Diffusion and homogenization limits with separate scales

Naoufel Ben Abdallah; Marjolaine Puel; Michael S. Vogelius

doi:10.1137/110828964

Diffusion and homogenization limits with separate scales

Naoufel Ben Abdallah, Marjolaine Puel, Michael S. Vogelius

School of Arts and Sciences, Mathematics

Research output: Contribution to journal › Article › peer-review

4 Scopus citations

Abstract

We consider the simultaneous diffusion and homogenization limit of the linear Boltzmann equation in a periodic medium in the regime where the mean free path is much smaller than the lattice constant. The resulting equation is a diffusion equation, with an averaged diffusion matrix that is formally obtained by first performing the diffusion limit and then the homogenization one. The rigorous proof relies on the use of two-scale limits, in combination with an asymptotic expansion of the equilibrium profile in powers of the ratio between the mean free path and the lattice constant.

Original language	English (US)
Pages (from-to)	1148-1179
Number of pages	32
Journal	Multiscale Modeling and Simulation
Volume	10
Issue number	4
DOIs	https://doi.org/10.1137/110828964
State	Published - 2012

All Science Journal Classification (ASJC) codes

General Chemistry
Modeling and Simulation
Ecological Modeling
General Physics and Astronomy
Computer Science Applications

Keywords

Diffusion approximation
Kinetic equations
Two-scale limits

Access to Document

10.1137/110828964

Cite this

@article{d13bffcef40042acbe19282efaf375d3,

title = "Diffusion and homogenization limits with separate scales",

abstract = "We consider the simultaneous diffusion and homogenization limit of the linear Boltzmann equation in a periodic medium in the regime where the mean free path is much smaller than the lattice constant. The resulting equation is a diffusion equation, with an averaged diffusion matrix that is formally obtained by first performing the diffusion limit and then the homogenization one. The rigorous proof relies on the use of two-scale limits, in combination with an asymptotic expansion of the equilibrium profile in powers of the ratio between the mean free path and the lattice constant.",

keywords = "Diffusion approximation, Kinetic equations, Two-scale limits",

author = "Abdallah, {Naoufel Ben} and Marjolaine Puel and Vogelius, {Michael S.}",

note = "Funding Information: Acknowledgements. The authors thank Danielle Lizotte and Claude Villeneuve for their technical assistance. This study was supported by a grant from The Kidney Foundation of Canada to RL. YD and MD are recipients of studentships from the Natural Sciences and Engineering Research Council of Canada.",

year = "2012",

doi = "10.1137/110828964",

language = "English (US)",

volume = "10",

pages = "1148--1179",

journal = "Multiscale Modeling and Simulation",

issn = "1540-3459",

publisher = "Society for Industrial and Applied Mathematics Publications",

number = "4",

}

TY - JOUR

T1 - Diffusion and homogenization limits with separate scales

AU - Abdallah, Naoufel Ben

AU - Puel, Marjolaine

AU - Vogelius, Michael S.

N1 - Funding Information: Acknowledgements. The authors thank Danielle Lizotte and Claude Villeneuve for their technical assistance. This study was supported by a grant from The Kidney Foundation of Canada to RL. YD and MD are recipients of studentships from the Natural Sciences and Engineering Research Council of Canada.

PY - 2012

Y1 - 2012

N2 - We consider the simultaneous diffusion and homogenization limit of the linear Boltzmann equation in a periodic medium in the regime where the mean free path is much smaller than the lattice constant. The resulting equation is a diffusion equation, with an averaged diffusion matrix that is formally obtained by first performing the diffusion limit and then the homogenization one. The rigorous proof relies on the use of two-scale limits, in combination with an asymptotic expansion of the equilibrium profile in powers of the ratio between the mean free path and the lattice constant.

AB - We consider the simultaneous diffusion and homogenization limit of the linear Boltzmann equation in a periodic medium in the regime where the mean free path is much smaller than the lattice constant. The resulting equation is a diffusion equation, with an averaged diffusion matrix that is formally obtained by first performing the diffusion limit and then the homogenization one. The rigorous proof relies on the use of two-scale limits, in combination with an asymptotic expansion of the equilibrium profile in powers of the ratio between the mean free path and the lattice constant.

KW - Diffusion approximation

KW - Kinetic equations

KW - Two-scale limits

UR - http://www.scopus.com/inward/record.url?scp=84871831594&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84871831594&partnerID=8YFLogxK

U2 - 10.1137/110828964

DO - 10.1137/110828964

M3 - Article

AN - SCOPUS:84871831594

SN - 1540-3459

VL - 10

SP - 1148

EP - 1179

JO - Multiscale Modeling and Simulation

JF - Multiscale Modeling and Simulation

IS - 4

ER -

Diffusion and homogenization limits with separate scales

Abstract

All Science Journal Classification (ASJC) codes

Keywords

Access to Document

Other files and links

Fingerprint

Cite this