Numerical reflections in hp-nonuniform Discontinuous Galerkin methods

The h and p adaptivity of Discontinuous Galerkin (DG) methods can reduce the cost of a simulation, but numerical reflections may occur at interfaces between regions of different resolution. We analyze eigensolutions of the DG method for linear hyperbolic systems and couple the solutions on either side of a change in resolution to find the amplitude of the reflections. We find that numerical reflections can be significant. Especially for high p, the reflections can be large at resolutions where the wave's propagation is resolved on the fine grid. We extend prior work to changes in h and p, and find that the reflections can be decreased by certain combinations of changes in h and p, but not eliminated entirely. In the 1D case, the reflections can still be eliminated by use of an exact characteristics-splitting flux.