Numerical Studies of Homogenization under a Fast Cellular Flow

Iyer, Gautam; Zygalakis, Konstantinos C.

Numerical Studies of Homogenization under a Fast Cellular Flow

Type

Article

Authors

Iyer, Gautam
Zygalakis, Konstantinos C.

KAUST Grant Number

KUK-C1-013-04

Online Publication Date

2012-09-13

Print Publication Date

2012-01

Date

2012-09-13

Abstract

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 →∞, then the problem homogenizes; this has been well studied. Also well studied is the limit when L is fixed and A→∞. In this case the solution averages along stream lines. The double limit as both the flow amplitude A→∞and the number of cells L 2 →∞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 = L 2. 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]. © 2012 Society for Industrial and Applied Mathematics.

Citation

Iyer G, Zygalakis KC (2012) Numerical Studies of Homogenization under a Fast Cellular Flow. Multiscale Model Simul 10: 1046–1058. Available: http://dx.doi.org/10.1137/120861308.

Acknowledgements

This work was partially supported by the Center for Nonlinear Analysis (NSF DMS-0405343 and DMS-0635983) and NSF PIRE grant OISE 0967140.This author's research was partially supported by NSF-DMS 1007914.This author's research was partially supported by award KUK-C1-013-04 of the King Abdullah University of Science and Technology (KAUST). It was partially carried out at Carnegie Mellon University, whose hospitality is gratefully acknowledged.