American Institute of Mathematical Sciences

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August  2016, 36(8): 4599-4618. doi: 10.3934/dcds.2016.36.4599

Numerical algorithms for stationary statistical properties of dissipative dynamical systems

Xiaoming Wang 1,

1. 

Department of Mathematics, The Florida State University, Tallahassee, FL 32306-4510

Received  May 2015 Revised  February 2016 Published  March 2016

It is well-known that physical laws for large chaotic dynamical systems are revealed statistically. The main concern of this manuscript is numerical methods for dissipative chaotic infinite dimensional dynamical systems that are able to capture the stationary statistical properties of the underlying dynamical systems. We first survey results on temporal and spatial approximations that enjoy the desired properties. We then present a new result on fully discretized approximations of infinite dimensional dissipative chaotic dynamical systems that are able to capture asymptotically the stationary statistical properties. The main ingredients in ensuring the convergence of the long time statistical properties of the numerical schemes are: (1) uniform dissipativity of the scheme in the sense that the union of the global attractors of the numerical approximations is pre-compact in the phase space; (2) convergence of the solutions of the numerical scheme to the solution of the continuous system on the unit time interval $[0,1]$ modulo an initial layer, uniformly with respect to initial data from the union of the global attractors. The two conditions are reminiscent of the Lax equivalence theorem where stability and consistency are needed for the convergence of a numerical scheme. Applications to the complex Ginzburg-Landau equation and the two-dimensional Navier-Stokes equations in a periodic box are discussed.
Keywords: invariant measure, finite difference, complex Ginzburg-Landau equation, Navier-Stokes equations., dissipative system, spatial discretization, spectral collocation method, Lax equivalence type conditions, Stationary statistical property, global attractor, uniformly dissipative schemes, time discretization.
Mathematics Subject Classification: Primary: 37M25, 65M99; Secondary: 65J08, 37L4.
Citation: Xiaoming Wang. Numerical algorithms for stationary statistical properties of dissipative dynamical systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (8) : 4599-4618. doi: 10.3934/dcds.2016.36.4599
References:
[1]

P. Billingsley, Weak Convergence of Measures: Applications in Probability,, SIAM, (1971).   Google Scholar

[2]

E. Cancs, E. Legoll and G. Stoltz, Theoretical and numerical comparison of some sampling methods for molecular dynamics,, ESAIM: Mathematical Modelling and Numerical Analysis, 41 (2007), 351.  doi: 10.1051/m2an:2007014.  Google Scholar

[3]

W. Cheng and X. Wang, A uniformly dissipative scheme for stationary statistical properties of the infinite prandtl number model,, Applied Mathematics Letters, 21 (2008), 1281.  doi: 10.1016/j.aml.2007.07.036.  Google Scholar

[4]

W. Cheng and X. Wang, A semi-implicit scheme for stationary statistical properties of the infinite Prandtl number model,, SIAM J. Num. Anal., 47 (2008), 250.  doi: 10.1137/080713501.  Google Scholar

[5]

C. Chiu, Q. Du and T. Y. Li, Error estimates of the Markov finite approximation of the Frobenius-Perron operator,, Nonlinear Anal., 19 (1992), 291.  doi: 10.1016/0362-546X(92)90175-E.  Google Scholar

[6]

A. Chorin, Vorticity and Turbulence,, Springer-Verlag, (1994).  doi: 10.1007/978-1-4419-8728-0.  Google Scholar

[7]

P. Constantin and C. Foias, Navier-Stokes Equations,, The University of Chicago Press, (1988).   Google Scholar

[8]

W. E and D. Li, The Andersen thermostat in molecular dynamics,, Comm. Pure Appl. Math., 61 (2008), 96.  doi: 10.1002/cpa.20198.  Google Scholar

[9]

C. Foias, M. Jolly, I. G. Kevrekidis and E. S. Titi, Dissipativity of numerical schemes,, Nonlinearity, 4 (1991), 591.  doi: 10.1088/0951-7715/4/3/001.  Google Scholar

