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Global attraction in a system of delay differential equations via compact and convex sets
Approximation of stationary statistical properties of the three dimensional autonomous planetary geostrophic equations of large-scale ocean circulation
1. | School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, China |
2. | Department of Mathematics, Nanjing University, Nanjing 210093, China |
The main objective of this paper is to study the semi-implicit semi-discrete scheme in time of the three dimensional autonomous planetary geostrophic equations of large-scale ocean circulation. We prove the global attractor and stationary statistical properties of the semi-implicit semi-discrete scheme in time of the three dimensional autonomous planetary geostrophic equations of large-scale ocean circulation converge to those of the three dimensional autonomous planetary geostrophic equations of large-scale ocean circulation as the time step goes to zero.
References:
[1] |
C. S. Cao and E. S. Titi,
Global well-posedness and finite-dimensional global attractor for a 3-D planetary geostrophic viscous model, Comm. Pure Appl. Math., 56 (2003), 198-233.
doi: 10.1002/cpa.10056. |
[2] |
W. Cheng and X. M. Wang,
A semi-implicit scheme for stationary statistical properties of the infinite Prandtl number model, SIAM J. Numer. Anal., 47 (2008), 250-270.
doi: 10.1137/080713501. |
[3] |
W. Cheng and X. M. Wang,
A uniformly dissipative scheme for stationary statistical properties of the infinite Prandtl number model, Appl. Math. Lett., 21 (2008), 1281-1285.
doi: 10.1016/j.aml.2007.07.036. |
[4] |
C. M. Elliott and A. M. Stuart,
The global dynamics of discrete semilinear parabolic equations, SIAM J. Numer. Anal., 30 (1993), 1622-1663.
doi: 10.1137/0730084. |
[5] |
C. Foias, M. Jolly, I. Kevrekidis and E. Titi,
Dissipativity of numerical schemes, Nonlinearity, 4 (1991), 591-613.
doi: 10.1088/0951-7715/4/3/001. |
[6] |
C. Foias, M. Jolly, I. Kevrekidis and E. Titi,
On some dissipative fully discrete nonlinear Galerkin schemes for the Kuramoto-Sivashinsky equation, Phys. Lett. A, 186 (1994), 87-96.
doi: 10.1016/0375-9601(94)90926-1. |
[7] |
C. Foias, O. Manley, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence, Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2001.
doi: 10.1017/CBO9780511546754.![]() ![]() ![]() |
[8] |
A. T. Hill and E. Suli,
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. |
[9] |
N. Ju,
On the global stability of a temporal discretization scheme for the Navier-Stokes equations, IMA J. Numer. Anal., 22 (2002), 577-597.
doi: 10.1093/imanum/22.4.577. |
[10] |
L. P. Kadanoff,
Turbulent heat flow: Structures and scaling, Phys. Today, 54 (2001), 34-39.
|
[11] |
A. Lasota and M. C. Mackey, Chaos, Fractals and Noise, Stochastic Aspects of Dynamics, Springer-Verlag, New York, 1994.
doi: 10.1007/978-1-4612-4286-4. |
[12] |
A. J. Majda and X. M. Wang, Nonlinear Dynamics and Statistical Theories for Basic Geophysical Flows, Cambridge University Press, Cambridge, 2006.
doi: 10.1017/CBO9780511616778.![]() ![]() ![]() |
[13] |
A. S. Monin and A. M. Yaglom, Statistical Fluid Mechanics: Mechanics of Turbulence. Vol. II., Dover Publications, Inc., Mineola, NY, 2007. |
[14] |
J. Pedlosky,
The equations for geostrophic motion in the ocean, J. Phys. Oceanogr., 14 (1984), 448-455.
|
[15] |
J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag, New York, 1987. |
[16] |
N. A. Phillips,
Geostrophic motion, Reviews of Geophysics, 1 (1963), 123-176.
|
[17] |
A. Robinson and H. Stommel,
The oceanic thermocline and associated thermohaline circulation, Tellus, 11 (1959), 295-308.
|
[18] |
R. M. Samelson, R. Temam and S. Wang,
Some mathematical properties of the planetary geostrophic equations for large-scale ocean circulation, Appl. Anal., 70 (1998), 147-173.
doi: 10.1080/00036819808840682. |
[19] |
R. M. Samelson, R. Temam and S. Wang,
Remarks on the planetary geostrophic model of gyre scale ocean circulation, Differential Integral Equations, 13 (2000), 1-14.
|
[20] |
R. M. Samelson and G. K. Vallis,
A simple friction and diffusion scheme for planetary geostrophic basin models, J. Phys. Oceanogr., 27 (1997), 186-194.
|
[21] |
J. Shen,
Convergence of approximate attractors for a fully discrete system for reaction-diffusion equations, Numer. Funct. Anal. Optim., 10 (1989), 1213-1234.
doi: 10.1080/01630568908816354. |
[22] |
J. Shen,
Long time stabilities and convergences for the fully discrete nonlinear Galerkin methods, Appl. Anal., 38 (1990), 201-229.
doi: 10.1080/00036819008839963. |
[23] |
A. M. Stuart and A. R. Humphries, Dynamical Systems and Numerical Analysis, Cambridge University Press, Cambridge, 1996.
![]() ![]() |
[24] |
R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997.
doi: 10.1007/978-1-4612-0645-3. |
[25] |
F. Tone,
On the long-time $H^2$-stability of the implicit Euler scheme for the 2D magnetohydrodynamics equations, J. Sci. Comput., 38 (2009), 331-348.
doi: 10.1007/s10915-008-9236-2. |
[26] |
F. Tone and X. M. Wang,
Approximation of the stationary statistical properties of the dynamical system generated by the two dimensional Rayleigh-Bénard convection problem, Anal. Appl., 9 (2011), 421-446.
doi: 10.1142/S0219530511001935. |
[27] |
F. Tone and D. Wirosoetisno,
On the long-time stability of the implicit Euler scheme for the 2D Navier-Stokes equations, SIAM J. Numer. Anal., 44 (2006), 29-40.
doi: 10.1137/040618527. |
[28] |
M. I. Vishik and A. V. Fursikov, Mathematical Problems of Statistical Hydromechanics, Kluwer Academic Publishers Group, Dordrecht, 1988.
doi: 10.1007/978-94-009-1423-0. |
[29] |
X. M. Wang,
Approximating stationary statistical properties, Chin. Ann. Math. Ser. B, 30 (2009), 831-844.
doi: 10.1007/s11401-009-0178-2. |
[30] |
X. M. Wang,
Approximation of stationary statistical properties of dissipative dynamical systems: Time discretization, Math. Comp., 79 (2010), 259-280.
doi: 10.1090/S0025-5718-09-02256-X. |
[31] |
X. M. Wang,
Numerical algorithms for stationary statistical properties of dissipative dynamical systems, Discrete Contin. Dyn. Syst., 36 (2016), 4599-4618.
doi: 10.3934/dcds.2016.36.4599. |
[32] |
P. Welander,
An advective model of the ocean thermocline, Numerical Algorithms, 11 (1959), 309-318.
|
[33] |
B. You,
Random attractors for the three dimensional stochastical planetary geostrophic equations of large-scale ocean circulation, Stochastics, 89 (2017), 766-785.
doi: 10.1080/17442508.2016.1276913. |
[34] |
B. You and F. Li,
The existence of a pullback attractor for the three dimensional non-autonomous planetary geostrophic viscous equations of large-scale ocean circulation, Nonlinear Anal., 112 (2015), 118-128.
doi: 10.1016/j.na.2014.08.018. |
[35] |
B. You and F. Li,
Random attractor for the three-dimensional planetary geostrophic equations of large-scale ocean circulation with small multiplicative noise, Stoch. Anal. Appl., 34 (2016), 278-292.
doi: 10.1080/07362994.2015.1126184. |
[36] |
B. You, C. K. Zhong and F. Li,
Pullback attractors for three dimensional non-autonomous planetary geostrophic viscous equations of large-scale ocean circulation, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1213-1226.
doi: 10.3934/dcdsb.2014.19.1213. |
show all references
References:
[1] |
C. S. Cao and E. S. Titi,
Global well-posedness and finite-dimensional global attractor for a 3-D planetary geostrophic viscous model, Comm. Pure Appl. Math., 56 (2003), 198-233.
doi: 10.1002/cpa.10056. |
[2] |
W. Cheng and X. M. Wang,
A semi-implicit scheme for stationary statistical properties of the infinite Prandtl number model, SIAM J. Numer. Anal., 47 (2008), 250-270.
doi: 10.1137/080713501. |
[3] |
W. Cheng and X. M. Wang,
A uniformly dissipative scheme for stationary statistical properties of the infinite Prandtl number model, Appl. Math. Lett., 21 (2008), 1281-1285.
doi: 10.1016/j.aml.2007.07.036. |
[4] |
C. M. Elliott and A. M. Stuart,
The global dynamics of discrete semilinear parabolic equations, SIAM J. Numer. Anal., 30 (1993), 1622-1663.
doi: 10.1137/0730084. |
[5] |
C. Foias, M. Jolly, I. Kevrekidis and E. Titi,
Dissipativity of numerical schemes, Nonlinearity, 4 (1991), 591-613.
doi: 10.1088/0951-7715/4/3/001. |
[6] |
C. Foias, M. Jolly, I. Kevrekidis and E. Titi,
On some dissipative fully discrete nonlinear Galerkin schemes for the Kuramoto-Sivashinsky equation, Phys. Lett. A, 186 (1994), 87-96.
doi: 10.1016/0375-9601(94)90926-1. |
[7] |
C. Foias, O. Manley, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence, Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2001.
doi: 10.1017/CBO9780511546754.![]() ![]() ![]() |
[8] |
A. T. Hill and E. Suli,
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. |
[9] |
N. Ju,
On the global stability of a temporal discretization scheme for the Navier-Stokes equations, IMA J. Numer. Anal., 22 (2002), 577-597.
doi: 10.1093/imanum/22.4.577. |
[10] |
L. P. Kadanoff,
Turbulent heat flow: Structures and scaling, Phys. Today, 54 (2001), 34-39.
|
[11] |
A. Lasota and M. C. Mackey, Chaos, Fractals and Noise, Stochastic Aspects of Dynamics, Springer-Verlag, New York, 1994.
doi: 10.1007/978-1-4612-4286-4. |
[12] |
A. J. Majda and X. M. Wang, Nonlinear Dynamics and Statistical Theories for Basic Geophysical Flows, Cambridge University Press, Cambridge, 2006.
doi: 10.1017/CBO9780511616778.![]() ![]() ![]() |
[13] |
A. S. Monin and A. M. Yaglom, Statistical Fluid Mechanics: Mechanics of Turbulence. Vol. II., Dover Publications, Inc., Mineola, NY, 2007. |
[14] |
J. Pedlosky,
The equations for geostrophic motion in the ocean, J. Phys. Oceanogr., 14 (1984), 448-455.
|
[15] |
J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag, New York, 1987. |
[16] |
N. A. Phillips,
Geostrophic motion, Reviews of Geophysics, 1 (1963), 123-176.
|
[17] |
A. Robinson and H. Stommel,
The oceanic thermocline and associated thermohaline circulation, Tellus, 11 (1959), 295-308.
|
[18] |
R. M. Samelson, R. Temam and S. Wang,
Some mathematical properties of the planetary geostrophic equations for large-scale ocean circulation, Appl. Anal., 70 (1998), 147-173.
