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Higher-order accuracy (order of $k+1$ in the $L^2$ norm) is one of the wellknown 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 cantake advantage of this superconvergence property by post-processing techniquesto enhance the accuracy of the DG solution. A popular class of post-processingtechniques to raise the convergence rate from order $k+1$ to order $2k+1$ inthe $L^2$ norm is the Smoothness-Increasing Accuracy-Conserving (SIAC)filtering. In addition to enhancing the accuracy, the SIAC filtering alsoincreases the inter-element smoothness of the DG solution. The SIAC filteringwas introduced for the DG method of the linear hyperbolic equation by Cockburnet al. in 2003. Since then, there are many generalizations of the SIACfiltering have been proposed. However, the development of SIAC filtering hasnever gone beyond the framework of using spline functions (mostly B-splines) toconstruct the filter function. In this paper, we first investigate the generalbasis function (beyond the spline functions) that can be used to construct theSIAC filter. The studies of the general basis function relax the SIAC filterstructure and provide more specific properties, such as extra smoothness, etc.Secondly, we study the basis functions' distribution and propose a new SIACfilter called compact SIAC filter that significantly reduces the original SIACfilter's support size while preserving (or even improving) its ability toenhance the accuracy of the DG solution. We show that the proofs of the newSIAC filters' ability to extract the superconvergence and provide numericalresults to confirm the theoretical results and demonstrate the new finding'sgood numerical performance.

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