Compact and Stable Discontinuous Galerkin Methods with Application to Atmospheric Flows

In this work we formulate the Compact Discontinuous Galerkin 2 (CDG2) method introduced in [8] for advection-diffusion problems. We present a proof of stability for the linear heat equation. Numerical results are shown for the compressible Navier-Stokes equation. We compare our new method numerically with two other well-established methods: the Compact Discontinuous Galerkin (CDG) and the Local Discontinuous Galerkin (LDG) method. In contrast to the LDG method, the primal formulation of the CDG2 method only involves the direct neighbors, making it more suitable for execution on parallel computers. The CDG method also has this compactness property, but the performance of the method is not as good as for the CDG2 method in terms of L2-error versus computation time