Compact and stable Discontinuous Galerkin methods for convection-diffusion problems

Abstract

We present a new scheme, the compact discontinuous Galerkin 2 (CDG2) method, for solving nonlinear convection-diffusion problems together with a detailed comparison to other well-accepted DG methods. The new CDG2 method is similar to the CDG method that was recently introduced in the work of Perraire and Persson for elliptic problems. One main feature of the CDG2 method is the compactness of the stencil which includes only neighboring elements, even for higher order approximation. Theoretical results showing coercivity and stability of CDG2 and CDG for the Poisson and the heat equation are given, providing computable bounds on any free parameters in the scheme. In numerical tests for an elliptic problem, a scalar convection-diffusion equation, and for the compressible Navier–Stokes equations, we demonstrate that the CDG2 method slightly outperforms similar methods in terms of $L^2$-accuracy and CPU time.