Capitalizing on Superconvergence for More Accurate Multi-Resolution Discontinuous Galerkin Methods

doi:10.1007/s42967-021-00121-w

Abstract

Abstract: This article focuses on exploiting superconvergence to obtain more accurate multi-resolution analysis. Specifcally, we concentrate on enhancing the quality of passing of information between scales by implementing the Smoothness-Increasing Accuracy-Conserving (SIAC) fltering combined with multi-wavelets. This allows for a more accurate approximation when passing information between meshes of diferent resolutions. Although this article presents the details of the SIAC flter using the standard discontinuous Galerkin method, these techniques are easily extendable to other types of data.

Key words: Multi-resolution analysis, Multi-wavelets, Discontinuous Galerkin, Smoothness-Increasing Accuracy-Conserving (SIAC), Post-processing, Superconvergence, Accuracy enhancement

CLC Number:

65M60

Jennifer K. Ryan. Capitalizing on Superconvergence for More Accurate Multi-Resolution Discontinuous Galerkin Methods[J]. Communications on Applied Mathematics and Computation, 2022, 4(2): 417-436.

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References

1.Alpert, B.K.:A class of bases in L² for the sparse representation of integral operators.SIAM J.Math.Anal.24, 246-262 (1993)
2.Bramble, J.H., Schatz, A.H.:Higher order local accuracy by averaging in the fnite element method.Math.Comput.31, 94-111 (1977)
3.Caviedes-Voulliéme, D., Gerhard, N., Sikstel, A., Müller, S.:Multiwavelet-based mesh adaptivity with discontinuous Galerkin schemes:exploring 2d shallow water problems.Adv.Water Resour.138, 103559 (2020)
4.Cockburn, B., Luskin, M., Shu, C.-W., Suli, E.:Enhanced accuracy by post-processing for fnite element methods for hyperbolic equations.Math.Comput.72, 577-606 (2003)
5.Docampo-Sánchez, J., Ryan, J.K., Mirzargar, M., Kirby, R.M.:Multi-dimensional fltering:reducing the dimension through rotation.SIAM J.Sci.Comput.39, A2179-A2200 (2017)
6.Gerhard, N., Iacono, F., May, G., Müller, R.S.S.:A high-order discontinuous Galerkin discretization with multiwavelet-based grid adaptation for compressible fows.J.Sci.Comput.62, 25-52 (2015)
7.Guo, W., Cheng, Y.:A sparse grid discontinuous Galerkin method for high-dimensional transport equations and its application to kinetic simulations.SIAM J.Sci.Comput.38, A3381-A3409 (2016)
8.Guo, W., Zhong, X., Qiu, J.-M.:Superconvergence of discontinuous Galerkin and local discontinuous Galerkin methods:eigen-structure analysis based on Fourier approach.J.Comput.Phys.235, 458-485 (2013)
9.Hovhannisyan, N., Müller, S., Schäfer, R.:Adaptive Multiresolution Discontinuous Galerkin Schemes for Conservation Laws.Report 311, Institut für Geometrie und Praktische Mathematik, Aachen (2010).http://www.igpm.rwth-aachen.de/en/reports2010
10.Ji, L., van Slingerland, P., Ryan, J.K., Vuik, K.:Superconvergent error estimates for position-dependent smoothness-increasing accuracy-conserving post-processing of discontinuous Galerkin solutions.Math.Comput.83, 2239-2262 (2014)
11.Ji, L., Yan, X., Ryan, J.K.:Accuracy enhancement for the linear convection-difusion equation in multiple dimensions.Math.Comput.81, 1929-1950 (2012)
12.Ji, L., Yan, X., Ryan, J.K.:Negative-order norm estimates for nonlinear hyperbolic conservation laws.J.Sci.Comput.54, 269-310 (2013)
13.Kesserwani, G., Caviedes-Voulliéme, D., Gerhard, N., Müller, S.:Multiwavelet discontinuous Galerkin h-adaptive shallow water model.Comput.Methods Appl.Mech.Eng.294, 56-71 (2015)
