Diffusion limit of the Lorentz model : asymptotic preserving schemes
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 36 (2002) no. 4, pp. 631-655.

This paper deals with the diffusion limit of a kinetic equation where the collisions are modeled by a Lorentz type operator. The main aim is to construct a discrete scheme to approximate this equation which gives for any value of the Knudsen number, and in particular at the diffusive limit, the right discrete diffusion equation with the same value of the diffusion coefficient as in the continuous case. We are also naturally interested with a discretization which can be used with few velocity discretization points, in order to reduce the cost of computation.

DOI : https://doi.org/10.1051/m2an:2002028
Classification : 82C70,  35B40,  65N06
Mots clés : Hilbert expansion, diffusion limit
@article{M2AN_2002__36_4_631_0,
author = {Buet, Christophe and Cordier, St\'ephane and Lucquin-Desreux, Brigitte and Mancini, Simona},
title = {Diffusion limit of the Lorentz model : asymptotic preserving schemes},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
pages = {631--655},
publisher = {EDP-Sciences},
volume = {36},
number = {4},
year = {2002},
doi = {10.1051/m2an:2002028},
zbl = {1062.82050},
mrnumber = {1932307},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an:2002028/}
}
Buet, Christophe; Cordier, Stéphane; Lucquin-Desreux, Brigitte; Mancini, Simona. Diffusion limit of the Lorentz model : asymptotic preserving schemes. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 36 (2002) no. 4, pp. 631-655. doi : 10.1051/m2an:2002028. http://www.numdam.org/articles/10.1051/m2an:2002028/

[1] M.L. Adams, Subcell balance methods for radiative transfer on arbitrary grids. Transport Theory Statist. Phys. 27 (1997) 385-431. | Zbl 0906.65136

[2] R. Botchorishvili, B. Perthame and A. Vasseur, Equilibrium schemes for scalar conservation laws with stiff sources. Inria report RR-3891 (2000), http://www.inria.fr/RRRT/RR-3891.html | Zbl 1017.65070

[3] C. Buet, S. Cordier and B. Lucquin-Desreux, The grazing collision limit for the Boltzmann-Lorentz model. Asymptot. Anal. 25 (2001) 93-107. | Zbl 1067.82051

[4] R.E. Caflisch, S. Jin and G. Russo, Uniformly accurate schemes for hyperbolic systems with relaxation. SIAM J. Numer. Anal. 34 (1997) 246-281. | Zbl 0868.35070

[5] G.Q. Chen, C.D. Levermore and T.P. Liu, Hyperbolic conservation laws with stiff relaxation terms and entropy. Comm. Pure Appl. Math. 47 (1994) 187-830. | Zbl 0806.35112

[6] S. Cordier, B. Lucquin-Desreux and A. Sabry, Numerical approximation of the Vlasov-Fokker-Planck-Lorentz model. ESAIM: Procced. CEMRACS 1999 (2001), http://www.emath.fr/Maths/Proc/Vol.10

[7] P. Degond and B. Lucquin-Desreux, The Fokker-Planck asymptotics of the Boltzmann collision operator in the Coulomb case. Math. Models Methods Appl. Sci. 2 (1992) 167-182. | Zbl 0755.35091

[8] P. Degond and B. Lucquin-Desreux, The asymptotics of collision operators for two species of particles of disparate masses. Math. Models Methods Appl. Sci. 6 (1996) 405-436. | Zbl 0853.76079

[9] L. Desvillettes, On asymptotics of the Boltzmann equation when the collisions become grazing. Transport Theory Statist. Phys. 21 (1992) 259-276. | Zbl 0769.76059

[10] J. Glimm, G. Marshall and B.J. Plohr, A generalized Riemann problem for quasi one dimensional gas flows. Adv. in Appl. Math. 5 (1984) 1-30. | Zbl 0566.76056

[11] E. Godlewski and P.A. Raviart, Numerical approximations of hyperbolic systems of conservation laws. Springer-Verlag, New York, Appl. Math. Sci. 118 (1996). | MR 1410987 | Zbl 0860.65075

[12] S. Goldstein, On diffusion by discontinuous movements, and on the telegraph equation. Quart. J. Mech. Appl. Math. 4 (1951) 129-156. | Zbl 0045.08102

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

[14] L. Gosse, A priori error estimate for a well-balanced scheme designed for inhomogeneous scalar conservation laws. C. R. Acad. Sci. Paris Sér. I Math. 327 (1998) 467-472. | Zbl 0909.65059

[15] L. Gosse, A well-balanced scheme using non-conservative products designed for hyperbolic systems of conservation laws with source terms. Math. Models Methods Appl. Sci. 11 (2001) 339-365. | Zbl 1018.65108

[16] L. Gosse and A.Y. Leroux, A well-balanced scheme designed for inhomogeneous scalar conservation laws. C. R. Acad. Sci. Paris Sér. I Math. I323 (1996) 543-546. | MR 1377240 | Zbl 0858.65091

[17] J.M. Greenberg and A.Y. Leroux, A well balanced scheme for the numerical processing of source terms in hyperbolic equations. SIAM J. Numer. Anal. 33 (1996) 1-16. | Zbl 0876.65064

[18] F. Hermeline, A finite volume method for the approximation of diffusion operators on distorted meshes. J. Comput. Phys 160 (2000) 481-499. | Zbl 0949.65101

[19] F. Hermeline, Two coupled particle-finite volume methods using Delaunay-Voronoï meshes for the approximation of Vlasov-Poisson and Vlasov-Maxwell equations. J. Comput. Phys 106 (1993). | Zbl 0777.65070

[20] S. Jin, Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations. SIAM J. Sci. Comput. 21 (1999) 441-454. | Zbl 0947.82008

[21] S. Jin, Numerical integrations of systems of conservation laws of mixed type. SIAM J. Appl. Math. 55 (1995) 1536-1551. | Zbl 0839.65092

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

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

[24] S. Jin and C.D. Levermore, Numerical schemes for hyperbolic conservation laws with stiff relaxation terms. J. Comput. Phys. 126 (1996) 449-467. | Zbl 0860.65089

[25] S. Jin and L. