Position-Dependent Smoothness-Increasing Accuracy-Conserving (SIAC) Filtering for Improving Discontinuous Galerkin Solutions

Position-dependent smoothness-increasing accuracy-conserving (SIAC) filtering is a promising technique not only in improving the order of the numerical solution obtained by a discontinuous Galerkin (DG) method but also in increasing the smoothness of the field and improving the magnitude of the errors. This was initially established as an accuracy enhancement technique by Cockburn et al. for linear hyperbolic equations to handle smooth solutions [Math. Comp., 72 (2003), pp. 577-606]. By implementing this technique, the quality of the solution can be improved from order $k+1$ to order $2k+1$ in the $L^2$-norm. Ryan and Shu used these ideas to extend this technique to be able to handle postprocessing near boundaries as well as discontinuities [Methods Appl. Anal., 10 (2003), pp. 295-307]. However, this presented difficulties as the resulting error had a stair-stepping effect and the errors themselves were not improved over those of the DG solution unless the mesh was suitably refined. In this paper, we discuss an improved filter for enhancing DG solutions that easily switches between one-sided postprocessing to handle boundaries or discontinuities and symmetric postprocessing for smooth regions. We numerically demonstrate that the magnitude of the errors using the modified postprocessor is roughly the same as that of the errors for the symmetric postprocessor itself, regardless of the boundary conditions.