[10]

C. Foias, M. Jolly, I. G. Kevrekidis and E. S. Titi, On some dissipative fully discrete nonlinear Galerkin schemes for the Kuramoto-Sivashinsky equation,, Phys. Lett. A, 186 (1994), 87.  doi: 10.1016/0375-9601(94)90926-1.  Google Scholar

[11]

C. Foias, O. Manley, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence,, Encyclopedia of Mathematics and its Applications, (2001).  doi: 10.1017/CBO9780511546754.  Google Scholar

[12]

T. Geveci, On the convergence of a time discretization scheme for the Navier-Stokes equations,, Math. Comp., 53 (1989), 43.  doi: 10.1090/S0025-5718-1989-0969488-5.  Google Scholar

[13]

S. Gottlieb, F. Tone, C. Wang, X. Wang and D. Wirosoetisno, Long time stability of a classical efficient scheme for two dimensional Navier-Stokes equations,, SIAM J. Numer. Anal., 50 (2012), 126.  doi: 10.1137/110834901.  Google Scholar

[14]

J. K. Hale, Asymptotic Behavior of Dissipative Systems,, Providence, (1988).   Google Scholar

[15]

J. G. Heywood and R. Rannacher, Finite element approximation of the nonstationary Navier-Stokes problem. II. Stability of solutions and error estimates uniform in time,, SIAM J. Numer. Anal., 23 (1986), 750.  doi: 10.1137/0723049.  Google Scholar

[16]

A. T. Hill and E. Süli, Approximation of the global attractor for the incompressible Navier-Stokes equation,, IMA J. Numer. Anal., 20 (2000), 663.  doi: 10.1093/imanum/20.4.633.  Google Scholar

[17]

D. A. Jones, A. M. Stuart and E. S. Titi, Persistence of invariant sets for dissipative evolution equations,, J. Math. Anal. Appl., 219 (1998), 479.  doi: 10.1006/jmaa.1997.5847.  Google Scholar

[18]

N. Ju, On the global stability of a temporal discretization scheme for the Navier-Stokes equations,, IMA J. Numer. Anal., 22 (2002), 577.  doi: 10.1093/imanum/22.4.577.  Google Scholar

[19]

L. P. Kadanoff, Turbulent heat flow: Structures and scaling,, Physics Today, 54 (2001), 34.  doi: 10.1063/1.1404847.  Google Scholar

[20]

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

[21]

A. Lasota and M. C. Mackey, Chaos, Fractals, and Noise, Stochastic Aspects of Dynamics, 2nd, ed., New York, (1994).  doi: 10.1007/978-1-4612-4286-4.  Google Scholar

[22]

P. D. Lax, Functional Analysis,, New York : Wiley, (2002).   Google Scholar

[23]

P. D. Lax and R. D. Richtmyer, Survey of the stability of linear finite difference equations,, Comm. Pure Appl. Math., 9 (1956), 267.  doi: 10.1002/cpa.3160090206.  Google Scholar

[24]

G. J. Lord, Attractors and inertial manifolds for finite-difference approximation of the complex Ginzburg-Landau equation,, SIAM J. Numer. Anal., 34 (1997), 1483.  doi: 10.1137/S003614299528554X.  Google Scholar

[25]

G. J. Lord and A. M. Stuart, Discrete Gevrey regularity, attractors and upper-semicontinuity for a finite-difference approximation to the Ginzburg-Landau equation,, Numer. Funct. Anal. Optim., 16 (1995), 1003.  doi: 10.1080/01630569508816658.  Google Scholar

[26]

A. J. Majda and A. Bertozzi, Vorticity and Incompressible Flow,, Cambridge University Press, (2002).   Google Scholar

[27]

A. J. Majda and X. Wang, Nonlinear Dynamics and Statistical Theory for Basic Geophysical Flows,, Cambridge University Press, (2006).  doi: 10.1017/CBO9780511616778.  Google Scholar

[28]

