The Direct Discontinuous Galerkin (DDG) Method for Diffusion with Interface Corrections (Communications in Computational Physics, 2010)
Based on a novel numerical flux involving jumps of even order derivativesof the numerical solution, a direct discontinuous Galerkin (DDG) method for diffusionproblems was introduced in [H. Liu and J. Yan, SIAM J. Numer. Anal. 47(1) (2009),475-698]. In this work, we show that higher order (k≥4) derivatives in the numericalflux can be avoided if some interface corrections are included in the weak formulationof the DDG method; still the jump of 2nd order derivatives is shown to be importantfor the method to be efficient with a fixed penalty parameter for all pk elements. Therefined DDG method with such numerical fluxes enjoys the optimal (k+1)th orderof accuracy. The developed method is also extended to solve convection diffusionproblems in both one- and two-dimensional settings. A series of numerical tests arepresented to demonstrate the high order accuracy of the method.