Accuracy-enhancement of discontinuous Galerkin solutions for convection-diffusion equations in multiple-dimensions

Abstract:

Discontinuous Galerkin (DG) methods exhibit “hidden accuracy” that makes superconvergence 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 $k+\frac {3}{2}$ for the convection-diffusion equation that focuses on a special projection using the upwind flux was presented by Cheng and Shu. In this paper we demonstrate that it is possible to extend the smoothness-increasing accuracy-conserving (SIAC) filter for use on the multi-dimensional linear convection-diffusion equation in order to obtain 2$k$+$m$ order of accuracy, where $m$ depends upon the flux and takes on the values $0, \frac {1}{2},$ or $1.$ The technique that we use to extract this hidden accuracy was initially introduced by Cockburn, Luskin, Shu, and Süli 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 Galerkin (LDG) method and show theoretically that it is possible to obtain $\mathcal {O}(h^{2k+m})$ 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 $\mathcal {O}(h^{k+1})$ to $\mathcal {O}(h^{2k+1})$ using alternating fluxes and that we actually obtain $\mathcal {O}(h^{2k+2})$ for diffusion-dominated problems.