Homogenization of a spectral equation with drift in linear transport

Guillaume Bal

ESAIM: Control, Optimisation and Calculus of Variations (2001)

  • Volume: 6, page 613-627
  • ISSN: 1292-8119

Abstract

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This paper deals with the homogenization of a spectral equation posed in a periodic domain in linear transport theory. The particle density at equilibrium is given by the unique normalized positive eigenvector of this spectral equation. The corresponding eigenvalue indicates the amount of particle creation necessary to reach this equilibrium. When the physical parameters satisfy some symmetry conditions, it is known that the eigenvectors of this equation can be approximated by the product of two term. The first one solves a local transport spectral equation posed in the periodicity cell and the second one a homogeneous spectral diffusion equation posed in the entire domain. This paper addresses the case where these symmetry conditions are not fulfilled. We show that the factorization remains valid with the diffusion equation replaced by a convection-diffusion equation with large drift. The asymptotic limit of the leading eigenvalue is also modified. The spectral equation treated in this paper can model the stability of nuclear reactor cores and describe the distribution of neutrons at equilibrium. The same techniques can also be applied to the time-dependent linear transport equation with drift, which appears in radiative transfer theory and which models the propagation of acoustic, electromagnetic, and elastic waves in heterogeneous media.

How to cite

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Bal, Guillaume. "Homogenization of a spectral equation with drift in linear transport." ESAIM: Control, Optimisation and Calculus of Variations 6 (2001): 613-627. <http://eudml.org/doc/90611>.

@article{Bal2001,
abstract = {This paper deals with the homogenization of a spectral equation posed in a periodic domain in linear transport theory. The particle density at equilibrium is given by the unique normalized positive eigenvector of this spectral equation. The corresponding eigenvalue indicates the amount of particle creation necessary to reach this equilibrium. When the physical parameters satisfy some symmetry conditions, it is known that the eigenvectors of this equation can be approximated by the product of two term. The first one solves a local transport spectral equation posed in the periodicity cell and the second one a homogeneous spectral diffusion equation posed in the entire domain. This paper addresses the case where these symmetry conditions are not fulfilled. We show that the factorization remains valid with the diffusion equation replaced by a convection-diffusion equation with large drift. The asymptotic limit of the leading eigenvalue is also modified. The spectral equation treated in this paper can model the stability of nuclear reactor cores and describe the distribution of neutrons at equilibrium. The same techniques can also be applied to the time-dependent linear transport equation with drift, which appears in radiative transfer theory and which models the propagation of acoustic, electromagnetic, and elastic waves in heterogeneous media.},
author = {Bal, Guillaume},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {homogenization; linear transport; eigenvalue problem; drift; periodic domain; stability of nuclear reactor cores; linear transport equation with drift; waves in heterogeneous media},
language = {eng},
pages = {613-627},
publisher = {EDP-Sciences},
title = {Homogenization of a spectral equation with drift in linear transport},
url = {http://eudml.org/doc/90611},
volume = {6},
year = {2001},
}

TY - JOUR
AU - Bal, Guillaume
TI - Homogenization of a spectral equation with drift in linear transport
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2001
PB - EDP-Sciences
VL - 6
SP - 613
EP - 627
AB - This paper deals with the homogenization of a spectral equation posed in a periodic domain in linear transport theory. The particle density at equilibrium is given by the unique normalized positive eigenvector of this spectral equation. The corresponding eigenvalue indicates the amount of particle creation necessary to reach this equilibrium. When the physical parameters satisfy some symmetry conditions, it is known that the eigenvectors of this equation can be approximated by the product of two term. The first one solves a local transport spectral equation posed in the periodicity cell and the second one a homogeneous spectral diffusion equation posed in the entire domain. This paper addresses the case where these symmetry conditions are not fulfilled. We show that the factorization remains valid with the diffusion equation replaced by a convection-diffusion equation with large drift. The asymptotic limit of the leading eigenvalue is also modified. The spectral equation treated in this paper can model the stability of nuclear reactor cores and describe the distribution of neutrons at equilibrium. The same techniques can also be applied to the time-dependent linear transport equation with drift, which appears in radiative transfer theory and which models the propagation of acoustic, electromagnetic, and elastic waves in heterogeneous media.
LA - eng
KW - homogenization; linear transport; eigenvalue problem; drift; periodic domain; stability of nuclear reactor cores; linear transport equation with drift; waves in heterogeneous media
UR - http://eudml.org/doc/90611
ER -

References

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  1. [1] G. Allaire, Homogenization and two-scale convergence. SIAM J. Math. Anal. 9 (1992) 1482-1518. Zbl0770.35005MR1185639
  2. [2] G. Allaire and G. Bal, Homogenization of the criticality spectral equation in neutron transport. ESAIM: M2AN 33 (1999) 721-746. Zbl0931.35010MR1726482
  3. [3] G. Bal, Couplage d’équations et homogénéisation en transport neutronique, Thèse de Doctorat de l’Université Paris 6 (1997). 
  4. [4] G. Bal, Boundary layer analysis in the homogenization of neutron transport equations in a cubic domain. Asymptot. Anal. 20 (1999) 213-239. Zbl1040.35508MR1715334
  5. [5] 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. Zbl0937.35007MR1718300
  6. [6] G. Bal, Diffusion Approximation of Radiative Transfer Equations in a Channel. Transport Theory Statist. Phys. (to appear). Zbl1031.86002MR1848597
  7. [7] P. Benoist, Théorie du coefficient de diffusion des neutrons dans un réseau comportant des cavités, Note CEA-R 2278 (1964). 
  8. [8] A. Bensoussan, J.L. Lions and G. Papanicolaou, Asymptotic analysis for periodic structures. North-Holland (1978). Zbl0404.35001MR503330
  9. [9] , Boundary Layers and Homogenization of Transport Processes. RIMS, Kyoto Univ. (1979). Zbl0408.60100
  10. [10] J. Bergh and L. Löfström, Interpolation spaces. Springer, New York (1976). Zbl0344.46071
  11. [11] J. Bussac and P. Reuss, Traité de neutronique. Hermann, Paris (1978). 
  12. [12] Y. Capdeboscq, Homogenization of a diffusion equation with drift. C. R. Acad. Sci. Paris Sér. I Math. 327 (1998) 807-812. Zbl0918.35135MR1663726
  13. [13] , Homogenization of a Neutronic Critical Diffusion Problem with Drift. Proc. Roy Soc. Edinburgh Sect. A (accepted). Zbl1066.82530
  14. [14] F. Chatelin, Spectral approximation of linear operators. Academic Press, Comp. Sci. Appl. Math. (1983). Zbl0517.65036MR716134
  15. [15] R. Dautray and J.L. Lions, Mathematical analysis and numerical methods for Science and Technology, Vol. 6. Springer Verlag, Berlin (1993). Zbl0802.35001MR1295030
  16. [16] V. Deniz, The theory of neutron leakage in reactor lattices, in Handbook of nuclear reactor calculations, Vol. II, edited by Y. Ronen (1968) 409-508. 
  17. [17] J. Garnier, Homogenization in a periodic and time dependent potential. SIAM J. Appl. Math. 57 (1997) 95-111. Zbl0872.35009MR1429379
  18. [18] 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. Zbl0652.47031MR923047
  19. [19] T. Kato, Perturbation theory for linear operators. Springer Verlag, Berlin (1976). Zbl0342.47009MR407617
  20. [20] M.L. Kleptsyna and A.L. Piatnitski, On large deviation asymptotics for homgenized SDE with a small diffusion. Probab. Theory Appl. (submitted). 
  21. [21] S. Kozlov, Reductibility of quasiperiodic differential operators and averaging. Trans. Moscow Math. Soc. 2 (1984) 101-136. Zbl0566.35036MR737902
  22. [22] E.W. Larsen, Neutron transport and diffusion in inhomogeneous media. I. J. Math. Phys. 16 (1975) 1421-1427. MR391839
  23. [23] , Neutron transport and diffusion in inhomogeneous media. II. Nuclear Sci. Engrg. 60 (1976) 357-368. 
  24. [24] E.W. Larsen and J.B. Keller, Asymptotic solution of neutron transport problems for small mean free paths. J. Math. Phys. 15 (1974) 75-81. MR339741
  25. [25] E.W. Larsen and M. Williams, Neutron Drift in Heterogeneous Media. Nuclear Sci. Engrg. 65 (1978) 290-302. 
  26. [26] M. Mokhtar–Kharroubi, Mathematical Topics in Neutron Transport Theory. World Scientific, Singapore (1997). Zbl0997.82047
  27. [27] J. Planchard, Méthodes mathématiques en neutronique, Collection de la Direction des Études et Recherches d’EDF. Eyrolles (1995). 
  28. [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. Zbl0954.74533MR1427483
  29. [29] R. Sentis, Study of the corrector of the eigenvalue of a transport operator. SIAM J. Math. Anal. 16 (1985) 151-166. Zbl0609.45002MR772875
  30. [30] M. Struwe, Variational methods: Applications to nonlinear partial differential equations and Hamiltonian systems. Springer, Berlin (1990). Zbl0746.49010MR1078018

Citations in EuDML Documents

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  1. Grégoire Allaire, Guillaume Bal, Vincent Siess, Homogenization and localization in locally periodic transport
  2. Grégoire Allaire, Guillaume Bal, Vincent Siess, Homogenization and localization in locally periodic transport
  3. Grégoire Allaire, Homogénéisation et limite de diffusion pour une équation de transport
  4. Thierry Goudon, Antoine Mellet, Homogenization and diffusion asymptotics of the linear Boltzmann equation
  5. Thierry Goudon, Antoine Mellet, Homogenization and Diffusion Asymptotics of the Linear Boltzmann Equation

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