Smoothness-Increasing Accuracy-Conserving (SIAC) Filters for Discontinuous Galerkin Solutions: Application to Structured Tetrahedral Meshes

Abstract

In this paper, we attempt to address the potential usefulness of smoothness-increasing accuracy-conserving (SIAC) filters when applied to real-world simulations. SIAC filters as a class of post-processors were initially developed in Bramble and Schatz (Math Comput 31:94, 1977) and later applied to discontinuous Galerkin (DG) solutions of linear hyperbolic partial differential equations by Cockburn et al. (Math Comput 72:577, 2003), and are successful in raising the order of accuracy from $k+1$ to $2k+1$ in the $L^2$—norm when applied to a locally translation-invariant mesh. While there have been several attempts to demonstrate the usefulness of this filtering technique to nontrivial mesh structures (Curtis et al. in SIAM J Sci Comput 30(1):272, 2007; Mirzaee et al. in SIAM J Numer Anal 49:1899, 2011; King et al. in J Sci Comput, 2012), the application of the SIAC filter never exceeded beyond two-space dimensions. As tetrahedral meshes are often the type considered in more realistic simulations, we contribute to the class of SIAC post-processors by demonstrating the effectiveness of SIAC filtering when applied to structured tetrahedral meshes. These types of meshes are generated by tetrahedralizing uniform hexahedra and therefore, while maintaining the structured nature of a hexahedral mesh, they exhibit an unstructured tessellation within each hexahedral element. Moreover, we address the computationally intensive task of performing numerical integrations when one considers tetrahedral elements for SIAC filtering and provide guidelines on how to ameliorate these challenges through the use of more general cubature rules. We consider two examples of a hyperbolic equation and confirm the usefulness of SIAC filters in obtaining the superconvergence accuracy of $2k+1$ when applied to structured tetrahedral meshes. Additionally, the DG methodology merely requires weak constraints on the fluxes between elements. As SIAC filters improve this weak continuity to $\mathcal{C }^{k-1}$—continuity at the element interfaces, we provide results that show how post-processing is useful in extracting smooth isosurfaces of DG fields.