How to Design A Generic Accuracy-Enhancing Filter for Discontinuous Galerkin Methods

Abstract: Higher-order accuracy (order of $k+1$ in the $L^2$ norm) is one of the well known beneficial properties of the discontinuous Galerkin (DG) method. Furthermore, many studies have demonstrated the superconvergence property (order of $2k+1$ in the negative norm) of the semi-discrete DG method. One can take advantage of this superconvergence property by post-processing techniques to enhance the accuracy of the DG solution. A popular class of post-processing techniques to raise the convergence rate from order $k+1$ to order $2k+1$ in the $L^2$ norm is the Smoothness-Increasing Accuracy-Conserving (SIAC) filtering. In addition to enhancing the accuracy, the SIAC filtering also increases the inter-element smoothness of the DG solution. The SIAC filtering was introduced for the DG method of the linear hyperbolic equation by Cockburn et al. in 2003. Since then, there are many generalizations of the SIAC filtering have been proposed. However, the development of SIAC filtering has never gone beyond the framework of using spline functions (mostly B-splines) to construct the filter function. In this paper, we first investigate the general basis function (beyond the spline functions) that can be used to construct the SIAC filter. The studies of the general basis function relax the SIAC filter structure and provide more specific properties, such as extra smoothness, etc. Secondly, we study the basis functions' distribution and propose a new SIAC filter called compact SIAC filter that significantly reduces the original SIAC filter's support size while preserving (or even improving) its ability to enhance the accuracy of the DG solution. We show that the proofs of the new SIAC filters' ability to extract the superconvergence and provide numerical results to confirm the theoretical results and demonstrate the new finding's good numerical performance.