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, doi: 10.3934/dcdsb.2018273
References:
[1]

R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.

[2]

A. Ait Ou Ammi and M. Marion, Nonlinear Galerkin methods and mixed finite elements: Two-grid algorithms for the Navier-Stokes equations, Numer. Math., 68 (1994), 189-213. doi: 10.1007/s002110050056.

[3]

D. N. ArnoldF. Brezzi and M. Fortin, A stable finite element for the Stokes equations, Calcolo, 21 (1984), 337-344. doi: 10.1007/BF02576171.

[4]

G. A. BakerV. A. Dougalis and O. A. Karakashian, On a high order accurate fully discrete Galerkin approximation to the Navier-Stokes equations, Math. Comp., 39 (1982), 339-375. doi: 10.1090/S0025-5718-1982-0669634-0.

[5]

J. Bercovier and O. Pironneau, Error estimates for finite element solution of the Stokes problem in the primitive variables, Numer. Math., 33 (1979), 211-224. doi: 10.1007/BF01399555.

[6]

P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.

[7]

J. F. Gerbeau, C. Le Bris and T. Lelièvre, Mathematical Method for the Magnetohydrodynamics of Liquid Metals, Oxford University Press, Oxford, 2006. doi: 10.1093/acprof:oso/9780198566656.001.0001.

[8]

V. Girault and P. A. Raviart, Finite Element Method for Navier-Stokes Equations: Theory and Algorithms, Springer-Verlag, Berlin, Heidelberg, 1986. doi: 10.1007/978-3-642-61623-5.

[9]

Y. N. He and K. T. Li, Nonlinear Galerkin method and two-step method for the Navier-Stokes equations, Numer. Methods for PDEs, 12 (1996), 283-305. doi: 10.1002/(SICI)1098-2426(199605)12:3<283::AID-NUM1>3.0.CO;2-K.

[10]

Y. N. He and K. T. Li, Convergence and stability of finite element nonlinear Galerkin method for the Navier-Stokes equations, Numer. Math., 79 (1998), 77-106. doi: 10.1007/s002110050332.

[11]

Y. N. He, Two-level method based on finite element and Crank-Nicolson extrapolation for the time-dependent Navier-Stokes equations, SIAM J. Numer. Anal., 41 (2003), 1263-1285. doi: 10.1137/S0036142901385659.

[12]

Y. N. HeH. L. MiaoR. M. M. Mattheij and Z. X. Chen, Numerical analysis of a modified finite element nonlinear Galerkin method, Numer. Math., 97 (2004), 725-756. doi: 10.1007/s00211-003-0516-3.

[13]

Y. N. He, Stability and error analysis for a spectral Galerkin method for the Navier-Stokes equations with H2 or H1 initial data, Numer. Methods for PDEs, 21 (2005), 875-904. doi: 10.1002/num.20065.

[14]

Y. N. He, Optimal error estimate of the penalty finite element method for the time-dependent Navier-Stokes problem, Math. Comp., 74 (2005), 1201-1216. doi: 10.1090/S0025-5718-05-01751-5.

[15]

Y. N. He and K. M. Liu, A multi-level finite element method for the time-dependent Navier-Stokes equations, Numer. Methods for PDEs, 21 (2005), 1052-1068.

[16]

Y. N. He and W. W. Sun, Stability and convegence of the Crank-Nicolson/Adams-Bashforth scheme for the time-dependent Navier-Stokes equations, SIAM J. Numer. Anal., 45 (2007), 837-869. doi: 10.1137/050639910.

[17]

Y. N. He, Stability and error analysis for a spectral Galerkin method for the Navier-Stokes equations with with L2 initial data, Numer. Methods for PDEs, 24 (2008), 79-103. doi: 10.1002/num.20234.

[18]

Y. N. He, Euler implicit/explicit scheme for the 2D time-dependent Navier-Stokes equations with smooth or non-smooth initial data, Math. Comp., 77 (2008), 2097-2124. doi: 10.1090/S0025-5718-08-02127-3.

[19]

Y. N. He, Stability and convegence of the Crank-Nicolson/Adams-Bashforth scheme for the time-dependent Navier-Stokes equations with non-smooth initial data, Numer. Methods for PDEs, 28 (2012), 155-187. doi: 10.1002/num.20613.

[20]

Y. N. HeP. Z. Huang and X. L. Feng, H2-stability of the first order finite element fully discrete schemes for the 2D time-dependent Navier-Stokes equations with smooth and non-smooth initial data, J. Sci. Comput., 62 (2015), 230-264. doi: 10.1007/s10915-014-9854-9.

[21]

J. G. Heywood and R. Rannacher, Finite-element approximations of the nonstationary Navier-Stokes problem. Part Ⅰ: Regularity of solutions and second-order spatial discretization, SIAM J. Numer. Anal., 19 (1982), 275-311. doi: 10.1137/0719018.

