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


Authors: Haijin Wang, Yunxian Liu, Qiang Zhang and Chi-Wang Shu
Journal: Math. Comp. 88 (2019), 91-121
MSC (2010): Primary 65M12, 65M15, 65M60
DOI: https://doi.org/10.1090/mcom/3312
Published electronically: March 16, 2018
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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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Additional Information

Haijin Wang
Affiliation: College of Science, Nanjing University of Posts and Telecommunications, Nanjing 210023, Jiangsu Province, People’s Republic of China
Email: hjwang@njupt.edu.cn

Yunxian Liu
Affiliation: School of Mathematics, Shandong University, Jinan 250100, Shandong Province, People’s Republic of China
Email: yxliu@sdu.edu.cn

Qiang Zhang
Affiliation: Department of Mathematics, Nanjing University, Nanjing 210093, Jiangsu Province, People’s Republic of China
Email: qzh@nju.edu.cn

Chi-Wang Shu
Affiliation: Division of Applied Mathematics, Brown University, Providence, Rhode Island 02912
Email: shu@dam.brown.edu

DOI: https://doi.org/10.1090/mcom/3312
Keywords: Local discontinuous Galerkin method, implicit-explicit scheme, incompressible flow, Oseen equation, Navier-Stokes, stability, error estimate
Received by editor(s): October 23, 2016
Received by editor(s) in revised form: June 15, 2017, and July 1, 2017
Published electronically: March 16, 2018
Additional Notes: The first author’s research was sponsored by NSFC grant 11601241, Natural Science Foundation of Jiangsu Province grant BK20160877, and NUPTSF grant NY215067.
The second author’s research was sponsored by NSFC grant 11471194.
The third author’s research was supported by NSFC grants 11671199, 11571290, and 11271187.
The fourth author’s research was supported by DOE grant DE-FG02-08ER25863 and NSF grant DMS-1418750.
Article copyright: © Copyright 2018 American Mathematical Society

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