Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 

 

The Euler implicit/explicit scheme for the 2D time-dependent Navier-Stokes equations with smooth or non-smooth initial data


Author: Yinnian He
Journal: Math. Comp. 77 (2008), 2097-2124
MSC (2000): Primary 35L70, 65N30, 76D06
DOI: https://doi.org/10.1090/S0025-5718-08-02127-3
Published electronically: May 8, 2008
MathSciNet review: 2429876
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This paper considers the stability and convergence results for the Euler implicit/explicit scheme applied to the spatially discretized two-dimensional (2D) time-dependent Navier-Stokes equations. A Galerkin finite element spatial discretization is assumed, and the temporal treatment is implicit/explict scheme, which is implicit for the linear terms and explicit for the nonlinear term. Here the stability condition depends on the smoothness of the initial data $ u_0\in H^\alpha$, i.e., the time step condition is $ \tau\leq C_0$ in the case of $ \alpha=2$, $ \tau\vert\log h\vert\leq C_0$ in the case of $ \alpha=1$ and $ \tau h^{-2}\leq C_0$ in the case of $ \alpha=0$ for mesh size $ h$ and some positive constant $ C_0$. We provide the $ H^2$-stability of the scheme under the stability condition with $ \alpha=0,1,2$ and obtain the optimal $ H^1-L^2$ error estimate of the numerical velocity and the optimal $ L^2$ error estimate of the numerical pressure under the stability condition with $ \alpha=1,2$.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 35L70, 65N30, 76D06

Retrieve articles in all journals with MSC (2000): 35L70, 65N30, 76D06


Additional Information

Yinnian He
Affiliation: Faculty of Science, Xi’an Jiaotong University, Xi’an 710049, People’s Republic of China
Email: heyn@mail.xjtu.edu.cn

DOI: https://doi.org/10.1090/S0025-5718-08-02127-3
Keywords: Navier-Stokes equations, mixed finite element, Euler implicit/explicit scheme, Smooth or non-smooth initial data
Received by editor(s): February 26, 2007
Received by editor(s) in revised form: September 17, 2007
Published electronically: May 8, 2008
Additional Notes: This research was subsidized by the NSF of China 10671154 and the National Basic Research Program under the grant 2005CB321703.
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.