TY - JOUR

T1 - Accuracy-enhancement of discontinuous galerkin solutions for convection-diffusion equations in multiple-dimensions

AU - Ji, Liangyue

AU - Xu, Yan

AU - Ryan, Jennifer K.

PY - 2012

Y1 - 2012

N2 - 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 + 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 multidimensional linear convection-diffusion equation in order to obtain 2k+m order of accuracy, where m 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 S̈uli 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 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(h2 k+1) using alternating fluxes and that we actually obtain O(h 2k+2) for diffusion-dominated problems.

AB - 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 + 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 multidimensional linear convection-diffusion equation in order to obtain 2k+m order of accuracy, where m 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 S̈uli 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 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(h2 k+1) using alternating fluxes and that we actually obtain O(h 2k+2) for diffusion-dominated problems.

KW - Accuracy enhancement

KW - Convection-diffusion equations

KW - Discontinuous Galerkin method

KW - Filtering

KW - Negative-order norm error estimates

KW - Post-processing

UR - http://www.scopus.com/inward/record.url?scp=84864398510&partnerID=8YFLogxK

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U2 - 10.1090/S0025-5718-2012-02586-5

DO - 10.1090/S0025-5718-2012-02586-5

M3 - Article

AN - SCOPUS:84864398510

VL - 81

SP - 1929

EP - 1950

JO - Mathematics of Computation

JF - Mathematics of Computation

SN - 0025-5718

IS - 280

ER -