A generalized framework for direct discontinuous Galerkin methods for nonlinear diffusion equations

TL;DR

This paper introduces a unified framework for direct discontinuous Galerkin methods applicable to nonlinear diffusion equations, simplifying flux definitions and demonstrating optimal convergence and stability through numerical experiments.

Contribution

The paper presents a novel, general framework for direct discontinuous Galerkin methods that does not require the antiderivative of the diffusion matrix, enabling broader applicability.

Findings

Symmetric and interface correction methods achieve optimal convergence.

Nonsymmetric version loses order with non-diagonal diffusion matrices.

Method effectively captures singular and blow-up solutions.

Abstract

In this study, we propose a unified, general framework for the direct discontinuous Galerkin methods. In the new framework, the antiderivative of the nonlinear diffusion matrix is not needed. This allows a simple definition of the numerical flux, which can be used for general diffusion equations with no further modification. We also present the nonlinear stability analyses of the new direct discontinuous Galerkin methods and perform several numerical experiments to evaluate their performance. The numerical tests show that the symmetric and the interface correction versions of the method achieve optimal convergence and are superior to the nonsymmetric version, which demonstrates optimal convergence only for problems with diagonal diffusion matrices but loses order for even degree polynomials with a non-diagonal diffusion matrix. Singular or blow-up solutions are also well captured with the new direct discontinuous Galerkin methods.