Efficient Implementation of Smoothness-Increasing Accuracy-Conserving (SIAC) Filters for Discontinuous Galerkin Solutions

Abstract

The discontinuous Galerkin (DG) methods provide a high-order extension of the finite volume method in much the same way as high-order or spectral/hp elements extend standard finite elements. However, lack of inter-element continuity is often contrary to the smoothness assumptions upon which many post-processing algorithms such as those used in visualization are based. Smoothness-increasing accuracy-conserving (SIAC) filters were proposed as a means of ameliorating the challenges introduced by the lack of regularity at element interfaces by eliminating the discontinuity between elements in a way that is consistent with the DG methodology; in particular, high-order accuracy is preserved and in many cases increased. The goal of this paper is to explicitly define the steps to efficient computation of this filtering technique as applied to both structured triangular and quadrilateral meshes. Furthermore, as the SIAC filter is a good candidate for parallelization, we provide, for the first time, results that confirm anticipated performance scaling when parallelized on a shared-memory multi-processor machine.

Smoothness-Increasing Accuracy-Conserving Filters for Discontinuous Galerkin Solutions over Unstructured Triangular Meshes

“…For example, when using the Gauss-Jacobi rule, we need a total of 3k+1 2 2 quadrature points. For more information regarding the Gaussian quadrature and the various mappings involved in the integration, consult [14,15,11]. We further add that in order to find the footprint of the kernel on the DG mesh, we first lay a regular grid over our unstructured mesh.…”

Smoothness-Increasing Accuracy-Conserving (SIAC) Filtering and Quasi-Interpolation: A Unified View

“…The compact support of the SIAC kernel along with its superconvergence properties in approximating the DG solutions is one of the main reasons for its popularity in simulation science [7,27,20]. We first focus on the symmetric SIAC kernel and then show how results from the previous section also generalize to the one-sided SIAC kernel.…”

“…The maximal approximation order of B-splines along with their minimal support made B-spline-based approximation techniques popular in a variety of applications, including signal processing, biomedical imaging, finite element methods and superconvergence-extraction techniques [33,34,31,30,7,11,4,29,24,20]. In this section, we introduce B-splines in the univariate case, with all the results easily extending to higher-dimensional Cartesian lattices (or uniform quadrilateral meshes) using tensor products.…”

“…SIAC filtering increases the inter-element continuity up to C k−1 and raises the convergence rate of the discontinuous Galerkin (DG) solution from order k + 1 to order 2k + 1 for linear hyperbolic equations solved over a uniform mesh [7]. Convergence properties of SIAC filtering and its effectiveness have been widely studied in the literature [7,16,4,[12][13][14].…”

“…While Theorem 2 uniquely specifies all the kernel coefficients of the SIAC kernel of order k, conventionally the computation of the kernel coefficients is accomplished by inverting a matrix-vector system with a large condition number [20]. By revisiting the symmetric SIAC kernel construction and using the results from Section 3, we introduce a new formulation of the symmetric SIAC kernel construction that entails a direct and exact computation of the kernel coefficients in terms of rational numbers.…”