ESAIM: Mathematical Modelling and Numerical Analysis

Research Article

Coupling of transport and diffusion models in linear transport theory

Bal, Guillaumea1 and Maday, Yvona2

a1 Department of Applied Physics & Applied Mathematics, Columbia University, New York, NY 10027, USA. gb2030@columbia.edu.

a2 Laboratoire d'Analyse Numérique, Université Paris VI, boîte courrier 187, 75252 Paris Cedex 5, France. maday@ann.jussieu.fr.

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.

(Received August 5 2001)

(Revised November 6 2001)

(Online publication April 15 2002)

Key Words:

  • Linear transport;
  • even parity formulation;
  • diffusion approximation;
  • domain decomposition;
  • diffuse tomography.

Mathematics Subject Classification:

  • 45K05;
  • 92C55;
  • 65N55
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