14.King, J., Mirzaee, H., Ryan, J.K., Kirby, R.M.:Smoothness-Increasing Accuracy-Conserving (SIAC) fltering for discontinuous Galerkin solutions:improved errors versus higher-order accuracy.J.Sci.Comput.53, 129-149 (2012)
15.Liu, Y., Cheng, Y., Chen, S., Zhang, Y.-T.:Krylov implicit integration factor discontinuous Galerkin methods on sparse grids for high dimensional reaction-difusion equations.J.Comput.Phys.388, 90-102 (2019)
16.Meng, X., Ryan, J.K.:Discontinuous Galerkin methods for nonlinear scalar hyperbolic conservation laws:divided diference estimates and accuracy enhancement.Numer.Math.136, 27-73 (2017)
17.Meng, X., Ryan, J.K.:Divided diference estimates and accuracy enhancement of discontinuous Galerkin methods for nonlinear symmetric systems of hyperbolic conservation laws.IMA J.Numer.Anal.38, 125-155 (2018)
18.Mirzaee, H., Ji, L., Ryan, J.K., Kirby, R.M.:Smoothness-Increasing Accuracy-Conserving (SIAC) post-processing for discontinuous Galerkin solutions over structured triangular meshes.SIAM J.Numer.Anal.49, 1899-1920 (2011)
19.Mirzaee, H., King, J., Ryan, J.K., Kirby, R.M.:Smoothness-Increasing Accuracy-Conserving (SIAC) flters for discontinuous Galerkin solutions over unstructured triangular meshes.SIAM J.Sci.Comput.35, A212-A230 (2013)
20.Mirzargar, M., Ryan, J.K., Kirby, R.M.:Smoothness-Increasing Accuracy-Conserving (SIAC) fltering and quasi-interpolation:a unifed view.J.Sci.Comput.67, 237-261 (2016)
21.Nyström, E.J.:Über die praktische aufösung von integralgleichungen mit anwendungen auf randwertaufgaben.Acta Math.54, 185-204 (1930)
22.Ryan, J.K., Cockburn, B.:Local derivative post-processing for the discontinuous Galerkin method.J.Comput.Phys.228, 8642-8664 (2009)
23.Ryan, J.K., Shu, C.-W.:One-sided post-processing for the discontinuous Galerkin method.Methods Appl.Anal.10, 295-307 (2003)
24.Ryan, J.K.:Exploiting superconvergence through Smoothness-Increasing Accuracy-Conserving (SIAC) fltering.In:Kirby, R., Berzins, M., Hesthaven, J.(eds.) Spectral and High Order Methods for Partial Diferential Equations ICOSAHOM 2014.Lecture Notes in Computational Science and Engineering, vol.106, pp.87-102.Springer, Cham (2015)
25.Tao, Z., Chenand, A., Zhang, M., Cheng, Y.:Sparse grid central discontinuous Galerkin method for linear hyperbolic systems in high dimensions.SIAM J.Sci.Comput.41, A1626-A1651 (2019)
26.Tao, Z., Guo, W., Cheng, Y.:Sparse grid discontinuous Galerkin methods for the Vlasov-Maxwell system.J.Comput.Phys.3, 100022 (2019)
27.Thomée, V.:High order local approximations to derivatives in the fnite element method.Math.Comput.31, 652-660 (1977)
28.Van Slingerland, P., Ryan, J.K., Vuik, K.:Position-dependent Smoothness-Increasing AccuracyConserving (SIAC) fltering for accuracy for improving discontinuous Galerkin solutions.SIAM J.Sci.Comput.33, 802-825 (2011)
29.Vuik, M.J., Ryan, J.K.:Multiwavelet troubled-cell indicator for discontinuity detection of discontinuous Galerkin schemes.J.Comput.Phys.270, 138-160 (2014)
30.Vuik, M.J., Ryan, J.K.:Multiwavelets and jumps in DG approximations.In:Kirby, R., Berzins, M., Hesthaven, J.(eds.) Spectral and High Order Methods for Partial Diferential Equations ICOSAHOM 2014.Lecture Notes in Computational Science and Engineering, vol.106, pp.503-511.Springer, Cham (2015)
31.Wang, Z., Tang, Q., Guo, W., Cheng, Y.:Sparse grid discontinuous Galerkin methods for high-dimensional elliptic equations.J.Comput.Phys.314, 244-263 (2016)

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