A. S. Monin and A. M. Yaglom, Statistical Fluid Mechanics; Mechanics of Turbulence,, English ed. updated, (1975).   Google Scholar

[29]

G. Raugel, Global attractors in partial differential equations,, in Handbook of dynamical systems, 2 (2002), 885.  doi: 10.1016/S1874-575X(02)80038-8.  Google Scholar

[30]

S. Reich, Backward error analysis for numerical integrators,, SIAM J. Numer. Anal., 36 (1999), 1549.  doi: 10.1137/S0036142997329797.  Google Scholar

[31]

J. Shen, Convergence of approximate attractors for a fully discrete system for reaction-diffusion equations,, Numer. Funct. Anal. and Optimiz., 10 (1989), 1213.  doi: 10.1080/01630568908816354.  Google Scholar

[32]

J. Shen, Long time stabilities and convergences for the fully discrete nonlinear Galerkin methods,, Appl. Anal., 38 (1990), 201.  doi: 10.1080/00036819008839963.  Google Scholar

[33]

H. Sigurgeirsson and A. M. Stuart, Statistics from computations,, in Foundations of Computational Mathematics, 284 (2001), 323.   Google Scholar

[34]

A. M. Stuart and A. R. Humphries, Dynamical Systems and Numerical Analysis,, Cambridge University Press, (1996).   Google Scholar

[35]

D. Talay, Simulation of stochastic differential systems,, in Probabilistic Methods in Applied Physics, 451 (1995), 54.  doi: 10.1007/3-540-60214-3_51.  Google Scholar

[36]

R. M. Temam, Sur l'approximation des solutions des équations de Navier-Stokes,, C.R. Acad. Sci., 262 (1966), 219.   Google Scholar

[37]

R. M. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, 2nd edition,, CBMS-SIAM, (1995).  doi: 10.1137/1.9781611970050.  Google Scholar

[38]

R. M. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics,, 2nd ed. Springer-Verlag, (1997).  doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[39]

F. Tone and X. Wang, Approximation of the stationary statistical properties of the dynamical systems generated by the two-dimensional Rayleigh-Benard convection problem,, Analysis and Applications, 9 (2011), 421.  doi: 10.1142/S0219530511001935.  Google Scholar

[40]

F. Tone, X. Wang and D. Wirosoetisno, Long-time dynamics of 2d double-diffusive convection: Analysis and/of numerics,, Numer. Math., 130 (2015), 541.  doi: 10.1007/s00211-014-0670-9.  Google Scholar

[41]

F. Tone and D. Wirosoetisno, On the long-time stability of the implicit Euler scheme for the two-dimensional Navier-Stokes equations,, SIAM J. Num. Anal., 44 (2006), 29.  doi: 10.1137/040618527.  Google Scholar

[42]

P. F. Tupper, Ergodicity and the numerical simulation of Hamiltonian systems,, SIAM J. Applied Dynamical Systems, 4 (2005), 563.  doi: 10.1137/040603802.  Google Scholar

[43]

M. I. Vishik and A. V. Fursikov, Mathematical Problems of Statistical Hydromechanics,, Kluwer Acad. Publishers, (1988).  doi: 10.1007/978-94-009-1423-0.  Google Scholar

[44]

P. Walters, An introduction to ergodic theory,, Springer-Verlag, (1982).   Google Scholar

[45]

X. Wang, Infinite Prandtl number limit of Rayleigh-Bénard convection,, Comm. Pure and Appl. Math., 57 (2004), 1265.  doi: 10.1002/cpa.3047.  Google Scholar

[46]

X. Wang, Stationary statistical properties of Rayleigh-Bénard convection at large Prandtl number,, Comm. Pure and Appl. Math., 61 (2008), 789.  doi: 10.1002/cpa.20214.  Google Scholar

[47]

X. Wang, Upper Semi-Continuity of Stationary Statistical Properties of Dissipative Systems,, Dedicated to Prof. Li Ta-Tsien on the occasion of his 70th birthday, 23 (2009), 521.  doi: 10.3934/dcds.2009.23.521.  Google Scholar

