A domain decomposition analysis for a two-scale linear transport problem

Golse, François; Jin, Shi; Levermore, C. David

ESAIM: Modélisation mathématique et analyse numérique, Tome 37 (2003) no. 6, pp. 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.

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

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

[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

[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

[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

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

[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

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

[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 | Zbl

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

[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

[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

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

[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 | Zbl

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

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

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

[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

[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

[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

[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

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

[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

[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

[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