One-Sided Position-Dependent Smoothness-Increasing Accuracy-Conserving (SIAC) Filtering Over Uniform and Non-Uniform Meshes

Ryan, Jennifer, Li, Xiaozhou, Kirby, Mike and Vuik, Kees (2015) One-Sided Position-Dependent Smoothness-Increasing Accuracy-Conserving (SIAC) Filtering Over Uniform and Non-Uniform Meshes. Journal of Scientific Computing, 64 (3). pp. 773-817. ISSN 1573-7691

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Abstract

In this paper, we introduce a new position-dependent Smoothness-Increasing Accuracy-Conserving (SIAC) filter that retains the benefits of position dependence while ameliorating some of its shortcomings. As in the previous position-dependent filter, our new filter can be applied near domain boundaries, near a discontinuity in the solution, or at the interface of different mesh sizes; and as before, in general, it numerically enhances the accuracy and increases the smoothness of approximations obtained using the discontinuous Galerkin (dG) method. However, the previously proposed position-dependent one-sided filter had two significant disadvantages: (1) increased computational cost (in terms of function evaluations), brought about by the use of $4k+1$ central B-splines near a boundary (leading to increased kernel support) and (2) increased numerical conditioning issues that necessitated the use of quadruple precision for polynomial degrees of $k\ge 3$ for the reported accuracy benefits to be realizable numerically. Our new filter addresses both of these issues --- maintaining the same support size and with similar function evaluation characteristicsas the symmetric filter in a way that has better numerical conditioning --- making it, unlike its predecessor, amenable for GPU computing. Our new filter was conceived by revisiting the original error analysis for superconvergence of SIAC filters and by examining the role of the B-splines and their weights in the SIAC filtering kernel. We demonstrate, in the uniform mesh case, that our new filter is globally superconvergent for $k=1$ and superconvergent in the interior (e.g., region excluding the boundary) for $k\ge2$. Furthermore, we present the first theoretical proof of superconvergence for postprocessing over smoothly varying meshes, and explain the accuracy-order conserving nature of this new filter when applied to certain non-uniform meshes cases. We provide numerical examples supporting our theoretical results and demonstrating that our new filter, in general, enhances the smoothness and accuracy of the solution. Numerical results are presented for solutions of both linear and nonlinear equation solved on both uniform and non-uniform one- and two-dimensional meshes.

Item Type: Article
Uncontrolled Keywords: discontinuous galerkin,post-processing,siac filtering,superconvergence,uniform meshes,smoothly-varying meshes,non-uniform meshes
Faculty \ School: Faculty of Science > School of Mathematics
Depositing User: Pure Connector
Date Deposited: 30 Apr 2015 14:58
Last Modified: 21 Apr 2020 23:34
URI: https://ueaeprints.uea.ac.uk/id/eprint/53335
DOI: 10.1007/s10915-014-9946-6

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