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__Numerical studies of homogenization under a fast cellular flow __

K. Zygalakis

**Abstract**

We discuss the problem of a particle diffusing in the presence of a fast two dimensional cellular flow in a finite domain. If the flow amplitude $A$ is held fixed and the number of cells $L^2$ goes to infinity, then the problem homogenizes, while if $L$ is held fixed and $A$ goes to infinity the solution averages along stream lines. More interesting is the case of the double limit when both $L$ and $A$ go to infinity, and this is the limit in which we focus here. In particular, in the first part of the talk we review some well-known results related to homogenization for Stochastic Differential Equations (SDEs) and discuss different approaches for calculating the effective diffusion matrix. We then describe the construction of numerical integrators used in our numerical investigations and finish the talk by presenting some numerical results for the double limit, focusing on the sharp transition between the homogenization and the averaging regimes when $A=L^4$.