doi: 10.1080/00036819808840682. |
[19] |
R. M. Samelson, R. Temam and S. Wang,
Remarks on the planetary geostrophic model of gyre scale ocean circulation, Differential Integral Equations, 13 (2000), 1-14.
|
[20] |
R. M. Samelson and G. K. Vallis,
A simple friction and diffusion scheme for planetary geostrophic basin models, J. Phys. Oceanogr., 27 (1997), 186-194.
|
[21] |
J. Shen,
Convergence of approximate attractors for a fully discrete system for reaction-diffusion equations, Numer. Funct. Anal. Optim., 10 (1989), 1213-1234.
doi: 10.1080/01630568908816354. |
[22] |
J. Shen,
Long time stabilities and convergences for the fully discrete nonlinear Galerkin methods, Appl. Anal., 38 (1990), 201-229.
doi: 10.1080/00036819008839963. |
[23] |
A. M. Stuart and A. R. Humphries, Dynamical Systems and Numerical Analysis, Cambridge University Press, Cambridge, 1996.
![]() ![]() |
[24] |
R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997.
doi: 10.1007/978-1-4612-0645-3. |
[25] |
F. Tone,
On the long-time $H^2$-stability of the implicit Euler scheme for the 2D magnetohydrodynamics equations, J. Sci. Comput., 38 (2009), 331-348.
doi: 10.1007/s10915-008-9236-2. |
[26] |
F. Tone and X. M. Wang,
Approximation of the stationary statistical properties of the dynamical system generated by the two dimensional Rayleigh-Bénard convection problem, Anal. Appl., 9 (2011), 421-446.
doi: 10.1142/S0219530511001935. |
[27] |
F. Tone and D. Wirosoetisno,
On the long-time stability of the implicit Euler scheme for the 2D Navier-Stokes equations, SIAM J. Numer. Anal., 44 (2006), 29-40.
doi: 10.1137/040618527. |
[28] |
M. I. Vishik and A. V. Fursikov, Mathematical Problems of Statistical Hydromechanics, Kluwer Academic Publishers Group, Dordrecht, 1988.
doi: 10.1007/978-94-009-1423-0. |
[29] |
X. M. Wang,
Approximating stationary statistical properties, Chin. Ann. Math. Ser. B, 30 (2009), 831-844.
doi: 10.1007/s11401-009-0178-2. |
[30] |
X. M. Wang,
Approximation of stationary statistical properties of dissipative dynamical systems: Time discretization, Math. Comp., 79 (2010), 259-280.
doi: 10.1090/S0025-5718-09-02256-X. |
[31] |
X. M. Wang,
Numerical algorithms for stationary statistical properties of dissipative dynamical systems, Discrete Contin. Dyn. Syst., 36 (2016), 4599-4618.
doi: 10.3934/dcds.2016.36.4599. |
[32] |
P. Welander,
An advective model of the ocean thermocline, Numerical Algorithms, 11 (1959), 309-318.
|
[33] |
B. You,
Random attractors for the three dimensional stochastical planetary geostrophic equations of large-scale ocean circulation, Stochastics, 89 (2017), 766-785.
doi: 10.1080/17442508.2016.1276913. |
[34] |
B. You and F. Li,
The existence of a pullback attractor for the three dimensional non-autonomous planetary geostrophic viscous equations of large-scale ocean circulation, Nonlinear Anal., 112 (2015), 118-128.
doi: 10.1016/j.na.2014.08.018. |
[35] |
B. You and F. Li,
Random attractor for the three-dimensional planetary geostrophic equations of large-scale ocean circulation with small multiplicative noise, Stoch. Anal. Appl., 34 (2016), 278-292.
doi: 10.1080/07362994.2015.1126184. |
[36] |
B. You, C. K. Zhong and F. Li,
Pullback attractors for three dimensional non-autonomous planetary geostrophic viscous equations of large-scale ocean circulation, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1213-1226.
doi: 10.3934/dcdsb.2014.19.1213. |
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