A domain decomposition analysis for a two-scale linear transport problem
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 37 (2003) no. 6, p. 869-892

We present a domain decomposition theory on an interface problem for the linear transport equation between a diffusive and a non-diffusive region. To leading order, i.e. up to an error of the order of the mean free path in the diffusive region, the solution in the non-diffusive region is independent of the density in the diffusive region. However, the diffusive and the non-diffusive regions are coupled at the interface at the next order of approximation. In particular, our algorithm avoids iterating the diffusion and transport solutions as is done in most other methods - see for example Bal-Maday (2002). Our analysis is based instead on an accurate description of the boundary layer at the interface matching the phase-space density of particles leaving the non-diffusive region to the bulk density that solves the diffusion equation.

DOI : https://doi.org/10.1051/m2an:2003059
Classification:  65N55,  82B40,  82B80,  82C40,  82C70,  76R50
Keywords: domain decomposition, transport equation, diffusion approximation, kinetic-fluid coupling
@article{M2AN_2003__37_6_869_0,
     author = {Golse, Fran\c cois and Jin, Shi and Levermore, C. David},
     title = {A domain decomposition analysis for a two-scale linear transport problem},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {37},
     number = {6},
     year = {2003},
     pages = {869-892},
     doi = {10.1051/m2an:2003059},
     zbl = {1078.65125},
     mrnumber = {2026400},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2003__37_6_869_0}
}
Golse, François; Jin, Shi; Levermore, C. David. A domain decomposition analysis for a two-scale linear transport problem. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 37 (2003) no. 6, pp. 869-892. doi : 10.1051/m2an:2003059. http://www.numdam.org/item/M2AN_2003__37_6_869_0/

[1] G. Bal and Y. Maday, Coupling of Transport and Diffusion Models in Linear Transport Theory. ESAIM: M2AN 36 (2002) 69-86. | Numdam | Zbl 0995.45008

[2] C. Bardos, R. Santos and R. Sentis, Diffusion approximation and computation of critical size. Trans. Amer. Math. Soc. 284 (1984) 617-649. | Zbl 0508.60067

[3] A. Bensoussan, J.-L. Lions and G.C. Papanicolaou, Boundary layers and homogenization of transport processes. Publ. Res. Inst. Math. Sci. 15 (1979) 53-157. | Zbl 0408.60100

[4] 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 (Como, 1992). Amer. Math. Soc., Providence, RI, Contemp. Math. 157 (1994) 377-398. | Zbl 0796.76063

[5] C. Buet, S. Cordier, B. Lucquin-Desreux and S. Mancini, Diffusion limit of the Lorentz model: asymptotic preserving schemes. ESAIM: M2AN 36 (2002) 631-655. | Numdam | Zbl 1062.82050

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

[7] R. Dautray and J.L. Lions, Analyse Mathèmatique et Calcul Numérique pour les Sciences et les Techniques. Collection du Commissariat à l'Énergie Atomique: Série Scientifique, Masson, Paris (1985). | Zbl 0642.35001

[8] P. Degond and C. Schmeiser, Kinetic boundary layers and fluid-kinetic coupling in semiconductors. Transport Theory Statist. Phys. 28 (1999) 31-55. | Zbl 0942.35162

[9] S. Dellacherie, Kinetic fluid coupling in the field of the atomic vapor laser isotopic separation: numerical results in the case of a mono-species perfect gas, presented at the 23rd International Symposium on Rarefied Gas Dynamics, Whistler (British Columbia), July (2002).

[10] F. Golse, Applications of the Boltzmann equation within the context of upper atmosphere vehicle aerodynamics. Comput. Methods Appl. Mech. Engrg. 75 (1989) 299-316. | MR 1035753 | Zbl 0687.76078

[11] F. Golse, Knudsen layers from a computational viewpoint. Transport Theory Statist. Phys. 21 (1992) 211-236. | Zbl 0754.76066

[12] F. Golse, S. Jin and C.D. Levermore, The convergence of numerical transfer schemes in diffusive regimes, I. The dicrete-ordinate method. SIAM J. Numer. Anal. 36 (1999) 1333-1369. | Zbl 1053.82030

[13] M. Günther, P. Le Tallec, J.-P. Perlat and J. Struckmeier, Numerical modeling of gas flows in the transition between rarefied and continuum regimes. Numerical flow simulation I, (Marseille, 1997). Vieweg, Braunschweig, Notes Numer. Fluid Mech. 66 (1998) 222-241.

[14] S. Jin and C.D. Levermore, The discrete-ordinate method in diffusive regimes. Transport Theory Statist. Phys. 20 (1991) 413-439. | Zbl 0760.65125

[15] S. Jin and C.D. Levermore, Fully discrete numerical transfer in diffusive regimes. Transport Theory Statist. Phys. 22 (1993) 739-791. | Zbl 0818.65141

[16] S. Jin, L. Pareschi and G. Toscani, Uniformly accurate diffusive relaxation schemes for multiscale transport equations. SIAM J. Numer. Anal. 38 (2000) 913-936. | MR 1781209 | Zbl 0976.65091

[17] A. Klar, Convergence of alternating domain decomposition schemes for kinetic and aerodynamic equations. Math. Methods Appl. Sci. 18 (1995) 649-670. | Zbl 0827.76057

[18] A. Klar, Asymptotic-induced domain decomposition methods for kinetic and drift-diffusion semiconductor equations. SIAM J. Sci. Comput. 19 (1998) 2032-2050. | Zbl 0918.65090

[19] A. Klar, An asymptotic-induced scheme for nonstationary transport equations in the diffusive limit. SIAM J. Numer. Anal. 35 (1998) 1073-1094. | MR 1619859 | Zbl 0918.65091

[20] A. Klar, H. Neunzert and J. Struckmeier, Transition from kinetic theory to macroscopic fluid equations: a problem for domain decomposition and a source for new algorithm. Transport Theory Statist. Phys. 29 (2000) 93-106. | Zbl 0956.82023

[21] A. Klar and N. Siedow, Boundary layers and domain decomposition for radiative heat transfer and diffusion equations: applications to glass manufacturing process. European J. Appl. Math. 9 (1998) 351-372. | Zbl 0927.45004

[22] E.W. Larsen, J.E. Morel and W.F. Miller, Jr., Asymptotic solutions of numerical transport problems in optically thick, diffusive regimes. J. Comput. Phys. 69 (1987) 283-324. | Zbl 0627.65146

[23] J. Lehner and G.M. Wing, On the spectrum of an unsymmetric operator arising in the transport theory of neutrons. Comm. Pure Appl. Math. 8 (1955) 217-234. | Zbl 0064.23004

[24] P. Le Tallec and F. Mallinger, Coupling Boltzmann and Navier-Stokes equations by half fluxes. J. Comput. Phys. 136 (1997) 51-67. | Zbl 0890.76042

[25] P. Le Tallec and M. Tidriri, Convergence analysis of domain decomposition algorithms with full overlapping for the advection-diffusion problems. Math. Comp. 68 (1999) 585-606. | Zbl 1043.65112

[26] M. Tidriri, New models for the solution of intermediate regimes in transport theory and radiative transfer: existence theory, positivity, asymptotic analysis, and approximations. J. Statist. Phys. 104 (2001) 291-325. | Zbl 1126.82334

[27] N. Wiener and E. Hopf, Über eine Klasse singulärer Integralgleichungen, Sitzber. Preuss. Akad. Wiss., Sitzung der phys.-math. Klasse, Berlin (1931) 696-706. | Zbl 0003.30701