Superconvergence of discontinuous Galerkin method for scalar nonlinear conservation laws in one space dimension

Xiong Meng Harbin Institute of Technology Chi-Wang Shu Brown University Qiang Zhang Nanjing University Boying Wu Harbin Institute of Technology

Numerical Analysis and Scientific Computing mathscidoc:1610.25071

SIAM Journal on Numerical Analysis, 50, 2336-2356, 2012
In this paper, an analysis of the superconvergence property of the semidiscrete discontinuous Galerkin (DG) method applied to one-dimensional time-dependent nonlinear scalar conservation laws is carried out. We prove that the error between the DG solution and a particular projection of the exact solution achieves $\left(k + \frac32\right)$-th order superconvergence when upwind fluxes are used. The results hold true for arbitrary nonuniform regular meshes and for piecewise polynomials of degree $k$ ($k \ge 1$), under the condition that $|f'(u)|$ possesses a uniform positive lower bound. Numerical experiments are provided to show that the superconvergence property actually holds true for nonlinear conservation laws with general flux functions, indicating that the restriction on $f(u)$ is artificial.
discontinuous Galerkin method, superconvergence, upwind flux, error estimates
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@inproceedings{xiong2012superconvergence,
  title={Superconvergence of discontinuous Galerkin method for scalar nonlinear conservation laws in one space dimension},
  author={Xiong Meng, Chi-Wang Shu, Qiang Zhang, and Boying Wu},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20161012105858556595189},
  booktitle={SIAM Journal on Numerical Analysis},
  volume={50},
  pages={2336-2356},
  year={2012},
}
Xiong Meng, Chi-Wang Shu, Qiang Zhang, and Boying Wu. Superconvergence of discontinuous Galerkin method for scalar nonlinear conservation laws in one space dimension. 2012. Vol. 50. In SIAM Journal on Numerical Analysis. pp.2336-2356. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20161012105858556595189.
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