Discontinuous Galerkin Approximation of Relaxation Models for Linear and Nonlinear Diffusion Equations

Abstract

In this work we present finite element approximations of relaxed systems for nonlinear diffusion problems, which can also tackle the cases of degenerate and strongly degenerate diffusion equations. Relaxation schemes take advantage of the replacement of the original partial differential equation (PDE) with a semilinear hyperbolic system of equations, with a stiff source term, tuned by a relaxation parameter $\epsilon$. When $\epsilon \rightarrow 0^+$, the system relaxes onto the original PDE: in this way, a consistent discretization of the relaxation system for vanishing $\epsilon$ yields a consistent discretization of the original PDE. The numerical schemes obtained with this procedure do not require solving implicit nonlinear problems and possess the robustness of upwind discretizations. The proposed approximations are based on a discontinuous Galerkin method in space and on suitable implicit-explicit integration in time. Then, in principle, we can achieve any order of accuracy and obtain stable solutions, even when the diffusion equation becomes degenerate and solution singularities develop. Moreover, when needed, we can easily incorporate slope limiters within our schemes in order to handle spurious oscillatory phenomena. Some preliminary theoretical results are given, along with several numerical tests in one and two space dimensions, both for linear and nonlinear diffusion problems, including a degenerate diffusion equation, that provide numerical evidence of the properties of the presented approach.