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Publication 13

Numerical Studies of Homogenization Under a Fast Cellular Flow


CMUGautam Iyer
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213

OxfordKonstantinos C. Zygalakis
Mathematical Institute, University of Oxford
24-29 St Giles', OX13LB

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 problem homogenizes, and 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 $A \to \infty$ and the number of cells $L^2 \to \infty$ was recently studied [Iyer, Komorowski, Novikov, Ryzhik; 2011], 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 [IKNR; 2011] using the numerical method devised in [Pavliotis, Stuart, Zygalakis; 2009]. Our treatment of the numerical method uses recent developments in the theory of modified equations for numerical integrators of SDEs [Zygalakis; 2011].
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