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

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.