Numerical Studies of Homogenization under a Fast Cellular Flow

We consider a two dimensional particle diffusing in the presence of a fast cellular flow confined to a finite domain. If the flow amplitude $A$ is held fixed and the number of cells $L^2\rightarrow\infty$, then the problem homogenizes; this has been well studied. Also well studied is the limit when $L$ is fixed and $A\rightarrow\infty$. In this case the solution averages along stream lines. The double limit as both the flow amplitude $A\rightarrow\infty$ and the number of cells $L^2\rightarrow\infty$ was recently studied [G. Iyer et al., preprint, arXiv:1108.0074]; one observes a sharp transition between the homogenization and averaging regimes occurring at $A\approx L^4$. This paper numerically studies a few theoretically unresolved aspects of this problem when both $A$ and $L$ are large that were left open in [G. Iyer et al., preprint, arXiv:1108.0074] using the numerical method devised in [G. A. Pavliotis, A. M. Stewart, and K. C. Zygalakis, J. Comput. Phys., 228 (2009), pp. 1030--1055]. Our treatment of the numerical method uses recent developments in the theory of modified equations for numerical integrators of stochastic differential equations [K. C. Zygalakis, SIAM J. Sci. Comput., 33 (2001), pp. 102--130].