Local discontinuous Galerkin methods with implicit-explicit time-marching for multi-dimensional convection-diffusion problems

Haijin Wang Nanjing University Shiping Wang Harbin Engineering University Qiang Zhang Nanjing University Chi-Wang Shu Brown University

Numerical Analysis and Scientific Computing mathscidoc:1610.25017

ESAIM: Mathematical Modelling and Numerical Analysis, 50, 1083-1105, 2016
The main purpose of this paper is to analyze the stability and error estimates of the local discontinuous Galerkin (LDG) methods coupled with implicit-explicit (IMEX) time discretization schemes, for solving multi-dimensional convection-diffusion equations with nonlinear convection. By establishing the important relationship between the gradient and the interface jump of the numerical solution with the independent numerical solution of the gradient in the LDG method, on both rectangular and triangular elements, we can obtain the same stability results as in the one-dimensional case, i.e, the IMEX LDG schemes are unconditionally stable for the multi-dimensional convection-diffusion problems, in the sense that the time-step $\dt$ is only required to be upper-bounded by a positive constant independent of the spatial mesh size $h$. Furthermore, by the aid of the so-called elliptic projection and the adjoint argument, we can also obtain optimal error estimates in both space and time, for the corresponding fully discrete IMEX LDG schemes, under the same condition, i.e, if piecewise polynomial of degree $k$ is adopted on either rectangular or triangular meshes, we can show the convergence accuracy is of order $\Ocal(h^{k+1}+\dt^s)$ for the $s$-th order IMEX LDG scheme ($s=1,2,3$) under consideration. Numerical experiments are also given to verify our main results.
local discontinuous Galerkin method, implicit-explicit scheme, convection-diffusion, stability, error estimate
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@inproceedings{haijin2016local,
  title={Local discontinuous Galerkin methods with implicit-explicit time-marching for multi-dimensional convection-diffusion problems},
  author={Haijin Wang, Shiping Wang, Qiang Zhang, and Chi-Wang Shu},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20161011105411260184131},
  booktitle={ESAIM: Mathematical Modelling and Numerical Analysis},
  volume={50},
  pages={1083-1105},
  year={2016},
}
Haijin Wang, Shiping Wang, Qiang Zhang, and Chi-Wang Shu. Local discontinuous Galerkin methods with implicit-explicit time-marching for multi-dimensional convection-diffusion problems. 2016. Vol. 50. In ESAIM: Mathematical Modelling and Numerical Analysis. pp.1083-1105. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20161011105411260184131.
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