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In this study, we propose a unified, general framework for the directdiscontinuous Galerkin methods. In the new framework, the antiderivative of thenonlinear diffusion matrix is not needed. This allows a simple definition ofthe numerical flux, which can be used for general diffusion equations with nofurther modification. We also present the nonlinear stability analyses of thenew direct discontinuous Galerkin methods and perform several numericalexperiments to evaluate their performance. The numerical tests show that thesymmetric and the interface correction versions of the method achieve optimalconvergence and are superior to the nonsymmetric version, which demonstratesoptimal convergence only for problems with diagonal diffusion matrices butloses order for even degree polynomials with a non-diagonal diffusion matrix.Singular or blow-up solutions are also well captured with the new directdiscontinuous Galerkin methods.

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