Accuracyenhancement of discontinuous Galerkin solutions for convectiondiffusion equations in multipledimensions
Authors:
Liangyue Ji, Yan Xu and Jennifer K. Ryan
Journal:
Math. Comp. 81 (2012), 19291950
MSC (2010):
Primary 65M60; Secondary 35K10, 35L02
Published electronically:
March 2, 2012
MathSciNet review:
2945143
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Abstract: Discontinuous Galerkin (DG) methods exhibit ``hidden accuracy'' that makes superconvergence of this method an increasing popular topic to address. Previous investigations have focused on the superconvergent properties of ordinary differential equations and linear hyperbolic equations. Additionally, superconvergence of order for the convectiondiffusion equation that focuses on a special projection using the upwind flux was presented by Cheng and Shu. In this paper we demonstrate that it is possible to extend the smoothnessincreasing accuracyconserving (SIAC) filter for use on the multidimensional linear convectiondiffusion equation in order to obtain 2+ order of accuracy, where depends upon the flux and takes on the values or The technique that we use to extract this hidden accuracy was initially introduced by Cockburn, Luskin, Shu, and Süli for linear hyperbolic equations and extended by Ryan et al. as a smoothnessincreasing accuracyconserving filter. We solve this convectiondiffusion equation using the local discontinuous Galerkin (LDG) method and show theoretically that it is possible to obtain in the negativeorder norm. By postprocessing the LDG solution to a linear convection equation using a specially designed kernel such as the one by Cockburn et al., we can compute this same order accuracy in the norm. Additionally, we present numerical studies that confirm that we can improve the LDG solution from to using alternating fluxes and that we actually obtain for diffusiondominated problems.
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 Y. Cheng and C.W. Shu, Superconvergence of discontinuous Galerkin and local discontinuous Galerkin schemes for linear hyperbolic and convectiondiffusion equations in one space dimension, SIAM Journal on Numerical Analysis, 47 (2010), pp. 40444072. MR 2585178 (2011e:65187)
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 B. Cockburn, Discontinuous Galerkin methods for methods for convectiondominated problems, in Highorder methods for computational physics, T.J. Barth and H. Deconinck, editors, Lecture Notes in Computational Science and Engineering, volume 9, Springer, 1999, pp. 69224. MR 1712278 (2000f:76095)
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Additional Information
Liangyue Ji
Affiliation:
Delft Institute of Applied Mathematics, Delft University of Technology, 2628 CD Delft, The Netherlands.
Address at time of publication:
Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, P.R. China.
Email:
jlyue@mail.ustc.edu.cn
Yan Xu
Affiliation:
Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China
Email:
yxu@ustc.edu.cn
Jennifer K. Ryan
Affiliation:
Delft Institute of Applied Mathematics, Delft University of Technology, 2628 CD Delft, The Netherlands
Email:
J.K.Ryan@tudelft.nl
DOI:
http://dx.doi.org/10.1090/S002557182012025865
Keywords:
discontinuous Galerkin method,
convectiondiffusion equations,
negativeorder norm error estimates,
filtering,
postprocessing,
accuracy enhancement.
Received by editor(s):
September 26, 2010
Received by editor(s) in revised form:
April 26, 2011, and July 1, 2011
Published electronically:
March 2, 2012
Additional Notes:
The research of the second author was supported by NSFC grant No.10971211, No. 11031007, FANEDD No. 200916, FANEDD of CAS, NCET No. 090922 and the Fundamental Research Funds for the Central Universities. Additional support was provided by the Alexander von HumboldtFoundation while the author was in residence at Freiburg University, Germany
Article copyright:
© Copyright 2012
American Mathematical Society
