Abstract:

We show that suitable temporal discretization of continuous in time dissipative chaotic dynamical systems are able to capture the stationary statistical properties asymptotically in the sense that the invariant measures of the discrete dynamical systems generated by the numerical scheme converge to those of the underlying continuous dynamical system at vanishing time-step. The main ingredients are the uniform dissipativity and uniform convergence on the unit interval of the numerical scheme. The methodology is applied to the infinite Prandtl number model for convection. Various numerical schemes that are able to capture stationary statistical properties of the infinite Prandtl number model will be presented and compared. Spatial approximation will be discussed as well.