[22]

J. G. Heywood and R. Rannacher, Finite-element approximations of the nonstationary Navier-Stokes problem. Part Ⅳ: Error estimates for second-order time discretization, SIAM J. Numer. Anal., 27 (1990), 353-384. doi: 10.1137/0727022.

[23]

A. T. Hill and E. Süli, Approximation of the global attractor for the incompressible Navier-Stokes equations, IMA J. Numer. Anal., 20 (2000), 633-667. doi: 10.1093/imanum/20.4.633.

[24]

H. Johnston and J. G. Liu, Accurate, stable and efficient Navier-Stokes slovers based on explicit treatment of the pressure term, J. Comput. Phys., 199 (2004), 221-259. doi: 10.1016/j.jcp.2004.02.009.

[25]

J. Kim and P. Moin, Application of a fractional-step method to incompressible Navier-Stokes equations, J. Comput. Phys., 59 (1985), 308-323. doi: 10.1016/0021-9991(85)90148-2.

[26]

R. B. Kellogg and J. E. Osborn, A regularity result for the Stokes problem in a convex polygon, J. Functional Anal., 21 (1976), 397-431. doi: 10.1016/0022-1236(76)90035-5.

[27]

S. Larsson, The long-time behavior of finite-element approximations of solutions to semilinear parabolic problems, SIAM J. Numer. Anal., 26 (1989), 348-365. doi: 10.1137/0726019.

[28]

M. Marion and R. Temam, Navier-Stokes equations: Theory and approximation, in: Handbook of Numerical Analysis, Vol. Ⅵ, pp. 503–688, North-Holland, Amsterdam, 1998.

[29]

J. Shen, Long time stability and convergence for fully discrete nonlinear Galerkin methods, Appl. Anal., 38 (1990), 201-229. doi: 10.1080/00036819008839963.

[30]

J. C. Simo and F. Armero, Unconditional stability and long-term behavior of transient algorithms for the incompressible Navier-Stokes and Euler equations, Comput. Methods Appl. Mech. Engrg., 111 (1994), 111-154. doi: 10.1016/0045-7825(94)90042-6.

[31]

R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, 3rd ed., North-Holland, Amsterdam, 1984.

[32]

F. Tone, Error analysis for a second scheme for the Navier-Stokes equations, Appl. Numer. Math., 50 (2004), 93-119. doi: 10.1016/j.apnum.2003.12.003.

show all references

References:
[1]

R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.

[2]

A. Ait Ou Ammi and M. Marion, Nonlinear Galerkin methods and mixed finite elements: Two-grid algorithms for the Navier-Stokes equations, Numer. Math., 68 (1994), 189-213. doi: 10.1007/s002110050056.

[3]

D. N. ArnoldF. Brezzi and M. Fortin, A stable finite element for the Stokes equations, Calcolo, 21 (1984), 337-344. doi: 10.1007/BF02576171.

[4]

G. A. BakerV. A. Dougalis and O. A. Karakashian, On a high order accurate fully discrete Galerkin approximation to the Navier-Stokes equations, Math. Comp., 39 (1982), 339-375. doi: 10.1090/S0025-5718-1982-0669634-0.

[5]

J. Bercovier and O. Pironneau, Error estimates for finite element solution of the Stokes problem in the primitive variables, Numer. Math., 33 (1979), 211-224. doi: 10.1007/BF01399555.

[6]

P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.

[7]

J. F. Gerbeau, C. Le Bris and T. Lelièvre, Mathematical Method for the Magnetohydrodynamics of Liquid Metals, Oxford University Press, Oxford, 2006. doi: 10.1093/acprof:oso/9780198566656.001.0001.

[8]

V. Girault and P. A. Raviart, Finite Element Method for Navier-Stokes Equations: Theory and Algorithms, Springer-Verlag, Berlin, Heidelberg, 1986. doi: 10.1007/978-3-642-61623-5.

[9]

Y. N. He and K. T. Li, Nonlinear Galerkin method and two-step method for the Navier-Stokes equations, Numer. Methods for PDEs, 12 (1996), 283-305. doi: 10.1002/(SICI)1098-2426(199605)12:3<283::AID-NUM1>3.0.CO;2-K.

[10]

Y. N. He and K. T. Li, Convergence and stability of finite element nonlinear Galerkin method for the Navier-Stokes equations, Numer. Math., 79 (1998), 77-106. doi: 10.1007/s002110050332.

[11]

Y. N. He, Two-level method based on finite element and Crank-Nicolson extrapolation for the time-dependent Navier-Stokes equations, SIAM J. Numer. Anal., 41 (2003), 1263-1285. doi: 10.1137/S0036142901385659.

[12]

Y. N. HeH. L. MiaoR. M. M. Mattheij and Z. X. Chen, Numerical analysis of a modified finite element nonlinear Galerkin method, Numer. Math., 97 (2004), 725-756. doi: 10.1007/s00211-003-0516-3.

