Numerical Studies of Homogenization under a Fast Cellular Flow

Gautam Iyer; Konstantinos C. Zygalakis

doi:10.1137/120861308

Numerical Studies of Homogenization under a Fast Cellular Flow

Gautam Iyer, Konstantinos C. Zygalakis

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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.

Original language	English (US)
Pages (from-to)	1046-1058
Number of pages	13
Journal	Multiscale Modeling & Simulation
Volume	10
Issue number	3
DOIs	https://doi.org/10.1137/120861308
State	Published - Sep 13 2012
Externally published	Yes

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): KUK-C1-013-04
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.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.

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10.1137/120861308

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@article{48403abe339347a7ac588f31e1dccb08,

title = "Numerical Studies of Homogenization under a Fast Cellular Flow",

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]. {\textcopyright} 2012 Society for Industrial and Applied Mathematics.",

author = "Gautam Iyer and Zygalakis, {Konstantinos C.}",

note = "KAUST Repository Item: Exported on 2020-10-01 Acknowledged KAUST grant number(s): KUK-C1-013-04 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. This publication acknowledges KAUST support, but has no KAUST affiliated authors.",

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AU - Zygalakis, Konstantinos C.

N1 - KAUST Repository Item: Exported on 2020-10-01 Acknowledged KAUST grant number(s): KUK-C1-013-04 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. This publication acknowledges KAUST support, but has no KAUST affiliated authors.

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N2 - 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.

AB - 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.

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DO - 10.1137/120861308

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EP - 1058

JO - Multiscale Modeling & Simulation

JF - Multiscale Modeling & Simulation

IS - 3

ER -

Numerical Studies of Homogenization under a Fast Cellular Flow

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