Coupling of transport and diffusion models in linear transport theory

Guillaume Bal; Yvon Maday

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 36, Issue: 1, page 69-86
  • ISSN: 0764-583X

Abstract

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This paper is concerned with the coupling of two models for the propagation of particles in scattering media. The first model is a linear transport equation of Boltzmann type posed in the phase space (position and velocity). It accurately describes the physics but is very expensive to solve. The second model is a diffusion equation posed in the physical space. It is only valid in areas of high scattering, weak absorption, and smooth physical coefficients, but its numerical solution is much cheaper than that of transport. We are interested in the case when the domain is diffusive everywhere except in some small areas, for instance non-scattering or oscillatory inclusions. We present a natural coupling of the two models that accounts for both the diffusive and non-diffusive regions. The interface separating the models is chosen so that the diffusive regime holds in its vicinity to avoid the calculation of boundary or interface layers. The coupled problem is analyzed theoretically and numerically. To simplify the presentation, the transport equation is written in the even parity form. Applications include, for instance, the treatment of clear or spatially inhomogeneous regions in near-infra-red spectroscopy, which is increasingly being used in medical imaging for monitoring certain properties of human tissues.

How to cite

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Bal, Guillaume, and Maday, Yvon. "Coupling of transport and diffusion models in linear transport theory." ESAIM: Mathematical Modelling and Numerical Analysis 36.1 (2010): 69-86. <http://eudml.org/doc/194097>.

@article{Bal2010,
abstract = { This paper is concerned with the coupling of two models for the propagation of particles in scattering media. The first model is a linear transport equation of Boltzmann type posed in the phase space (position and velocity). It accurately describes the physics but is very expensive to solve. The second model is a diffusion equation posed in the physical space. It is only valid in areas of high scattering, weak absorption, and smooth physical coefficients, but its numerical solution is much cheaper than that of transport. We are interested in the case when the domain is diffusive everywhere except in some small areas, for instance non-scattering or oscillatory inclusions. We present a natural coupling of the two models that accounts for both the diffusive and non-diffusive regions. The interface separating the models is chosen so that the diffusive regime holds in its vicinity to avoid the calculation of boundary or interface layers. The coupled problem is analyzed theoretically and numerically. To simplify the presentation, the transport equation is written in the even parity form. Applications include, for instance, the treatment of clear or spatially inhomogeneous regions in near-infra-red spectroscopy, which is increasingly being used in medical imaging for monitoring certain properties of human tissues. },
author = {Bal, Guillaume, Maday, Yvon},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Linear transport; even parity formulation; diffusion approximation; domain decomposition; diffuse tomography.; Boltzmann equation; linear transport; diffusion approximation; diffuse tomography; diffusion equation},
language = {eng},
month = {3},
number = {1},
pages = {69-86},
publisher = {EDP Sciences},
title = {Coupling of transport and diffusion models in linear transport theory},
url = {http://eudml.org/doc/194097},
volume = {36},
year = {2010},
}

TY - JOUR
AU - Bal, Guillaume
AU - Maday, Yvon
TI - Coupling of transport and diffusion models in linear transport theory
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 36
IS - 1
SP - 69
EP - 86
AB - This paper is concerned with the coupling of two models for the propagation of particles in scattering media. The first model is a linear transport equation of Boltzmann type posed in the phase space (position and velocity). It accurately describes the physics but is very expensive to solve. The second model is a diffusion equation posed in the physical space. It is only valid in areas of high scattering, weak absorption, and smooth physical coefficients, but its numerical solution is much cheaper than that of transport. We are interested in the case when the domain is diffusive everywhere except in some small areas, for instance non-scattering or oscillatory inclusions. We present a natural coupling of the two models that accounts for both the diffusive and non-diffusive regions. The interface separating the models is chosen so that the diffusive regime holds in its vicinity to avoid the calculation of boundary or interface layers. The coupled problem is analyzed theoretically and numerically. To simplify the presentation, the transport equation is written in the even parity form. Applications include, for instance, the treatment of clear or spatially inhomogeneous regions in near-infra-red spectroscopy, which is increasingly being used in medical imaging for monitoring certain properties of human tissues.
LA - eng
KW - Linear transport; even parity formulation; diffusion approximation; domain decomposition; diffuse tomography.; Boltzmann equation; linear transport; diffusion approximation; diffuse tomography; diffusion equation
UR - http://eudml.org/doc/194097
ER -

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