A numerical analysis of the nodal Discontinuous Galerkin scheme via Flux Reconstruction for the advection-diffusion equation

A von Neumann analysis of the Flux Reconstruction (FR) formulation is performed for the linear advection-diffusion equation to investigate the stability, dissipation and dispersion associated with the nodal Discontinuous Galerkin (DG) scheme. We show that the maximum stable time step for advection-diffusion is stricter than that for pure-advection or pure-diffusion individually. However, the simple harmonic sum of the maximum stable time steps for advection and diffusion provides a suitable estimate for the linear advection-diffusion and Navier-Stokes equations on unstructured, tensor product elements. The estimates are accurate within 50% error on all test cases and are conservative on tests with Cartesian grids but not always on unstructured grids. We show that the CFL limit is strongly influenced by the choice of interface fluxes and, in general, the limit for a scheme using centered values is much higher than that which has one-sided values. We also verify that schemes with centered values produce less error for well resolved solutions while schemes with one-sided values produce less error for solutions that are under-resolved.