[13]

Y. N. He, Stability and error analysis for a spectral Galerkin method for the Navier-Stokes equations with H2 or H1 initial data, Numer. Methods for PDEs, 21 (2005), 875-904. doi: 10.1002/num.20065.

[14]

Y. N. He, Optimal error estimate of the penalty finite element method for the time-dependent Navier-Stokes problem, Math. Comp., 74 (2005), 1201-1216. doi: 10.1090/S0025-5718-05-01751-5.

[15]

Y. N. He and K. M. Liu, A multi-level finite element method for the time-dependent Navier-Stokes equations, Numer. Methods for PDEs, 21 (2005), 1052-1068.

[16]

Y. N. He and W. W. Sun, Stability and convegence of the Crank-Nicolson/Adams-Bashforth scheme for the time-dependent Navier-Stokes equations, SIAM J. Numer. Anal., 45 (2007), 837-869. doi: 10.1137/050639910.

[17]

Y. N. He, Stability and error analysis for a spectral Galerkin method for the Navier-Stokes equations with with L2 initial data, Numer. Methods for PDEs, 24 (2008), 79-103. doi: 10.1002/num.20234.

[18]

Y. N. He, Euler implicit/explicit scheme for the 2D time-dependent Navier-Stokes equations with smooth or non-smooth initial data, Math. Comp., 77 (2008), 2097-2124. doi: 10.1090/S0025-5718-08-02127-3.

[19]

Y. N. He, Stability and convegence of the Crank-Nicolson/Adams-Bashforth scheme for the time-dependent Navier-Stokes equations with non-smooth initial data, Numer. Methods for PDEs, 28 (2012), 155-187. doi: 10.1002/num.20613.

[20]

Y. N. HeP. Z. Huang and X. L. Feng, H2-stability of the first order finite element fully discrete schemes for the 2D time-dependent Navier-Stokes equations with smooth and non-smooth initial data, J. Sci. Comput., 62 (2015), 230-264. doi: 10.1007/s10915-014-9854-9.

[21]

J. G. Heywood and R. Rannacher, Finite-element approximations of the nonstationary Navier-Stokes problem. Part Ⅰ: Regularity of solutions and second-order spatial discretization, SIAM J. Numer. Anal., 19 (1982), 275-311. doi: 10.1137/0719018.

[22]

J. G. Heywood and R. Rannacher, Finite-element approximations of the nonstationary Navier-Stokes problem. Part Ⅳ: Error estimates for second-order time discretization, SIAM J. Numer. Anal., 27 (1990), 353-384. doi: 10.1137/0727022.

[23]

A. T. Hill and E. Süli, Approximation of the global attractor for the incompressible Navier-Stokes equations, IMA J. Numer. Anal., 20 (2000), 633-667. doi: 10.1093/imanum/20.4.633.

[24]

H. Johnston and J. G. Liu, Accurate, stable and efficient Navier-Stokes slovers based on explicit treatment of the pressure term, J. Comput. Phys., 199 (2004), 221-259. doi: 10.1016/j.jcp.2004.02.009.

[25]

J. Kim and P. Moin, Application of a fractional-step method to incompressible Navier-Stokes equations, J. Comput. Phys., 59 (1985), 308-323. doi: 10.1016/0021-9991(85)90148-2.

[26]

R. B. Kellogg and J. E. Osborn, A regularity result for the Stokes problem in a convex polygon, J. Functional Anal., 21 (1976), 397-431. doi: 10.1016/0022-1236(76)90035-5.

[27]

S. Larsson, The long-time behavior of finite-element approximations of solutions to semilinear parabolic problems, SIAM J. Numer. Anal., 26 (1989), 348-365. doi: 10.1137/0726019.

[28]

M. Marion and R. Temam, Navier-Stokes equations: Theory and approximation, in: Handbook of Numerical Analysis, Vol. Ⅵ, pp. 503–688, North-Holland, Amsterdam, 1998.

[29]

J. Shen, Long time stability and convergence for fully discrete nonlinear Galerkin methods, Appl. Anal., 38 (1990), 201-229. doi: 10.1080/00036819008839963.

[30]

J. C. Simo and F. Armero, Unconditional stability and long-term behavior of transient algorithms for the incompressible Navier-Stokes and Euler equations, Comput. Methods Appl. Mech. Engrg., 111 (1994), 111-154. doi: 10.1016/0045-7825(94)90042-6.

[31]

R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, 3rd ed., North-Holland, Amsterdam, 1984.

[32]

F. Tone, Error analysis for a second scheme for the Navier-Stokes equations, Appl. Numer. Math., 50 (2004), 93-119. doi: 10.1016/j.apnum.2003.12.003.

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$-1$
$\tau$ 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$-1$
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$-1$
$\tau$ 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$-1$
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
$t$ 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
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$-1$
$\tau$ 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$-1$
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$-1$
$\tau$ 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$-1$
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
$t$ 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
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$-1$
$\tau$ 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$-1$
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$-1$
$\tau$ 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$-1$
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
$t$ 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
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