Optimal error estimates for discontinuous Galerkin methods based on upwind-biased fluxes for linear hyperbolic equations

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

Numerical Analysis and Scientific Computing mathscidoc:1610.25023

Mathematics of Computation, 85, 1225-1261, 2016
We analyze discontinuous Galerkin methods using upwind-biased numerical fluxes for time-dependent linear conservation laws. In one dimension, optimal a priori error estimates of order $k+1$ are obtained for the semidiscrete scheme when piecewise polynomials of degree at most $k$ $(k \ge 0)$ are used. Our analysis is valid for arbitrary nonuniform regular meshes and for both periodic boundary conditions and for initial-boundary value problems. We extend the analysis to the multidimensional case on Cartesian meshes when piecewise tensor product polynomials are used, and to the fully discrete scheme with explicit Runge--Kutta time discretization. Numerical experiments are shown to demonstrate the theoretical results.
discontinuous Galerkin method, upwind-biased flux, conservation laws, error estimates
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@inproceedings{xiong2016optimal,
  title={Optimal error estimates for discontinuous Galerkin methods based on upwind-biased fluxes for linear hyperbolic equations},
  author={Xiong Meng, Chi-Wang Shu, and Boying Wu},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20161011114243397704137},
  booktitle={Mathematics of Computation},
  volume={85},
  pages={1225-1261},
  year={2016},
}
Xiong Meng, Chi-Wang Shu, and Boying Wu. Optimal error estimates for discontinuous Galerkin methods based on upwind-biased fluxes for linear hyperbolic equations. 2016. Vol. 85. In Mathematics of Computation. pp.1225-1261. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20161011114243397704137.
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