Stability analysis and error estimates of local discontinuous Galerkin methods with implicit-explicit time-marching for the time-dependent fourth order PDEs

Haijin Wang Qiang Zhang Chi-Wang Shu Brown University

Numerical Analysis and Scientific Computing mathscidoc:1804.25010

ESAIM: Mathematical Modelling and Numerical Analysis, 51, 1931-1955, 2017
The main purpose of this paper is to give stability analysis and error estimates of the local discontinuous Galerkin (LDG) methods coupled with three specific implicit-explicit (IMEX) Runge-Kutta time discretization methods up to third order accuracy, for solving one-dimensional time-dependent linear fourth order partial differential equations. In the time discretization, all the lower order derivative terms are treated explicitly and the fourth order derivative term is treated implicitly. By the aid of energy analysis, we show that the IMEX-LDG schemes are unconditionally energy stable, in the sense that the time step $\dt$ is only required to be upper-bounded by a constant which is independent of the mesh size $h$. The optimal error estimate is also derived by the aid of the elliptic projection and the adjoint argument. Numerical experiments are given to verify that the corresponding IMEX-LDG schemes can achieve optimal error accuracy.
local discontinuous Galerkin method, implicit-explicit time-marching scheme, time-dependent fourth order equations, stability, error estimates, energy method
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@inproceedings{haijin2017stability,
  title={Stability analysis and error estimates of local discontinuous Galerkin methods with implicit-explicit time-marching for the time-dependent fourth order PDEs},
  author={Haijin Wang, Qiang Zhang, and Chi-Wang Shu},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180416094445040022045},
  booktitle={ESAIM: Mathematical Modelling and Numerical Analysis},
  volume={51},
  pages={1931-1955},
  year={2017},
}
Haijin Wang, Qiang Zhang, and Chi-Wang Shu. Stability analysis and error estimates of local discontinuous Galerkin methods with implicit-explicit time-marching for the time-dependent fourth order PDEs. 2017. Vol. 51. In ESAIM: Mathematical Modelling and Numerical Analysis. pp.1931-1955. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180416094445040022045.
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