Local discontinuous Galerkin methods with implicit-explicit time-marching for time-dependent incompressible fluid flow

Haijin Wang Yunxian Liu Qiang Zhang Chi-Wang Shu Brown University

Numerical Analysis and Scientific Computing mathscidoc:1804.25021

Mathematics of Computation, 2018
The main purpose of this paper is to study the stability and error estimates of the local discontinuous Galerkin (LDG) methods coupled with multi-step implicit-explicit (IMEX) time discretization schemes, for solving time-dependent incompressible fluid flows. We will give theoretical analysis for the Oseen equation, and assess the performance of the schemes for incompressible Navier-Stokes equations numerically. For the Oseen equation, using first order IMEX time discretization as an example, we show that the IMEX-LDG scheme is unconditionally stable for $\Qcal_k$ elements on cartesian meshes, in the sense that the time-step $\dt$ is only required to be {bounded from above} by a positive constant independent of the spatial mesh size $h$. Furthermore, by the aid of the \emph{Stokes projection} and an elaborate energy analysis, we obtain the $L^{\infty}(L^2)$ optimal error estimates for both the velocity and the stress (gradient of velocity), in both space and time. By the \emph{inf-sup} argument, we also obtain the $L^{\infty}(L^2)$ optimal error estimates for the pressure. Numerical experiments are given to validate our main results.
local discontinuous Galerkin method, implicit-explicit scheme, incompressible flow, Oseen equation, Navier-Stokes, stability, error estimate
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@inproceedings{haijin2018local,
  title={Local discontinuous Galerkin methods with implicit-explicit time-marching for time-dependent incompressible fluid flow},
  author={Haijin Wang, Yunxian Liu, Qiang Zhang, and Chi-Wang Shu},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180416104154751487056},
  booktitle={Mathematics of Computation},
  year={2018},
}
Haijin Wang, Yunxian Liu, Qiang Zhang, and Chi-Wang Shu. Local discontinuous Galerkin methods with implicit-explicit time-marching for time-dependent incompressible fluid flow. 2018. In Mathematics of Computation. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180416104154751487056.
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