Hexagonal Smoothness-Increasing Accuracy-Conserving Filtering

Mirzargar, Mahsa, Jallepalli, Ashok, Ryan, Jennifer K. and Kirby, Robert M. (2017) Hexagonal Smoothness-Increasing Accuracy-Conserving Filtering. Journal of Scientific Computing, 73 (2-3). 1072–1093. ISSN 0885-7474

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Abstract

Discontinuous Galerkin (DG) methods are a popular class of numerical techniques to solve partial differential equations due to their higher order of accuracy. However, the inter-element discontinuity of a DG solution hinders its utility in various applications, including visualization and feature extraction. This shortcoming can be alleviated by postprocessing of DG solutions to increase the inter-element smoothness. A class of postprocessing techniques proposed to increase the inter-element smoothness is SIAC filtering. In addition to increasing the inter-element continuity, SIAC filtering also raises the convergence rate from order k+1k+1 to order 2k+12k+1 . Since the introduction of SIAC filtering for univariate hyperbolic equations by Cockburn et al. (Math Comput 72(242):577–606, 2003), many generalizations of SIAC filtering have been proposed. Recently, the idea of dimensionality reduction through rotation has been the focus of studies in which a univariate SIAC kernel has been used to postprocess a two-dimensional DG solution (Docampo-Sánchez et al. in Multi-dimensional filtering: reducing the dimension through rotation, 2016. arXiv preprint arXiv:1610.02317). However, the scope of theoretical development of multidimensional SIAC filters has never gone beyond the usage of tensor product multidimensional B-splines or the reduction of the filter dimension. In this paper, we define a new SIAC filter called hexagonal SIAC (HSIAC) that uses a nonseparable class of two-dimensional spline functions called hex splines. In addition to relaxing the separability assumption, the proposed HSIAC filter provides more symmetry to its tensor-product counterpart. We prove that the superconvergence property holds for a specific class of structured triangular meshes using HSIAC filtering and provide numerical results to demonstrate and validate our theoretical results.

Item Type: Article
Uncontrolled Keywords: b-splines,hex splines,box splines,smoothness-increasing accuracy-conserving (siac) filtering,quasi-interpolation,approximation theory,discontinuous galerkin
Faculty \ School: Faculty of Science > School of Mathematics
Related URLs:
Depositing User: Pure Connector
Date Deposited: 31 Aug 2017 05:07
Last Modified: 11 Jun 2020 00:32
URI: https://ueaeprints.uea.ac.uk/id/eprint/64704
DOI: 10.1007/s10915-017-0517-5

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