Uniformly dissipative approximations of stationary statistical properties of infinite dimensional dissipative complex systems
Wenfang (Wendy) Cheng, Xiaoming Wang
We show that a class of numerical schemes, uniformly dissipative approximations, which uniformly preserves the dissipativity of the continuous infinite dimensional dissipative complex (chaotic) systems possess highly desirable properties in terms of approximating stationary statistics properties. In particular, the stationary statistical properties of these uniformly dissipative schemes converge to those of the continuous in time dynamical system at vanishing mesh size. The idea is illustrated on the infinite Prandtl number model for convection and semi-discrete in time discretization although the general strategy works for a broad class of dissipative complex systems and fully discretized approximations. So far as we know, this is the first result on rigorous validation of numerical schemes for approximating stationary statistical properties of general infinite dimensional dissipative complex systems.