Error estimates to smooth solutions of semi-discrete discontinuous Galerkin methods with quadrature rules for scalar conservation laws

Juntao Huang Tsinghua University Chi-Wang Shu Brown University

Numerical Analysis and Scientific Computing mathscidoc:1610.25004

Numerical Methods for Partial Differential Equations, 33, 467-488, 2017
In this paper, we focus on error estimates to smooth solutions of semi-discrete discontinuous Galerkin (DG) methods with quadrature rules for scalar conservation laws. The main techniques we use are energy estimate and Taylor expansion first introduced by Zhang and Shu. We show that, with ${P}^k$ (piecewise polynomials of degree $k$) finite elements in 1D problems, if the quadrature over elements is exact for polynomials of degree $(2k)$, error estimates of $O(h^{k+1/2})$ are obtained for general monotone fluxes, and optimal estimates of $O(h^{k+1})$ are obtained for upwind fluxes. For multidimensional problems, if in addition quadrature over edges is exact for polynomials of degree $(2k+1)$, error estimates of $O(h^k)$ are obtained for general monotone fluxes, and $O(h^{k+1/2})$ are obtained for monotone and sufficiently smooth numerical fluxes. Numerical results validate our analysis.
Discontinuous Galerkin; error estimate; quadrature rules; conservation laws; semi-discrete
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@inproceedings{juntao2017error,
  title={ Error estimates to smooth solutions of semi-discrete discontinuous Galerkin methods with quadrature rules for scalar conservation laws},
  author={Juntao Huang, and Chi-Wang Shu},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20161011092230625772118},
  booktitle={Numerical Methods for Partial Differential Equations},
  volume={33},
  pages={467-488},
  year={2017},
}
Juntao Huang, and Chi-Wang Shu. Error estimates to smooth solutions of semi-discrete discontinuous Galerkin methods with quadrature rules for scalar conservation laws. 2017. Vol. 33. In Numerical Methods for Partial Differential Equations. pp.467-488. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20161011092230625772118.
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