Geodesic
Superconvergence of Discontinuous Galerkin methods based on upwind-biased fluxes for 1D linear hyperbolic equations
Cao, Waixiang1 ; Li, Dongfang2 ; Yang, Yang3 ; Zhang, Zhimin1, 4
1 Beijing Computational Science Research Center, Zhongguancun Software Park II, No. 10 West Dongbeiwang Road, Haidian District, Beijing 100094, P.R. China.
2 School of Mathematics and Statistics, Huazhong University of Science and Technology, 1037 Luoyu Rd, Hongshan, Wuhan, Hubei 430074, P.R. China.
3 Department of Mathematical Sciences, Michigan Technological University, 1400 Townsend Drive, Houghton, MI 49931, USA.
4 Department of Mathematics, Wayne State University, 42 W. Warren Ave. Detroit, MI 48202, USA.
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 2, pp. 467-486

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In this paper, we study superconvergence properties of the discontinuous Galerkin method using upwind-biased numerical fluxes for one-dimensional linear hyperbolic equations. A (2k+1)th order superconvergence rate of the DG approximation at the numerical fluxes and for the cell average is obtained under quasi-uniform meshes and some suitable initial discretization, when piecewise polynomials of degree k are used. Furthermore, surprisingly, we find that the derivative and function value approximation of the DG solution are superconvergent at a class of special points, with an order k+1 and k+2, respectively. These superconvergent points can be regarded as the generalized Radau points. All theoretical findings are confirmed by numerical experiments.

DOI : 10.1051/m2an/2016026
Classification : 65M15, 65M60, 65N30
Keywords: Discontinuous Galerkin methods, superconvergence, generalized Radau points, upwind-biased fluxes

Cao, Waixiang 1 ; Li, Dongfang 2 ; Yang, Yang 3 ; Zhang, Zhimin 1, 4

1 Beijing Computational Science Research Center, Zhongguancun Software Park II, No. 10 West Dongbeiwang Road, Haidian District, Beijing 100094, P.R. China.
2 School of Mathematics and Statistics, Huazhong University of Science and Technology, 1037 Luoyu Rd, Hongshan, Wuhan, Hubei 430074, P.R. China.
3 Department of Mathematical Sciences, Michigan Technological University, 1400 Townsend Drive, Houghton, MI 49931, USA.
4 Department of Mathematics, Wayne State University, 42 W. Warren Ave. Detroit, MI 48202, USA. 
@article{M2AN_2017__51_2_467_0,
     author = {Cao, Waixiang and Li, Dongfang and Yang, Yang and Zhang, Zhimin},
     title = {Superconvergence of {Discontinuous} {Galerkin} methods based on upwind-biased fluxes for {1D} linear hyperbolic equations},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {467--486},
     publisher = {EDP-Sciences},
     volume = {51},
     number = {2},
     year = {2017},
     doi = {10.1051/m2an/2016026},
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     zbl = {1367.65127},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an/2016026/}
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Cao, Waixiang; Li, Dongfang; Yang, Yang; Zhang, Zhimin. Superconvergence of Discontinuous Galerkin methods based on upwind-biased fluxes for 1D linear hyperbolic equations. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 2, pp. 467-486. doi: 10.1051/m2an/2016026

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