Smoothness-Increasing Accuracy-Conserving (SIAC) Filtering for Discontinuous Galerkin Solutions over Nonuniform Meshes: Superconvergence and Optimal Accuracy

Abstract

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.

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

“…For non-uniform meshes, the simple choice is choosing the scaling as the local element size [17] or the largest element size [8]. For more recent research results, one can determine the scaling according to the structure of the given non-uniform mesh to obtain the optimal accuracy, see [13].…”

“…Since the introduction of the SIAC filtering for the DG method, many generalizations of SIAC filtering have been proposed from various perspectives. To name a few, it has been extended to the boundary (position-dependent) filtering [17,18,20], the derivative filtering [11,16] as well as the extension to nonuniform meshes [8,13]. It also has shown useful for applications in visualisation [19,22], shock capturing [1,23], etc.…”

Multi-element SIAC Filter for Shock Capturing Applied to High-Order Discontinuous Galerkin Spectral Element Methods

“…If the filter recovers a smaller family of polynomial orders, m < N , then the accuracy of the overall approximation is bound by the filter accuracy. Typically, such SIAC filters were designed to obtain super-convergence in a post-processing step by a convolution of the numerical approximation against a specifically designed kernel function once at the final time [5,9,19,26,29,33]. However, recent work has applied the SIAC filter as a shock capturing and/or regularization of general discontinuities strategy during the computation of the approximate solution for global spectral collocation methods [34,37].…”