Numerical algorithms for stationary statistical properties of dissipative dynamical systems

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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

Abstract
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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 and Continuous Dynamical Systems, 2016, 36 (8) : 4599-4618. doi: 10.3934/dcds.2016.36.4599

References:

[1]	P. Billingsley, Weak Convergence of Measures: Applications in Probability, SIAM, Philadelphia, 1971.
[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), 351-389. doi: 10.1051/m2an:2007014.
[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), 1281-1285. doi: 10.1016/j.aml.2007.07.036.
[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), 250-270. doi: 10.1137/080713501.
[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), 291-308. doi: 10.1016/0362-546X(92)90175-E.
[6]	A. Chorin, Vorticity and Turbulence, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4419-8728-0.
[7]	P. Constantin and C. Foias, Navier-Stokes Equations, The University of Chicago Press, Chicago, 1988.
[8]	W. E and D. Li, The Andersen thermostat in molecular dynamics, Comm. Pure Appl. Math., 61 (2008), 96-136. doi: 10.1002/cpa.20198.
[9]	C. Foias, M. Jolly, I. G. Kevrekidis and E. S. Titi, Dissipativity of numerical schemes, Nonlinearity, 4 (1991), 591-613. doi: 10.1088/0951-7715/4/3/001.
[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), 87-96. doi: 10.1016/0375-9601(94)90926-1.
[11]	C. Foias, O. Manley, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence, Encyclopedia of Mathematics and its Applications, 83. Cambridge University Press, Cambridge, 2001. doi: 10.1017/CBO9780511546754.
[12]	T. Geveci, On the convergence of a time discretization scheme for the Navier-Stokes equations, Math. Comp., 53 (1989), 43-53. doi: 10.1090/S0025-5718-1989-0969488-5.
[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), 126-150. doi: 10.1137/110834901.
[14]	J. K. Hale, Asymptotic Behavior of Dissipative Systems, Providence, R.I. : American Mathematical Society, 1988.
[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), 750-777. doi: 10.1137/0723049.
[16]	A. T. Hill and E. Süli, Approximation of the global attractor for the incompressible Navier-Stokes equation, IMA J. Numer. Anal., 20 (2000), 663-667. doi: 10.1093/imanum/20.4.633.
[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), 479-502. doi: 10.1006/jmaa.1997.5847.
[18]	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.
[19]	L. P. Kadanoff, Turbulent heat flow: Structures and scaling, Physics Today, 54 (2001), 34-39. doi: 10.1063/1.1404847.
[20]	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.
[21]	A. Lasota and M. C. Mackey, Chaos, Fractals, and Noise, Stochastic Aspects of Dynamics, 2nd, ed. New York, Springer-Verlag, 1994. doi: 10.1007/978-1-4612-4286-4.
[22]	P. D. Lax, Functional Analysis, New York : Wiley, 2002.
[23]	P. D. Lax and R. D. Richtmyer, Survey of the stability of linear finite difference equations, Comm. Pure Appl. Math., 9 (1956), 267-293. doi: 10.1002/cpa.3160090206.
[24]	G. J. Lord, Attractors and inertial manifolds for finite-difference approximation of the complex Ginzburg-Landau equation, SIAM J. Numer. Anal., 34 (1997), 1483-1512. doi: 10.1137/S003614299528554X.
[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), 1003-1047. doi: 10.1080/01630569508816658.
[26]	A. J. Majda and A. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, Cambridge, England, 2002.
[27]	A. J. Majda and X. Wang, Nonlinear Dynamics and Statistical Theory for Basic Geophysical Flows, Cambridge University Press, Cambridge, England, 2006. doi: 10.1017/CBO9780511616778.
[28]	A. S. Monin and A. M. Yaglom, Statistical Fluid Mechanics; Mechanics of Turbulence, English ed. updated, augmented and rev. by the authors. MIT Press, Cambridge, Mass., 1975.
[29]	G. Raugel, Global attractors in partial differential equations, in Handbook of dynamical systems, North-Holland, Amsterdam, 2 (2002), 885-982. doi: 10.1016/S1874-575X(02)80038-8.
[30]	S. Reich, Backward error analysis for numerical integrators, SIAM J. Numer. Anal., 36 (1999), 1549-1570. doi: 10.1137/S0036142997329797.
[31]	J. Shen, Convergence of approximate attractors for a fully discrete system for reaction-diffusion equations, Numer. Funct. Anal. and Optimiz., 10 (1989), 1213-1234. doi: 10.1080/01630568908816354.
[32]	J. Shen, Long time stabilities and convergences for the fully discrete nonlinear Galerkin methods, Appl. Anal., 38 (1990), 201-229. doi: 10.1080/00036819008839963.
[33]	H. Sigurgeirsson and A. M. Stuart, Statistics from computations, in Foundations of Computational Mathematics, Edited by R. Devore, A. Iserles and E. Suli, Cambridge University Press, 284 (2001), 323-344.
[34]	A. M. Stuart and A. R. Humphries, Dynamical Systems and Numerical Analysis, Cambridge University Press, 1996.
[35]	D. Talay, Simulation of stochastic differential systems, in Probabilistic Methods in Applied Physics, P. Kree and W. Wedig (Eds), Lecture Notes in Physics, Springer-Verlag, 451 (1995), 54-96. doi: 10.1007/3-540-60214-3_51.
[36]	R. M. Temam, Sur l'approximation des solutions des équations de Navier-Stokes, C.R. Acad. Sci., Paris, Serie A, 262 (1966), 219-221.
[37]	R. M. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, 2nd edition, CBMS-SIAM, SIAM, 1995. doi: 10.1137/1.9781611970050.
[38]	R. M. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2nd ed. Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.
[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), 421-446. doi: 10.1142/S0219530511001935.
[40]	F. Tone, X. Wang and D. Wirosoetisno, Long-time dynamics of 2d double-diffusive convection: Analysis and/of numerics, Numer. Math., 130 (2015), 541-566. doi: 10.1007/s00211-014-0670-9.
[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), 29-40. doi: 10.1137/040618527.
[42]	P. F. Tupper, Ergodicity and the numerical simulation of Hamiltonian systems, SIAM J. Applied Dynamical Systems, 4 (2005), 563-587. doi: 10.1137/040603802.
[43]	M. I. Vishik and A. V. Fursikov, Mathematical Problems of Statistical Hydromechanics, Kluwer Acad. Publishers, Dordrecht/Boston/London, 1988. doi: 10.1007/978-94-009-1423-0.
[44]	P. Walters, An introduction to ergodic theory, Springer-Verlag, New York, 1982.
[45]	X. Wang, Infinite Prandtl number limit of Rayleigh-Bénard convection, Comm. Pure and Appl. Math., 57 (2004), 1265-1282. doi: 10.1002/cpa.3047.
[46]	X. Wang, Stationary statistical properties of Rayleigh-Bénard convection at large Prandtl number, Comm. Pure and Appl. Math., 61 (2008), 789-815. doi: 10.1002/cpa.20214.
[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, Discrete and Continuous Dynamical Systems, A, 23 (2009), 521-540. doi: 10.3934/dcds.2009.23.521.
[48]	X. Wang, Approximating stationary statistical properties, Chinese Ann. Math. Series B, (an invited article in a special issue dedicated to Andy Majda), 30 (2009), 831-844. doi: 10.1007/s11401-009-0178-2.
[49]	X. 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.
[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), 753-779. doi: 10.1007/s00211-012-0450-3.
[51]	Y. Yan, Attractors and error estimates for discretizations of incompressible Navier-Stokes equations, SIAM J. Numer. Anal., 33 (1996), 1451-1472. doi: 10.1137/S0036142993248092.

