We show that a class of numerical schemes, uniformly dissipative
approximations, which uniformly preserve 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. Our result may be also viewed as a
Lax type result in the sense that consistency and uniform
dissipativity implies the convergence of stationary statistical
properties.
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