[48]

X. Wang, Approximating stationary statistical properties,, Chinese Ann. Math. Series B, 30 (2009), 831.  doi: 10.1007/s11401-009-0178-2.  Google Scholar

[49]

X. Wang, Approximation of stationary statistical properties of dissipative dynamical systems: Time discretization,, Math. Comp., 79 (2010), 259.  doi: 10.1090/S0025-5718-09-02256-X.  Google Scholar

[50]

X. Wang, An efficient second order in time scheme for approximating long time statistical properties of the two dimensional Navier-Stokes equations,, Numer. Math., 121 (2012), 753.  doi: 10.1007/s00211-012-0450-3.  Google Scholar

[51]

Y. Yan, Attractors and error estimates for discretizations of incompressible Navier-Stokes equations,, SIAM J. Numer. Anal., 33 (1996), 1451.  doi: 10.1137/S0036142993248092.  Google Scholar

show all references

References:
[1]

P. Billingsley, Weak Convergence of Measures: Applications in Probability,, SIAM, (1971).   Google Scholar

[2]

E. Cancs, E. Legoll and G. Stoltz, Theoretical and numerical comparison of some sampling methods for molecular dynamics,, ESAIM: Mathematical Modelling and Numerical Analysis, 41 (2007), 351.  doi: 10.1051/m2an:2007014.  Google Scholar

[3]

W. Cheng and X. Wang, A uniformly dissipative scheme for stationary statistical properties of the infinite prandtl number model,, Applied Mathematics Letters, 21 (2008), 1281.  doi: 10.1016/j.aml.2007.07.036.  Google Scholar

[4]

W. Cheng and X. Wang, A semi-implicit scheme for stationary statistical properties of the infinite Prandtl number model,, SIAM J. Num. Anal., 47 (2008), 250.  doi: 10.1137/080713501.  Google Scholar

[5]

C. Chiu, Q. Du and T. Y. Li, Error estimates of the Markov finite approximation of the Frobenius-Perron operator,, Nonlinear Anal., 19 (1992), 291.  doi: 10.1016/0362-546X(92)90175-E.  Google Scholar

[6]

A. Chorin, Vorticity and Turbulence,, Springer-Verlag, (1994).  doi: 10.1007/978-1-4419-8728-0.  Google Scholar

[7]

P. Constantin and C. Foias, Navier-Stokes Equations,, The University of Chicago Press, (1988).   Google Scholar

[8]

W. E and D. Li, The Andersen thermostat in molecular dynamics,, Comm. Pure Appl. Math., 61 (2008), 96.  doi: 10.1002/cpa.20198.  Google Scholar

[9]

C. Foias, M. Jolly, I. G. Kevrekidis and E. S. Titi, Dissipativity of numerical schemes,, Nonlinearity, 4 (1991), 591.  doi: 10.1088/0951-7715/4/3/001.  Google Scholar

[10]

C. Foias, M. Jolly, I. G. Kevrekidis and E. S. Titi, On some dissipative fully discrete nonlinear Galerkin schemes for the Kuramoto-Sivashinsky equation,, Phys. Lett. A, 186 (1994), 87.  doi: 10.1016/0375-9601(94)90926-1.  Google Scholar

[11]

C. Foias, O. Manley, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence,, Encyclopedia of Mathematics and its Applications, (2001).  doi: 10.1017/CBO9780511546754.  Google Scholar

[12]

T. Geveci, On the convergence of a time discretization scheme for the Navier-Stokes equations,, Math. Comp., 53 (1989), 43.  doi: 10.1090/S0025-5718-1989-0969488-5.  Google Scholar

[13]

S. Gottlieb, F. Tone, C. Wang, X. Wang and D. Wirosoetisno, Long time stability of a classical efficient scheme for two dimensional Navier-Stokes equations,, SIAM J. Numer. Anal., 50 (2012), 126.  doi: 10.1137/110834901.  Google Scholar

