Accuracy comparison for discontinuous Galerkin schemes in the case of wave and vortex

This study considers discontinuous Galerkin schemes with Legendre polynomial basis of degree $K=0,\dots,5$. The scheme build for convection equation is studied in sense of amplitude and dispersion error. The relation between wave number and cell size is determined to provide the desired solution accuracy in task with wave. The computations for acoustic wave are performed with the solver for full Euler equations. Despite different problem statement, the analytical estimates for the relation prove practical applicability. The other test considers inviscid vortex flow. The practical estimate of mesh density is based on the solution of model vortex flow with viscosity. The relation between scheme order and mesh density is pointed for all testcases. The results of this paper can be used in building meshes for tasks intended to be solved with discontinuous Galerkin approach.