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June  2019, 24(6): 2745-2780. doi: 10.3934/dcdsb.2018273

## H2-stability of some second order fully discrete schemes for the Navier-Stokes equations

 1 School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, China 2 College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, China 3 Department of Mathematics, School of Arts and Sciences, Shaanxi University of Science and Technology, Xi'an 710021, China 4 Department of Mathematics, Baoji University of Arts and Sciences, Baoji 721013, China

* Corresponding author: Pengzhan Huang

Received  March 2018 Revised  May 2018 Published  October 2018

Fund Project: Supported by the Major Research and Development Program of China (Grant No.2016YFB0200901), the NSF of China (Grant Nos. 11861067, 11771348 and 11771259) and the NSF of Xinjiang Province (Grant No. 2017D01C052)

This paper considers the $H^2$-stability results for the second order fully discrete schemes based on the mixed finite element method for the 2D time-dependent Navier-Stokes equations with the initial data $u_0∈ H^α,$ where $α = 0, ~1$ and 2. A mixed finite element method is used to the spatial discretization of the Navier-Stokes equations, and the temporal treatments of the spatial discrete Navier-Stokes equations are the second order semi-implicit, implicit/explict and explicit schemes. The $H^2$-stability results of the schemes are provided, where the second order semi-implicit and implicit/explicit schemes are almost unconditionally $H^2$-stable, the second order explicit scheme is conditionally $H^2$-stable in the case of $\alpha = 2$, and the semi-implicit, implicit/explicit and explicit schemes are conditionally $H^2$-stable in the case of $\alpha = 1, ~0$. Finally, some numerical tests are made to verify the above theoretical results.

