ACCURACY-ENHANCEMENT OF DISCONTINUOUS GALERKIN SOLUTIONS FOR CONVECTION-DIFFUSION EQUATIONS IN MULTIPLE-DIMENSIONS

Discontinuous Glerkin (DG) methods exhibit. "hidden accuracy" that makes superconvergenee of this method an increasing popular topic to address. Previous investigations have focused on the superconvergent properties of ordinary differential equations and linear hyperbolic equations. Additionally, superconvergence of order k + 3/2 for the convection-diffusion equation that focuses on a special projection using the upwind flux was presented by Cheng and Situ. In this paper we demonstrate that it is possible to extend the smoothness-increasing accuracy-conserving (SIAC) filter for use on the multidimensional linear convection-diffusion equation in order to obtain 2k+m order of accuracy, where in depends upon the flux and takes on the values 0, 1/2, or 1. The technique that we use to extract this hidden accuracy was initially introduced by Cockburn, Luskin. Shu, and Still for linear hyperbolic equations and extended by Ryan et al. as a smoothness-increasing accuracy-conserving filter. We solve this convection-diffusion equation using the local discontinuous Glerkin (LDG) method and show theoretically that, it. is possible to obtain O(h(2k+m)) in the negative-order norm. By post-processing the LDG solution to a linear convection equation using a specially designed kernel such as the one by Cockburn et: al., we can compute this same order accuracy in the L-2-norm Additionally. we present numerical studies that. confirm that we can improve the LDG solution from O(h(k+1)) to O(h(2k+1)) using alternating fluxes and that we actually obtain O(h(2k+2)) for diffusion-dominated problems.