[14]

J. K. Hale, Asymptotic Behavior of Dissipative Systems,, Providence, (1988).   Google Scholar

[15]

J. G. Heywood and R. Rannacher, Finite element approximation of the nonstationary Navier-Stokes problem. II. Stability of solutions and error estimates uniform in time,, SIAM J. Numer. Anal., 23 (1986), 750.  doi: 10.1137/0723049.  Google Scholar

[16]

A. T. Hill and E. Süli, Approximation of the global attractor for the incompressible Navier-Stokes equation,, IMA J. Numer. Anal., 20 (2000), 663.  doi: 10.1093/imanum/20.4.633.  Google Scholar

[17]

D. A. Jones, A. M. Stuart and E. S. Titi, Persistence of invariant sets for dissipative evolution equations,, J. Math. Anal. Appl., 219 (1998), 479.  doi: 10.1006/jmaa.1997.5847.  Google Scholar

[18]

N. Ju, On the global stability of a temporal discretization scheme for the Navier-Stokes equations,, IMA J. Numer. Anal., 22 (2002), 577.  doi: 10.1093/imanum/22.4.577.  Google Scholar

[19]

L. P. Kadanoff, Turbulent heat flow: Structures and scaling,, Physics Today, 54 (2001), 34.  doi: 10.1063/1.1404847.  Google Scholar

[20]

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

[21]

A. Lasota and M. C. Mackey, Chaos, Fractals, and Noise, Stochastic Aspects of Dynamics, 2nd, ed., New York, (1994).  doi: 10.1007/978-1-4612-4286-4.  Google Scholar

[22]

P. D. Lax, Functional Analysis,, New York : Wiley, (2002).   Google Scholar

[23]

P. D. Lax and R. D. Richtmyer, Survey of the stability of linear finite difference equations,, Comm. Pure Appl. Math., 9 (1956), 267.  doi: 10.1002/cpa.3160090206.  Google Scholar

[24]

G. J. Lord, Attractors and inertial manifolds for finite-difference approximation of the complex Ginzburg-Landau equation,, SIAM J. Numer. Anal., 34 (1997), 1483.  doi: 10.1137/S003614299528554X.  Google Scholar

[25]

G. J. Lord and A. M. Stuart, Discrete Gevrey regularity, attractors and upper-semicontinuity for a finite-difference approximation to the Ginzburg-Landau equation,, Numer. Funct. Anal. Optim., 16 (1995), 1003.  doi: 10.1080/01630569508816658.  Google Scholar

[26]

A. J. Majda and A. Bertozzi, Vorticity and Incompressible Flow,, Cambridge University Press, (2002).   Google Scholar

[27]

A. J. Majda and X. Wang, Nonlinear Dynamics and Statistical Theory for Basic Geophysical Flows,, Cambridge University Press, (2006).  doi: 10.1017/CBO9780511616778.  Google Scholar

[28]

A. S. Monin and A. M. Yaglom, Statistical Fluid Mechanics; Mechanics of Turbulence,, English ed. updated, (1975).   Google Scholar

[29]

G. Raugel, Global attractors in partial differential equations,, in Handbook of dynamical systems, 2 (2002), 885.  doi: 10.1016/S1874-575X(02)80038-8.  Google Scholar

[30]

S. Reich, Backward error analysis for numerical integrators,, SIAM J. Numer. Anal., 36 (1999), 1549.  doi: 10.1137/S0036142997329797.  Google Scholar

[31]

J. Shen, Convergence of approximate attractors for a fully discrete system for reaction-diffusion equations,, Numer. Funct. Anal. and Optimiz., 10 (1989), 1213.  doi: 10.1080/01630568908816354.  Google Scholar

[32]

J. Shen, Long time stabilities and convergences for the fully discrete nonlinear Galerkin methods,, Appl. Anal., 38 (1990), 201.  doi: 10.1080/00036819008839963.  Google Scholar

[33]

