Superconvergence of discontinuous Galerkin methods for 2-D hyperbolic equations

Waixiang Cao Beijing Computational Science Research Center Chi-Wang Shu Brown University Yang Yang Michigan Technological University Zhimin Zhang Wayne State University

Numerical Analysis and Scientific Computing mathscidoc:1610.25026

SIAM Journal on Numerical Analysis, 53, 1651-1671, 2015
This paper is concerned with superconvergence properties of discontinuous Galerkin (DG) methods for 2-D linear hyperbolic conservation laws over rectangular meshes when upwind fluxes are used. We prove, under some suitable initial and boundary discretizations, the ($2k+1$)-th order superconvergence rate of the DG approximation at the downwind points and for the cell averages, when piecewise tensor-product polynomials of degree $k$ are used. Moreover, we prove that the gradient of the DG solution is superconvergent with a rate of ($k+1$)-th order at all interior left Radau points; and the function value approximation is superconvergent at all right Radau points with a rate of ($k+2$)-th order. Numerical experiments indicate that the aforementioned superconvergence rates are sharp.
Discontinuous Galerkin method, superconvergence, hyperbolic equations, Radau points, cell averages, initial and boundary discretizations
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@inproceedings{waixiang2015superconvergence,
  title={Superconvergence of discontinuous Galerkin methods for 2-D hyperbolic equations},
  author={Waixiang Cao, Chi-Wang Shu, Yang Yang, and Zhimin Zhang},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20161011120845685078140},
  booktitle={SIAM Journal on Numerical Analysis},
  volume={53},
  pages={1651-1671},
  year={2015},
}
Waixiang Cao, Chi-Wang Shu, Yang Yang, and Zhimin Zhang. Superconvergence of discontinuous Galerkin methods for 2-D hyperbolic equations. 2015. Vol. 53. In SIAM Journal on Numerical Analysis. pp.1651-1671. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20161011120845685078140.
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