Citation: Yinnian He, Pengzhan Huang, Jian Li. H2-stability of some second order fully discrete schemes for the Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2745-2780. doi: 10.3934/dcdsb.2018273
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##### References:
The energy computed by semi-implicit scheme (left) and explicit scheme (right) based on several different time steps chosen with $u_0(x, y)\in H^2.$
The energy computed by semi-implicit scheme (left) and explicit scheme (right) based on several different time steps chosen with $u_0(x, y)\in H^1.$
The energy computed by semi-implicit scheme (left) and explicit scheme (right) based on several different time steps chosen with $u_0(x, y)\in H^0.$
The errors based on the semi-implicit scheme with $u_0(x, y)\in H^2.$
 $\tau$ 0.005 0.01 0.02 0.05 0.5 ${\| u- u_{h}^n \|_0}$ 1.753E$-4$ 1.752E$-4$ 1.752E$-4$ 1.842E$-4$ 5.788E$-4$ ${\| \nabla(u- u_{h}^n)\|_0}$ 7.330E$-3$ 7.330E$-3$ 7.331E$-3$ 8.480E$-3$ 8.917E$-3$ ${\| p- p_{h}^n \|_0}$ 1.166E$-2$ 1.839E$-2$ 3.357E$-2$ 8.126E$-2$ 8.204E$-1$
 $\tau$ 0.005 0.01 0.02 0.05 0.5 ${\| u- u_{h}^n \|_0}$ 1.753E$-4$ 1.752E$-4$ 1.752E$-4$ 1.842E$-4$ 5.788E$-4$ ${\| \nabla(u- u_{h}^n)\|_0}$ 7.330E$-3$ 7.330E$-3$ 7.331E$-3$ 8.480E$-3$ 8.917E$-3$ ${\| p- p_{h}^n \|_0}$ 1.166E$-2$ 1.839E$-2$ 3.357E$-2$ 8.126E$-2$ 8.204E$-1$
The errors based on the implicit/explicit scheme with $u_0(x, y)\in H^2.$
 $\tau$ 0.005 0.01 0.02 0.05 0.5 ${\| u- u_{h}^n \|_0}$ 1.753E$-4$ 1.752E$-4$ 1.752E$-4$ 1.842E$-4$ 5.925E$-4$ ${\| \nabla(u- u_{h}^n)\|_0}$ 7.330E$-3$ 7.330E$-3$ 7.331E$-3$ 8.480E$-3$ 9.291E$-3$ ${\| p- p_{h}^n \|_0}$ 1.166E$-2$ 1.839E$-2$ 3.357E$-2$ 8.126E$-2$ 8.204E$-1$
 $\tau$ 0.005 0.01 0.02 0.05 0.5 ${\| u- u_{h}^n \|_0}$ 1.753E$-4$ 1.752E$-4$ 1.752E$-4$ 1.842E$-4$ 5.925E$-4$ ${\| \nabla(u- u_{h}^n)\|_0}$ 7.330E$-3$ 7.330E$-3$ 7.331E$-3$ 8.480E$-3$ 9.291E$-3$ ${\| p- p_{h}^n \|_0}$ 1.166E$-2$ 1.839E$-2$ 3.357E$-2$ 8.126E$-2$ 8.204E$-1$
The energy based on the implicit/explicit scheme with $u_0(x, y)\in H^2.$
 $t$ 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 $\tau=0.5$ 2.935 0.579 0.1638 0.873 1.423 3.564 2.019 1.74 0.082 0.14 $\tau=0.05$ 3.512 1.641 0.379 0.475 1.537 2.391 2.158 1.048 0.133 0.332 $\tau=0.02$ 1.886 1.035 0.038 0.467 1.754 2.654 2.378 1.158 0.121 0.218 $\tau=0.01$ 2.152 0.792 0.014 0.469 1.739 2.655 2.376 1.157 0.12 0.218 $\tau=0.005$ 2.086 0.791 0.014 0.469 1.739 2.655 2.376 1.157 0.12 0.218
 $t$ 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 $\tau=0.5$ 2.935 0.579 0.1638 0.873 1.423 3.564 2.019 1.74 0.082 0.14 $\tau=0.05$ 3.512 1.641 0.379 0.475 1.537 2.391 2.158 1.048 0.133 0.332 $\tau=0.02$ 1.886 1.035 0.038 0.467 1.754 2.654 2.378 1.158 0.121 0.218 $\tau=0.01$ 2.152 0.792 0.014 0.469 1.739 2.655 2.376 1.157 0.12 0.218 $\tau=0.005$ 2.086 0.791 0.014 0.469 1.739 2.655 2.376 1.157 0.12 0.218
The errors based on the semi-implicit scheme with $u_0(x, y)\in H^1.$
 $\tau$ 0.0005 0.005 0.05 0.5 ${\| u- u_{h}^n \|_0}$ 1.894E$-4$ 1.894E$-4$ 1.980E$-4$ 4.376E$-4$ ${\| \nabla(u- u_{h}^n)\|_0}$ 6.442E$-3$ 6.442E$-3$ 7.719E$-3$ 8.131E$-3$ ${\| p- p_{h}^n \|_0}$ 8.554E$-3$ 1.355E$-2$ 8.366E$-2$ 8.208E$-1$
 $\tau$ 0.0005 0.005 0.05 0.5 ${\| u- u_{h}^n \|_0}$ 1.894E$-4$ 1.894E$-4$ 1.980E$-4$ 4.376E$-4$ ${\| \nabla(u- u_{h}^n)\|_0}$ 6.442E$-3$ 6.442E$-3$ 7.719E$-3$ 8.131E$-3$ ${\| p- p_{h}^n \|_0}$ 8.554E$-3$ 1.355E$-2$ 8.366E$-2$ 8.208E$-1$