H. Sigurgeirsson and A. M. Stuart, Statistics from computations,, in Foundations of Computational Mathematics, 284 (2001), 323.   Google Scholar

[34]

A. M. Stuart and A. R. Humphries, Dynamical Systems and Numerical Analysis,, Cambridge University Press, (1996).   Google Scholar

[35]

D. Talay, Simulation of stochastic differential systems,, in Probabilistic Methods in Applied Physics, 451 (1995), 54.  doi: 10.1007/3-540-60214-3_51.  Google Scholar

[36]

R. M. Temam, Sur l'approximation des solutions des équations de Navier-Stokes,, C.R. Acad. Sci., 262 (1966), 219.   Google Scholar

[37]

R. M. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, 2nd edition,, CBMS-SIAM, (1995).  doi: 10.1137/1.9781611970050.  Google Scholar

[38]

R. M. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics,, 2nd ed. Springer-Verlag, (1997).  doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[39]

F. Tone and X. Wang, Approximation of the stationary statistical properties of the dynamical systems generated by the two-dimensional Rayleigh-Benard convection problem,, Analysis and Applications, 9 (2011), 421.  doi: 10.1142/S0219530511001935.  Google Scholar

[40]

F. Tone, X. Wang and D. Wirosoetisno, Long-time dynamics of 2d double-diffusive convection: Analysis and/of numerics,, Numer. Math., 130 (2015), 541.  doi: 10.1007/s00211-014-0670-9.  Google Scholar

[41]

F. Tone and D. Wirosoetisno, On the long-time stability of the implicit Euler scheme for the two-dimensional Navier-Stokes equations,, SIAM J. Num. Anal., 44 (2006), 29.  doi: 10.1137/040618527.  Google Scholar

[42]

P. F. Tupper, Ergodicity and the numerical simulation of Hamiltonian systems,, SIAM J. Applied Dynamical Systems, 4 (2005), 563.  doi: 10.1137/040603802.  Google Scholar

[43]

M. I. Vishik and A. V. Fursikov, Mathematical Problems of Statistical Hydromechanics,, Kluwer Acad. Publishers, (1988).  doi: 10.1007/978-94-009-1423-0.  Google Scholar

[44]

P. Walters, An introduction to ergodic theory,, Springer-Verlag, (1982).   Google Scholar

[45]

X. Wang, Infinite Prandtl number limit of Rayleigh-Bénard convection,, Comm. Pure and Appl. Math., 57 (2004), 1265.  doi: 10.1002/cpa.3047.  Google Scholar

[46]

X. Wang, Stationary statistical properties of Rayleigh-Bénard convection at large Prandtl number,, Comm. Pure and Appl. Math., 61 (2008), 789.  doi: 10.1002/cpa.20214.  Google Scholar

[47]

X. Wang, Upper Semi-Continuity of Stationary Statistical Properties of Dissipative Systems,, Dedicated to Prof. Li Ta-Tsien on the occasion of his 70th birthday, 23 (2009), 521.  doi: 10.3934/dcds.2009.23.521.  Google Scholar

[48]

X. Wang, Approximating stationary statistical properties,, Chinese Ann. Math. Series B, 30 (2009), 831.  doi: 10.1007/s11401-009-0178-2.  Google Scholar

[49]

X. Wang, Approximation of stationary statistical properties of dissipative dynamical systems: Time discretization,, Math. Comp., 79 (2010), 259.  doi: 10.1090/S0025-5718-09-02256-X.  Google Scholar

[50]

X. Wang, An efficient second order in time scheme for approximating long time statistical properties of the two dimensional Navier-Stokes equations,, Numer. Math., 121 (2012), 753.  doi: 10.1007/s00211-012-0450-3.  Google Scholar

[51]

Y. Yan, Attractors and error estimates for discretizations of incompressible Navier-Stokes equations,, SIAM J. Numer. Anal., 33 (1996), 1451.  doi: 10.1137/S0036142993248092.  Google Scholar

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Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079

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