The Euler implicit/explicit scheme for the 2D time-dependent Navier-Stokes equations with smooth or non-smooth initial data
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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 $\alpha =2$, $\tau |\log h|\leq C_0$ 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
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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
- 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.
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 77 (2008), 2097-2124
- MSC (2000): Primary 35L70, 65N30, 76D06
- DOI: https://doi.org/10.1090/S0025-5718-08-02127-3
- MathSciNet review: 2429876