Coupling of transport and diffusion models in linear transport theory
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 36 (2002) no. 1, pp. 69-86.

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.

Classification : 45K05,  92C55,  65N55
Mots clés : linear transport, even parity formulation, diffusion approximation, domain decomposition, diffuse tomography
     author = {Bal, Guillaume and Maday, Yvon},
     title = {Coupling of transport and diffusion models in linear transport theory},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     pages = {69--86},
     publisher = {EDP-Sciences},
     volume = {36},
     number = {1},
     year = {2002},
     doi = {10.1051/m2an:2002007},
     zbl = {0995.45008},
     mrnumber = {1916293},
     language = {en},
     url = {}
AU  - Bal, Guillaume
AU  - Maday, Yvon
TI  - Coupling of transport and diffusion models in linear transport theory
JO  - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY  - 2002
DA  - 2002///
SP  - 69
EP  - 86
VL  - 36
IS  - 1
PB  - EDP-Sciences
UR  -
UR  -
UR  -
UR  -
DO  - 10.1051/m2an:2002007
LA  - en
ID  - M2AN_2002__36_1_69_0
ER  - 
Bal, Guillaume; Maday, Yvon. Coupling of transport and diffusion models in linear transport theory. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 36 (2002) no. 1, pp. 69-86. doi : 10.1051/m2an:2002007.

[1] M.L. Adams and E.W. Larsen, Fast iterative methods for deterministic particle transport computations. Preprint (2001).

[2] R.E. Alcouffe, Diffusion synthetic acceleration methods for the diamond-differenced discrete-ordinates equations. Nucl. Sci. Eng. 64 (1977) 344.

[3] G. Allaire and G. Bal, Homogenization of the criticality spectral equation in neutron transport. ESAIM: M2AN 33 (1999) 721-746. | Numdam | Zbl 0931.35010

[4] S.R. Arridge, H. Dehghani, M. Schweiger, and E. Okada, 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.

[5] G. Bal, Couplage d'équations et homogénéisation en transport neutronique. Thèse de Doctorat de l'Université Paris 6 (1997). In French.

[6] G. Bal, 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. | Zbl 0937.35007

[7] G. Bal,Spatially varying discrete ordinates methods in XY-geometry. M 3 AS (Math. Models Methods Appl. Sci.) 10 (2000) 1277-1303. | Zbl 0969.65126

[8] G. Bal, Transport through diffusive and non-diffusive regions, embedded objects, and clear layers. To appear in SIAM J. Appl. Math. | MR 1918572 | Zbl 1020.45004

[9] G. Bal, V. Freilikher, G. Papanicolaou, and L. Ryzhik, Wave transport along surfaces with random impedance. Phys. Rev. B 6 (2000) 6228-6240.

[10] G. Bal and L. Ryzhik, Diffusion approximation of radiative transfer problems with interfaces. SIAM J. Appl. Math. 60 (2000) 1887-1912. | Zbl 0976.45008

[11] 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. | Zbl 0408.60100

[12] 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. | Zbl 0796.76063

[13] M. Cessenat, 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. | Zbl 0568.46030

[14] S. Chandrasekhar, Radiative Transfer. Dover Publications, New York (1960). | MR 111583

[15] P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland Publishing Company, Amsterdam, New York, Oxford (1978). | MR 520174 | Zbl 0383.65058

[16] R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology 6. Springer-Verlag, Berlin (1993). | MR 1295030 | Zbl 0755.35001

[17] J.J. Duderstadt and W.R. Martin, Transport Theory. Wiley-Interscience, New York (1979). | MR 551868 | Zbl 0407.76001

[18] M. Firbank, S.A. Arridge, M. Schweiger, and D.T. Delpy, An investigation of light transport through scattering bodies with non-scattering regions. Phys. Med. Biol. 41 (1996) 767-783.

[19] 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. | Zbl 0709.65092

[20] F. Golse, P.-L. Lions, B. Perthame, and R. Sentis, Regularity of the moments of the solution of a transport equation. J. Funct. Anal. 76 (1988) 110-125. | Zbl 0652.47031

[21] A.H. Hielscher, R.E. Alcouffe, and R.L. Barbour, Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues. Phys. Med. Biol. 43 (1998) 1285-1302.

[22] A. Ishimaru, Wave Propagation and Scattering in Random Media. Academics, New York (1978). | Zbl 0873.65115

[23] 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). | MR 1621249 | Zbl 0886.65124

[24] C.D. Levermore, W.J. Morokoff and B.T. Nadiga, Moment realizability and the validity of the Navier-Stokes equations for rarefied gas dynamics. Phys. Fluids 10 (1998) 3214-3226.

[25] E.E. Lewis and W.F. Miller Jr., Computational Methods of Neutron Transport. John Wiley & Sons, New York (1984). | Zbl 0594.65096

[26] L.D. Marini and A. Quarteroni, A Relaxation procedure for domain decomposition methods using finite elements. Numer. Math. 55 (1989) 575-598. | Zbl 0661.65111

[27] J. Planchard, Méthodes mathématiques en neutronique, in Collection de la Direction des Études et Recherches d'EDF, Eyrolles (1995). In French.

[28] L. Ryzhik, G. Papanicolaou, and J.B. Keller, Transport equations for elastic and other waves in random media. Wave Motion 24 (1996) 327-370. | Zbl 0954.74533

[29] 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). | MR 1488700 | Zbl 0894.73001

[30] M. Tidriri, Asymptotic analysis of a coupled system of kinetic equations. C. R. Acad. Sci. Paris Sér. I 328 (1999) 637-642. | Zbl 0945.76074

[31] S. Tiwari, Application of moment realizability criteria for the coupling of the Boltzmann and Euler equations. Transport Theory Statist. Phys. 29 (2000) 759-783.

Cité par Sources :