Nonlinear advection-diffusion equation: ADER-DG penalty vs. relaxation schemes

Abstract

This paper proposes to solve numerically the two dimensional nonlinear advection-diffusion equation. The space discretization relies on a classical Discontinuous Galerkin (DG) method. This scheme is combined together with an Arbitrary high order DERivatives (ADER) approach to ensure the same high order of accuracy in time compared to the precision in space. More precisely, two different methods are compared regarding the computational cost, the error and the order of convergence: the Symmetric Interior Penalty Galerkin (SIPG) and the Cattaneo relaxation methods. The viscosity of the medium, the mesh and the approximation degree being fixed, we aim at determining whether the penalty or the relaxation scheme is to be preferred. Numerical examples are provided to illustrate and quantify this comparison. We show that both approaches ensure to reach an arbitrary high precision and present an interest from an implementation perspective.