In this paper, we study superconvergence properties of the discontinuous Galerkin method using upwind-biased numerical fluxes for one-dimensional linear hyperbolic equations. A 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 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 and , respectively. These superconvergent points can be regarded as the generalized Radau points. All theoretical findings are confirmed by numerical experiments.
Mots-clés : Discontinuous Galerkin methods, superconvergence, generalized Radau points, upwind-biased fluxes
@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},
mrnumber = {3626407},
zbl = {1367.65127},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/m2an/2016026/}
}
TY - JOUR AU - Cao, Waixiang AU - Li, Dongfang AU - Yang, Yang AU - Zhang, Zhimin TI - Superconvergence of Discontinuous Galerkin methods based on upwind-biased fluxes for 1D linear hyperbolic equations JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2017 SP - 467 EP - 486 VL - 51 IS - 2 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an/2016026/ DO - 10.1051/m2an/2016026 LA - en ID - M2AN_2017__51_2_467_0 ER -
%0 Journal Article %A Cao, Waixiang %A Li, Dongfang %A Yang, Yang %A Zhang, Zhimin %T Superconvergence of Discontinuous Galerkin methods based on upwind-biased fluxes for 1D linear hyperbolic equations %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2017 %P 467-486 %V 51 %N 2 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/m2an/2016026/ %R 10.1051/m2an/2016026 %G en %F M2AN_2017__51_2_467_0
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. http://archive.numdam.org/articles/10.1051/m2an/2016026/
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