the split flux formulation (conservative, nonconservative, and skew-symmetric) and interelement numerical fluxes (upwind or central) for nonconstant coefficient problems are also studied. Our analysis demonstrates that schemes that appear stable when analyzed using the classic (constant speed) von Neumann technique may reveal instabilities in cases with nonconstant advection speeds (e.g., DG with Gauss-Lobatto points and central fluxes). Additionally, our analysis shows that other schemes (nonconservative DG with central fluxes and Gauss-Lobatto nodes) are stable for both constant and nonconstant advection speeds.