Smoothness-Increasing Accuracy-Conserving (SIAC) Postprocessing for Discontinuous Galerkin Solutions over Structured Triangular Meshes

Abstract

Abstract. Theoretically and computationally, it is possible to demonstrate that the order of accuracy of a discontinuous Galerkin (DG) solution for linear hyperbolic equations can be improved from order k+1 to 2k+1 through the use of smoothness-increasing accuracy-conserving (SIAC) filtering. However, it is a computationally complex task to perform this in an efficient manner, which becomes an even greater issue considering nonquadrilateral mesh structures. In this paper, we present an extension of this SIAC filter to structured triangular meshes. The basic theoretical assumption in the previous implementations of the postprocessor limits the use to numerical solutions solved over a quadrilateral mesh. However, this assumption is restrictive, which in turn complicates the application of this postprocessing technique to general tessellations. Additionally, moving from quadrilateral meshes to triangulated ones introduces more complexity in the calculations as the number of integrations required increases. In this paper, we extend the current theoretical results to variable coefficient hyperbolic equations over structured triangular meshes and demonstrate the effectiveness of the application of this postprocessor to structured triangular meshes as well as exploring the effect of using inexact quadrature. We show that there is a direct theoretical extension to structured triangular meshes for hyperbolic equations with bounded variable coefficients. This is a challenging first step toward implementing SIAC filters for unstructured tessellations. We show that by using the usual B-spline implementation, we are able to improve on the order of accuracy as well as decrease the magnitude of the errors. These results are valid regardless of whether exact or inexact integration is used. The results here demonstrate that it is still possible, both theoretically and computationally, to improve to 2k+1 over the DG solution itself for structured triangular meshes.

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.…”

“…In [14], we thoroughly discussed the extension of the SIAC filter to structured triangular meshes. Here, we simply take the existing implementation of the SIAC filter and apply the same ideas to unstructured triangular meshes.…”

“…For random meshes there was no clear order improvement, which may have been due to an incorrect kernel scaling. To further expand the applicability of the SIAC filter, the authors in [14] extended the postprocessing results, both theoretically and numerically, to structured triangular meshes. However, the outlook for triangular meshes was actually much better than that presented in [14].…”

“…To further expand the applicability of the SIAC filter, the authors in [14] extended the postprocessing results, both theoretically and numerically, to structured triangular meshes. However, the outlook for triangular meshes was actually much better than that presented in [14]. Indeed, the order improvement was not clear for filtering over a Union Jack mesh when a scaling H = h equal to the uniform mesh spacing was used.…”

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

“…A SIAC filter has the ability to extract a superconvergent solution from a DG approximation for different element types including quadrilateral, structured triangular, tetrahedral and even unstructured triangular meshes [17,21,18]. One-sided SIAC kernels have been proposed as an extension of this convolution-based postprocessing for simulations involving boundaries or sharp discontinuities such as shocks [24,35,26].…”