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Approximation of stationary statistical properties of dissipative dynamical systems: Time discretization

Author: Xiaoming Wang
Journal: Math. Comp. 79 (2010), 259-280
MSC (2000): Primary 65P99, 37M25, 65M12, 37L40, 76F35, 76F20, 37L30, 37N10, 35Q35
Published electronically: April 20, 2009
MathSciNet review: 2552226
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Abstract: We consider temporal approximation of stationary statistical properties of dissipative infinite-dimensional dynamical systems. We demonstrate that stationary statistical properties of the time discrete approximations, i.e., numerical scheme, converge to those of the underlying continuous dissipative infinite-dimensional dynamical system under three very natural assumptions as the time step approaches zero. The three conditions that are sufficient for the convergence of the stationary statistical properties are: (1) uniform dissipativity of the scheme in the sense that the union of the global attractors for 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]$ uniformly with respect to initial data from the union of the global attractors; (3) uniform continuity of the solutions to the continuous dynamical system on the unit time interval $ [0,1]$ uniformly for initial data from the union of the global attractors. The convergence of the global attractors is established under weaker assumptions. An application to the infinite Prandtl number model for convection is discussed.

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Additional Information

Xiaoming Wang
Affiliation: Department of Mathematics, Florida State University, Tallahassee, Florida 32306 and School of Mathematics, Fudan University, Shanghai, China 200433

Keywords: Stationary statistical property, invariant measure, global attractor, dissipative system, time discretization, uniformly dissipative scheme, infinite Prandtl number model for convection, Nusselt number
Received by editor(s): September 12, 2008
Received by editor(s) in revised form: December 23, 2008
Published electronically: April 20, 2009
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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