Smoothness-Increasing Accuracy-Conserving (SIAC) filtering for discontinuous Galerkin solutions over nonuniform meshes: Superconvergence and optimal accuracy

Smoothness-Increasing Accuracy-Conserving (SIAC) filtering is an area of increasing interest because it can extract the ``hidden accuracy" in discontinuous Galerkin (DG) solutions. It has been shown that by applying a SIAC filter to a DG solution, the accuracy order of the DG solution improves from order $k+1$ to order $2k+1$ for linear hyperbolic equations over uniform meshes. However, applying a SIAC filter over nonuniform meshes is difficult, and the quality of filtered solutions is usually unsatisfactory applied to approximations defined on nonuniform meshes. The applicability to such approximations over nonuniform meshes is the biggest obstacle to the development of a SIAC filter.
The purpose of this paper is twofold: to study the connection between the error of the filtered solution and the nonuniform mesh and to develop a filter scaling that approximates the optimal error reduction. First, through analyzing the error estimates for SIAC filtering, we computationally establish for the first time a relation between the filtered solutions and the unstructuredness of nonuniform meshes. Further, we demonstrate that there exists an optimal accuracy of the filtered solution for a given nonuniform mesh and that it is possible to obtain this optimal accuracy by the method we propose, an optimal filter scaling. By applying the newly designed filter scaling over nonuniform meshes, the filtered solution has demonstrated improvement in accuracy order as well as reducing the error compared to the original DG solution. Finally, we apply the proposed methods over a large number of nonuniform meshes and compare the performance with existing methods to demonstrate the superiority of our method.