show all references

References:

[1]	P. Billingsley, Weak Convergence of Measures: Applications in Probability, SIAM, Philadelphia, 1971.
[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), 351-389. doi: 10.1051/m2an:2007014.
[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), 1281-1285. doi: 10.1016/j.aml.2007.07.036.
[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), 250-270. doi: 10.1137/080713501.
[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), 291-308. doi: 10.1016/0362-546X(92)90175-E.
[6]	A. Chorin, Vorticity and Turbulence, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4419-8728-0.
[7]	P. Constantin and C. Foias, Navier-Stokes Equations, The University of Chicago Press, Chicago, 1988.
[8]	W. E and D. Li, The Andersen thermostat in molecular dynamics, Comm. Pure Appl. Math., 61 (2008), 96-136. doi: 10.1002/cpa.20198.
[9]	C. Foias, M. Jolly, I. G. Kevrekidis and E. S. Titi, Dissipativity of numerical schemes, Nonlinearity, 4 (1991), 591-613. doi: 10.1088/0951-7715/4/3/001.
[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), 87-96. doi: 10.1016/0375-9601(94)90926-1.
[11]	C. Foias, O. Manley, R. Rosa and R. Temam, Navier-Stokes Equations and Turbulence, Encyclopedia of Mathematics and its Applications, 83. Cambridge University Press, Cambridge, 2001. doi: 10.1017/CBO9780511546754.
[12]	T. Geveci, On the convergence of a time discretization scheme for the Navier-Stokes equations, Math. Comp., 53 (1989), 43-53. doi: 10.1090/S0025-5718-1989-0969488-5.
[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), 126-150. doi: 10.1137/110834901.
[14]	J. K. Hale, Asymptotic Behavior of Dissipative Systems, Providence, R.I. : American Mathematical Society, 1988.
[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), 750-777. doi: 10.1137/0723049.
[16]	A. T. Hill and E. Süli, Approximation of the global attractor for the incompressible Navier-Stokes equation, IMA J. Numer. Anal., 20 (2000), 663-667. doi: 10.1093/imanum/20.4.633.
[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), 479-502. doi: 10.1006/jmaa.1997.5847.
[18]	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.
[19]	L. P. Kadanoff, Turbulent heat flow: Structures and scaling, Physics Today, 54 (2001), 34-39. doi: 10.1063/1.1404847.
[20]	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.
[21]	A. Lasota and M. C. Mackey, Chaos, Fractals, and Noise, Stochastic Aspects of Dynamics, 2nd, ed. New York, Springer-Verlag, 1994. doi: 10.1007/978-1-4612-4286-4.
[22]	P. D. Lax, Functional Analysis, New York : Wiley, 2002.
[23]	P. D. Lax and R. D. Richtmyer, Survey of the stability of linear finite difference equations, Comm. Pure Appl. Math., 9 (1956), 267-293. doi: 10.1002/cpa.3160090206.
[24]	G. J. Lord, Attractors and inertial manifolds for finite-difference approximation of the complex Ginzburg-Landau equation, SIAM J. Numer. Anal., 34 (1997), 1483-1512. doi: 10.1137/S003614299528554X.
[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), 1003-1047. doi: 10.1080/01630569508816658.
[26]	A. J. Majda and A. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, Cambridge, England, 2002.
[27]	A. J. Majda and X. Wang, Nonlinear Dynamics and Statistical Theory for Basic Geophysical Flows, Cambridge University Press, Cambridge, England, 2006. doi: 10.1017/CBO9780511616778.
[28]	A. S. Monin and A. M. Yaglom, Statistical Fluid Mechanics; Mechanics of Turbulence, English ed. updated, augmented and rev. by the authors. MIT Press, Cambridge, Mass., 1975.
[29]	G. Raugel, Global attractors in partial differential equations, in Handbook of dynamical systems, North-Holland, Amsterdam, 2 (2002), 885-982. doi: 10.1016/S1874-575X(02)80038-8.
[30]	S. Reich, Backward error analysis for numerical integrators, SIAM J. Numer. Anal., 36 (1999), 1549-1570. doi: 10.1137/S0036142997329797.
[31]	J. Shen, Convergence of approximate attractors for a fully discrete system for reaction-diffusion equations, Numer. Funct. Anal. and Optimiz., 10 (1989), 1213-1234. doi: 10.1080/01630568908816354.
[32]	J. Shen, Long time stabilities and convergences for the fully discrete nonlinear Galerkin methods, Appl. Anal., 38 (1990), 201-229. doi: 10.1080/00036819008839963.
[33]	H. Sigurgeirsson and A. M. Stuart, Statistics from computations, in Foundations of Computational Mathematics, Edited by R. Devore, A. Iserles and E. Suli, Cambridge University Press, 284 (2001), 323-344.
[34]	A. M. Stuart and A. R. Humphries, Dynamical Systems and Numerical Analysis, Cambridge University Press, 1996.
[35]	D. Talay, Simulation of stochastic differential systems, in Probabilistic Methods in Applied Physics, P. Kree and W. Wedig (Eds), Lecture Notes in Physics, Springer-Verlag, 451 (1995), 54-96. doi: 10.1007/3-540-60214-3_51.
[36]	R. M. Temam, Sur l'approximation des solutions des équations de Navier-Stokes, C.R. Acad. Sci., Paris, Serie A, 262 (1966), 219-221.
[37]	R. M. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, 2nd edition, CBMS-SIAM, SIAM, 1995. doi: 10.1137/1.9781611970050.
[38]	R. M. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2nd ed. Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.
[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), 421-446. doi: 10.1142/S0219530511001935.
[40]	F. Tone, X. Wang and D. Wirosoetisno, Long-time dynamics of 2d double-diffusive convection: Analysis and/of numerics, Numer. Math., 130 (2015), 541-566. doi: 10.1007/s00211-014-0670-9.
[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), 29-40. doi: 10.1137/040618527.
[42]	P. F. Tupper, Ergodicity and the numerical simulation of Hamiltonian systems, SIAM J. Applied Dynamical Systems, 4 (2005), 563-587. doi: 10.1137/040603802.
[43]	M. I. Vishik and A. V. Fursikov, Mathematical Problems of Statistical Hydromechanics, Kluwer Acad. Publishers, Dordrecht/Boston/London, 1988. doi: 10.1007/978-94-009-1423-0.
[44]	P. Walters, An introduction to ergodic theory, Springer-Verlag, New York, 1982.
[45]	X. Wang, Infinite Prandtl number limit of Rayleigh-Bénard convection, Comm. Pure and Appl. Math., 57 (2004), 1265-1282. doi: 10.1002/cpa.3047.
[46]	X. Wang, Stationary statistical properties of Rayleigh-Bénard convection at large Prandtl number, Comm. Pure and Appl. Math., 61 (2008), 789-815. doi: 10.1002/cpa.20214.
[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, Discrete and Continuous Dynamical Systems, A, 23 (2009), 521-540. doi: 10.3934/dcds.2009.23.521.
[48]	X. Wang, Approximating stationary statistical properties, Chinese Ann. Math. Series B, (an invited article in a special issue dedicated to Andy Majda), 30 (2009), 831-844. doi: 10.1007/s11401-009-0178-2.
[49]	X. 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.
[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), 753-779. doi: 10.1007/s00211-012-0450-3.
[51]	Y. Yan, Attractors and error estimates for discretizations of incompressible Navier-Stokes equations, SIAM J. Numer. Anal., 33 (1996), 1451-1472. doi: 10.1137/S0036142993248092.

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