Skip to main content Accessibility help
Hostname: page-component-684899dbb8-rbzxz Total loading time: 0.223 Render date: 2022-05-29T03:23:19.461Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "useNewApi": true }

Coupling of transport and diffusion models in linear transport theory

Published online by Cambridge University Press:  15 April 2002

Guillaume Bal
Department of Applied Physics & Applied Mathematics, Columbia University, New York, NY 10027, USA.
Yvon Maday
Laboratoire d'Analyse Numérique, Université Paris VI, boîte courrier 187, 75252 Paris Cedex 5, France.
Get access


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.

Research Article
© EDP Sciences, SMAI, 2002

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)


M.L. Adams and E.W. Larsen, Fast iterative methods for deterministic particle transport computations. Preprint (2001).
Alcouffe, R.E., Diffusion synthetic acceleration methods for the diamond-differenced discrete-ordinates equations. Nucl. Sci. Eng. 64 (1977) 344. CrossRef
Allaire, G. and Bal, G., Homogenization of the criticality spectral equation in neutron transport. ESAIM: M2AN 33 (1999) 721-746. CrossRef
Arridge, S.R., Dehghani, H., Schweiger, M., and Okada, E., The finite element model for the propagation of light in scattering media: A direct method for domains with nonscattering regions. Med. Phys. 27 (2000) 252-264. CrossRef
G. Bal, Couplage d'équations et homogénéisation en transport neutronique. Thèse de Doctorat de l'Université Paris 6 (1997). In French.
Bal, G., First-order corrector for the homogenization of the criticality eigenvalue problem in the even parity formulation of the neutron transport. SIAM J. Math. Anal. 30 (1999) 1208-1240. CrossRef
G. Bal,Spatially varying discrete ordinates methods in XY-geometry. M 3 AS (Math. Models Methods Appl. Sci.) 10 (2000) 1277-1303.
G. Bal, Transport through diffusive and non-diffusive regions, embedded objects, and clear layers. To appear in SIAM J. Appl. Math.
Bal, G., Freilikher, V., Papanicolaou, G., and Ryzhik, L., Wave transport along surfaces with random impedance. Phys. Rev. B 6 (2000) 6228-6240. CrossRef
Bal, G. and Ryzhik, L., Diffusion approximation of radiative transfer problems with interfaces. SIAM J. Appl. Math. 60 (2000) 1887-1912. CrossRef
A. Bensoussan, J.-L. Lions, and G. Papanicolaou, Boundary layers and homogenization of transport processes. Res. Inst. Math. Sci. Kyoto Univ. 15 (1979)53-157.
J.-F. Bourgat, P. Le Tallec, B. Perthame, and Y. Qiu, Coupling Boltzmann and Euler equations without overlapping, in Domain Decomposition Methods in Science and Engineering, The Sixth International Conference on Domain Decomposition, Como, Italy, June 15-19, 1992, Contemp. Math. 157, American Mathematical Society, Providence, RI (1994) 377-398.
Cessenat, M., Théorèmes de trace L p pour des espaces de fonctions de la neutronique. C. R. Acad. Sci. Paris Sér. I Math. 299 (1984) 831-834.
S. Chandrasekhar, Radiative Transfer. Dover Publications, New York (1960).
P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland Publishing Company, Amsterdam, New York, Oxford (1978).
R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology 6. Springer-Verlag, Berlin (1993).
J.J. Duderstadt and W.R. Martin, Transport Theory. Wiley-Interscience, New York (1979).
Firbank, M., Arridge, S.A., Schweiger, M., and Delpy, D.T., An investigation of light transport through scattering bodies with non-scattering regions. Phys. Med. Biol. 41 (1996) 767-783. CrossRef
F. Gastaldi, A. Quarteroni, and G. Sacchi Landriani, On the coupling of two-dimensional hyperbolic and elliptic equations: analytical and numerical approach, in Domain Decomposition Methods for Partial Differential Equations, Third International Symposium on Domain Decomposition Methods for Partial Differential Equations, Houston, TX, 1989, SIAM, Philadelphia, PA (1990) 22-63.
Golse, F., Lions, P.-L., Perthame, B., and Sentis, R., Regularity of the moments of the solution of a transport equation. J. Funct. Anal. 76 (1988) 110-125. CrossRef
Hielscher, A.H., Alcouffe, R.E., and Barbour, R.L., Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues. Phys. Med. Biol. 43 (1998) 1285-1302. CrossRef
A. Ishimaru, Wave Propagation and Scattering in Random Media. Academics, New York (1978).
B. Lapeyre, E. Pardoux and R. Sentis, Méthodes de Monte-Carlo pour les équations de transport et de diffusion, in Mathématiques & Applications 29, Springer-Verlag, Berlin (1998).
Levermore, C.D., Morokoff, W.J. and Nadiga, B.T., Moment realizability and the validity of the Navier-Stokes equations for rarefied gas dynamics. Phys. Fluids 10 (1998) 3214-3226. CrossRef
E.E. Lewis and W.F. Miller Jr., Computational Methods of Neutron Transport. John Wiley & Sons, New York (1984).
Marini, L.D. and Quarteroni, A., Relaxation, A procedure for domain decomposition methods using finite elements. Numer. Math. 55 (1989) 575-598. CrossRef
J. Planchard, Méthodes mathématiques en neutronique, in Collection de la Direction des Études et Recherches d'EDF, Eyrolles (1995). In French.
Ryzhik, L., Papanicolaou, G., and Keller, J.B., Transport equations for elastic and other waves in random media. Wave Motion 24 (1996) 327-370. CrossRef
H. Sato and M.C. Fehler, Seismic Wave Propagation and Scattering in the Heterogeneous Earth, in AIP Series in Modern Acoustics and Signal Processing, AIP Press, Springer, New York (1998).
Tidriri, M., Asymptotic analysis of a coupled system of kinetic equations. C. R. Acad. Sci. Paris Sér. I 328 (1999) 637-642. CrossRef
Tiwari, S., Application of moment realizability criteria for the coupling of the Boltzmann and Euler equations. Transport Theory Statist. Phys. 29 (2000) 759-783. CrossRef

Save article to Kindle

To save this article to your Kindle, first ensure is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the or variations. ‘’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Coupling of transport and diffusion models in linear transport theory
Available formats

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

Coupling of transport and diffusion models in linear transport theory
Available formats

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

Coupling of transport and diffusion models in linear transport theory
Available formats

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *