Stability analysis and error estimates of Lax-Wendroff discontinuous Galerkin methods for linear conservation laws

Zheng Sun Brown University Chi-Wang Shu Brown University

Numerical Analysis and Scientific Computing mathscidoc:1610.25009

ESAIM: Mathematical Modelling and Numerical Analysis, 51, 1063-1087, 2017
In this paper, we analyze the Lax-Wendroff discontinuous Galerkin (LWDG) method for solving linear conservation laws. The method was originally proposed by Guo et al., where they applied local discontinuous Galerkin (LDG) techniques to approximate high order spatial derivatives in the Lax-Wendroff time discretization. We show that, under the standard CFL condition $\tau\leq \lambda h$ (where $\tau$ and $h$ are the time step and the maximum element length respectively and $\lambda>0$ is a constant) and uniform or non-increasing time steps, the second order schemes with piecewise linear elements and the third order schemes with arbitrary piecewise polynomial elements are stable in the $L^2$ norm. The specific type of stability may differ with different choices of numerical fluxes. Our stability analysis includes multidimensional problems with divergence-free coefficients. Besides solving the equation itself, the LWDG method also gives approximations to its time derivative simultaneously. We obtain optimal error estimates for both the solution $u$ and its first order time derivative $u_t$ in one dimension, and numerical examples are given to validate our analysis.
Discontinuous Galerkin method; Lax-Wendroff time discretization; linear conservation laws; $L^2$-stability; error estimates
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@inproceedings{zheng2017stability,
  title={Stability analysis and error estimates of Lax-Wendroff discontinuous Galerkin methods for linear conservation laws},
  author={Zheng Sun, and Chi-Wang Shu},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20161011101114478099123},
  booktitle={ESAIM: Mathematical Modelling and Numerical Analysis},
  volume={51},
  pages={1063-1087},
  year={2017},
}
Zheng Sun, and Chi-Wang Shu. Stability analysis and error estimates of Lax-Wendroff discontinuous Galerkin methods for linear conservation laws. 2017. Vol. 51. In ESAIM: Mathematical Modelling and Numerical Analysis. pp.1063-1087. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20161011101114478099123.
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