A Generalized Framework for Direct Discontinuous Galerkin Methods for Nonlinear Diffusion Equations

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.