Abstract
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.
Highlights
The long-time dynamics of many infinite-dimensional dynamical systems are very complex with abundant chaotic/turbulent behaviors
Invariant measure for a discrete dynamical system generated by a map Sdiscrete on a Banach space H is defined in a similar fashion with the continuous time t replaced by discrete time n = 0, 1, 2
As a by-product, we present an abstract result on criteria that ensure the convergence of the global attractors of the numerical schemes
Summary
The long-time dynamics of many infinite-dimensional dynamical systems are very complex with abundant chaotic/turbulent behaviors. Invariant measure (stationary statistical solution) for a discrete dynamical system generated by a map Sdiscrete on a Banach space H is defined in a similar fashion with the continuous time t replaced by discrete time n = 0, 1, 2,. One may show via the so-called Bogliubov-Krylov argument that these generalized long time averages over trajectory lead to invariant measures (may depend on the chosen Banach limit and initial datum u) of the system for appropriate dissipative dynamical systems, and the spatial and temporal averages are equivalent We will illustrate below that the key ingredient in algorithms that are able to capture the long-time statistical properties is the uniform dissipativity and the uniform convergence on the unit time interval (modulo an initial layer) for initial data coming out of a compact subset of the phase space.
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