Hexagonal Smoothness-Increasing Accuracy-Conserving Filtering
- 181 Downloads
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 \(k + 1\) to order \(2k + 1\). 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.
KeywordsB-splines Hex splines Box splines Smoothness-Increasing Accuracy-Conserving (SIAC) filtering Quasi-interpolation Approximation theory Discontinuous Galerkin
This work was sponsored in part by the Air Force Office of Scientific Research (AFOSR), Computational Mathematics Program (Program Manager: Dr. Jean-Luc Cambier), under Grant Nos. FA9550-12-1-0428 (first and fourth author), FA8655-13-1-3017 (third author). The second and fourth authors are sponsored in part by the Army Research Office (Program manager: Dr. Mike Coyle) under Grant No. W911NF-15-1-0222. In addition, the authors would like to acknowledge the anonymous reviewers for providing comments that improved the manuscript.
Compliance with Ethical Standards
Conflict of interest
The authors declare that they have no conflict of interest.
- 10.Docampo-Sánchez, J., Ryan, J.K., Mirzargar, M., Kirby, R.M.: Multi-dimensional filtering: reducing the dimension through rotation (2016). arXiv preprint arXiv:1610.02317
- 11.Entezari, A.: Optimal Sampling Lattices and Trivariate Box Splines. Ph.D. thesis, Simon Fraser University, Burnaby, BC, Canada, Canada (2007)Google Scholar
- 12.Jallepalli, A., Docampo-Sánchez, J., Ryan, J.K., Haimes, R., Kirby, R.M.: On the treatment of field quantities and elemental continuity in fem solutions. IEEE Trans. Vis. Comput. Graph. (2017) (accepted) Google Scholar
- 19.Meng, X., Ryan, J.K.: Divided difference estimates and accuracy enhancement of discontinuous Galerkin methods for nonlinear symmetric systems of hyperbolic conservation laws. IMA J. Numer. Anal. (2016). doi: 10.1093/imanum/drw072
- 20.Mirzaee, H.: Smoothness-Increasing Accuracy-Conserving filters (SIAC) for Discontinuous Galerkin Solutions. Ph.D. thesis, University of Utah, Salt Lake City, Utah, USA (2012)Google Scholar
- 23.Mirzaee, H., Ryan, J.K., Kirby, R.M.: Quantification of errors introduced in the numerical approximation and implementation of Smoothness-Increasing Accuracy-Conserving (SIAC) filtering of discontinuous Galerkin (DG) fields. J. Sci. Comput. 45(1–3), 447–470 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
- 26.Mirzargar, M.: A Reconstruction Framework for Common Sampling Lattices. Ph.D. thesis, University of Florida, Gainesville, Florida, USA (2012)Google Scholar
- 30.Nektar++ (2016). http://www.nektar.info