The errors based on the implicit/explicit scheme with $u_0(x, y)\in H^1.$
 $\tau$ 0.0005 0.005 0.05 0.5 ${\| u- u_{h}^n \|_0}$ 1.894E$-4$ 1.894E$-4$ 1.980E$-4$ 4.564E$-4$ ${\| \nabla(u- u_{h}^n)\|_0}$ 6.442E$-3$ 6.442E$-3$ 7.718E$-3$ 8.541E$-3$ ${\| p- p_{h}^n \|_0}$ 8.554E$-3$ 1.355E$-2$ 8.366E$-2$ 8.208E$-1$
 $\tau$ 0.0005 0.005 0.05 0.5 ${\| u- u_{h}^n \|_0}$ 1.894E$-4$ 1.894E$-4$ 1.980E$-4$ 4.564E$-4$ ${\| \nabla(u- u_{h}^n)\|_0}$ 6.442E$-3$ 6.442E$-3$ 7.718E$-3$ 8.541E$-3$ ${\| p- p_{h}^n \|_0}$ 8.554E$-3$ 1.355E$-2$ 8.366E$-2$ 8.208E$-1$
The energy based on the implicit/explicit scheme with $u_0(x, y)\in H^1.$
 $t$ 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 $\tau=0.5$ 2.627 0.516 0.156 0.798 1.248 3.217 1.768 1.587 0.073 0.12 $\tau=0.05$ 3.219 1.523 0.373 0.437 1.37 2.125 1.919 0.934 0.124 0.303 $\tau=0.005$ 1.856 0.704 0.012 0.417 1.546 2.362 2.114 1.03 0.112 0.194 $\tau=0.0005$ 1.857 0.704 0.012 0.417 1.546 2.362 2.114 1.03 0.107 0.194
 $t$ 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 $\tau=0.5$ 2.627 0.516 0.156 0.798 1.248 3.217 1.768 1.587 0.073 0.12 $\tau=0.05$ 3.219 1.523 0.373 0.437 1.37 2.125 1.919 0.934 0.124 0.303 $\tau=0.005$ 1.856 0.704 0.012 0.417 1.546 2.362 2.114 1.03 0.112 0.194 $\tau=0.0005$ 1.857 0.704 0.012 0.417 1.546 2.362 2.114 1.03 0.107 0.194
The errors based on the semi-implicit scheme with $u_0(x, y)\in H^0.$
 $\tau$ 0.0005 0.005 0.05 0.5 ${\| u- u_{h}^n \|_0}$ 2.979E$-3$ 2.979E$-3$ 2.977E$-3$ 3.768E$-3$ ${\| \nabla(u- u_{h}^n)\|_0}$ 3.474E$-2$ 3.474E$-2$ 3.492E$-2$ 3.589E$-2$ ${\| p- p_{h}^n \|_0}$ 2.847E$-2$ 2.204E$-2$ 5.737E$-2$ 8.168E$-1$
 $\tau$ 0.0005 0.005 0.05 0.5 ${\| u- u_{h}^n \|_0}$ 2.979E$-3$ 2.979E$-3$ 2.977E$-3$ 3.768E$-3$ ${\| \nabla(u- u_{h}^n)\|_0}$ 3.474E$-2$ 3.474E$-2$ 3.492E$-2$ 3.589E$-2$ ${\| p- p_{h}^n \|_0}$ 2.847E$-2$ 2.204E$-2$ 5.737E$-2$ 8.168E$-1$
The errors based on the implicit/explicit scheme with $u_0(x, y)\in H^0.$
 $\tau$ 0.0005 0.005 0.05 0.5 ${\| u- u_{h}^n \|_0}$ 2.979E$-3$ 2.979E$-3$ 2.977E$-3$ 3.768E$-3$ ${\| \nabla(u- u_{h}^n)\|_0}$ 3.474E$-2$ 3.474E$-2$ 3.492E$-2$ 3.599E$-2$ ${\| p- p_{h}^n \|_0}$ 2.847E$-2$ 2.204E$-2$ 5.737E$-2$ 8.168E$-1$
 $\tau$ 0.0005 0.005 0.05 0.5 ${\| u- u_{h}^n \|_0}$ 2.979E$-3$ 2.979E$-3$ 2.977E$-3$ 3.768E$-3$ ${\| \nabla(u- u_{h}^n)\|_0}$ 3.474E$-2$ 3.474E$-2$ 3.492E$-2$ 3.599E$-2$ ${\| p- p_{h}^n \|_0}$ 2.847E$-2$ 2.204E$-2$ 5.737E$-2$ 8.168E$-1$
The energy based on the implicit/explicit scheme with $u_0(x, y)\in H^0.$
 $t$ 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 $\tau=0.5$ 30.267 4.347 1.423 9.108 12.374 36.539 18.046 17.898 0.223 0.714 $\tau=0.05$ 20.143 7.701 0.409 5.202 17.862 26.862 23.947 11.731 1.319 2.133 $\tau=0.005$ 20.51 7.75 0.128 4.653 17.166 26.162 23.37 11.358 1.168 2.17 $\tau=0.0005$ 20.512 7.75 0.128 4.653 17.166 26.162 23.37 11.357 1.168 2.17
 $t$ 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 $\tau=0.5$ 30.267 4.347 1.423 9.108 12.374 36.539 18.046 17.898 0.223 0.714 $\tau=0.05$ 20.143 7.701 0.409 5.202 17.862 26.862 23.947 11.731 1.319 2.133 $\tau=0.005$ 20.51 7.75 0.128 4.653 17.166 26.162 23.37 11.358 1.168 2.17 $\tau=0.0005$ 20.512 7.75 0.128 4.653 17.166 26.162 23.37 11.357 1.168 2.17
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