A uniformly dissipative scheme for stationary statistical properties of the infinite Prandtl number model

The purpose of this short communication is to announce that a class of numerical schemes, uniformly dissipative approximations, which uniformly preserve the dissipativity of the continuous infinite dimensional dissipative complex (chaotic) systems possess 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 system at vanishing mesh size. The idea is illustrated on the infinite Prandtl number model for convection and semi-discretization in time, although the general strategy works for a broad class of dissipative complex systems and fully discretized approximations. As 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. (C) 2008 Elsevier Ltd. All rights reserved.