Calculating Effective Diffusivities in the Limit of Vanishing Molecular Diffusion

In this paper we study the problem of the numerical calculation (by Monte Carlo Methods) of the effective diffusivity for a particle moving in a periodic divergent-free velocity filed, in the limit of vanishing molecular diffusion. In this limit traditional numerical methods typically fail, since they do not represent accurately the geometry of the underlying deterministic dynamics. We propose a stochastic splitting method that takes into account the volume preserving property of the equations motion in the absence of noise, and when inertial effects can be neglected. An extension of the method is then proposed for the cases where the noise has a non trivial time-correlation structure and when inertial effects cannot be neglected. Modified equations are used to perform backward error analysis. The new stochastic geometric integrators are shown to outperform standard Euler-based integrators. Various asymptotic limits of physical interest are investigated by means of numerical experiments, using the new integrators.