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

We consider temporal approximation of stationary statistical properties of dissipative complex dynamical systems. We demonstrate that stationary statistical properties of the time discrete approximations (numerical scheme) converge to those of the underlying continuous dissipative complex 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 of pre-compactness of the union of the global attractors for the numerical approximations); (2) uniform (with respect to initial data from the union of the global attractors) convergence of the solutions of the numerical scheme to the solution of the continuous system on the unit time interval [0, 1]; and (3) the uniform (with respect to initial data from the union of the global attractors) continuity of the solutions to the continuous dynamical system on the unit time interval [0, 1]. The convergence of the global attractors is established under weaker assumptions. Application to the infinite Prandtl number model for convection is discussed. keywords: stationary statistical property, invariant measure, global attractor, dissipative system, time discretization, uniformly dissipative scheme, infinite Prandtl number model for convection, Nusselt number