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

Wang, Haijin; Liu, Yunxian; Zhang, Qiang; Shu, Chi-Wang

doi:10.1090/mcom/3312

Local discontinuous Galerkin methods with implicit-explicit time-marching for time-dependent incompressible fluid flow
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by Haijin Wang, Yunxian Liu, Qiang Zhang and Chi-Wang Shu HTML | PDF

Math. Comp. 88 (2019), 91-121 Request permission

Abstract:

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 $\mathcal {Q}_k$ elements on cartesian meshes, in the sense that the time-step $\tau$ 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 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 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.

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