Eigenanalysis and non-modal analysis of collocated discontinuous Galerkin discretizations with the summation-by-parts property

Abstract

Guided by the von Neumann and non-modal analyses, we investigate the dispersion and diffusion properties of collocated discontinuous Galerkin methods with the summation-by-parts property coupled with the simultaneous approximation technique. We use the linear advection and linear advection-diffusion equations as model problems. The analysis is carried out by varying the order of the spatial discretization, the Péclet number, and looking at the effect of the upwind term. The eigenanalysis is verified to provide insights into the numerical errors based on the behavior of the primary mode. The dispersion and diffusion errors associated with the spatial discretization are shown to behave according to the primary- or physical- mode in the range of low wavenumbers. In this context, the discretization of the model problems is verified to be stable for all flow regimes and independent of the solution polynomial degree. The effect of the upwind term shows that its effect decreases by increasing the accuracy of the discretization. Further, two analyses that includes all modes are used to get better insights into the diffusion and robustness of the scheme. From the non-modal analysis, the short-term diffusion is computed for different flow regimes and solution polynomial degrees. The energy decay or long-term diffusion based on all eigenmodes based on the eigenmodes matrix is analyzed for t > 0 and different spatial discretizations. The results are validated against under-resolved turbulence simulations of the Taylor–Green vortex at two Reynolds numbers, i.e. , Re = 100 and 1600, and a Mach number Ma = 0.1 , and the decaying of compressible homogeneous isotropic turbulence at a Reynolds number based on the Taylor microscale of Re λ = 100 and a turbulent Mach number of Ma t = 0.6 .