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

  • * Corresponding author: Pengzhan Huang

    * Corresponding author: Pengzhan Huang 

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)

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  • 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.

    Mathematics Subject Classification: 35Q30, 65N30.

    Citation:

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  • Figure 1.  The energy computed by semi-implicit scheme (left) and explicit scheme (right) based on several different time steps chosen with $u_0(x, y)\in H^2.$

    Figure 2.  The energy computed by semi-implicit scheme (left) and explicit scheme (right) based on several different time steps chosen with $u_0(x, y)\in H^1.$

    Figure 3.  The energy computed by semi-implicit scheme (left) and explicit scheme (right) based on several different time steps chosen with $u_0(x, y)\in H^0.$

    Table 1.  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$-1$
     | Show Table
    DownLoad: CSV

    Table 2.  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$-1$
     | Show Table
    DownLoad: CSV

    Table 3.  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 1.740 0.082 0.140
    $ \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 0.120 0.218
    $ \tau=0.005$ 2.086 0.791 0.014 0.469 1.739 2.655 2.376 1.157 0.120 0.218
     | Show Table
    DownLoad: CSV

    Table 4.  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$-1$
     | Show Table
    DownLoad: CSV

    Table 5.  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$-1$
     | Show Table
    DownLoad: CSV

    Table 6.  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 0.120
    $ \tau=0.05 $ 3.219 1.523 0.373 0.437 1.370 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 1.030 0.112 0.194
    $ \tau=0.0005$ 1.857 0.704 0.012 0.417 1.546 2.362 2.114 1.030 0.107 0.194
     | Show Table
    DownLoad: CSV

    Table 7.  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$-1$
     | Show Table
    DownLoad: CSV

    Table 8.  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$-1$
     | Show Table
    DownLoad: CSV

    Table 9.  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$ 20.510 7.750 0.128 4.653 17.166 26.162 23.370 11.358 1.168 2.170
    $ \tau=0.0005$ 20.512 7.750 0.128 4.653 17.166 26.162 23.370 11.357 1.168 2.170
     | Show Table
    DownLoad: CSV
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