Superconvergence of discontinuous Galerkin methods for 1-D linear hyperbolic equations with degenerate variable coefficients

Waixiang Cao Chi-Wang Shu Brown University Zhimin Zhang

Numerical Analysis and Scientific Computing mathscidoc:1804.25013

ESAIM: Mathematical Modelling and Numerical Analysis, 51, 2213-2235, 2017
In this paper, we study the superconvergence behavior of discontinuous Galerkin methods using upwind numerical fluxes for one-dimensional linear hyperbolic equations with degenerate variable coefficients. The study establishes superconvergence results for the flux function approximation as well as for the DG solution itself. To be more precise, we first prove that the DG flux function is superconvergent towards a particular flux function of the exact solution, with an order of $O(h^{k+2})$, when piecewise polynomials of degree $k$ are used. We then prove that the highest superconvergence rate of the DG solution itself is $O(h^{k+\frac 32})$ as the variable coefficient degenerates or achieves the value zero in the domain. As byproducts, we obtain superconvergence properties for the DG solution and the DG flux function at special points and for cell averages. All theoretical findings are confirmed by numerical experiments.
Discontinuous Galerkin methods, superconvergence, degenerate variable coefficients, Radau points, upwind fluxes
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@inproceedings{waixiang2017superconvergence,
  title={Superconvergence of discontinuous Galerkin methods for 1-D linear hyperbolic equations with degenerate variable coefficients},
  author={Waixiang Cao, Chi-Wang Shu, and Zhimin Zhang},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180416101842719691048},
  booktitle={ESAIM: Mathematical Modelling and Numerical Analysis},
  volume={51},
  pages={2213-2235},
  year={2017},
}
Waixiang Cao, Chi-Wang Shu, and Zhimin Zhang. Superconvergence of discontinuous Galerkin methods for 1-D linear hyperbolic equations with degenerate variable coefficients. 2017. Vol. 51. In ESAIM: Mathematical Modelling and Numerical Analysis. pp.2213-2235. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180416101842719691048.
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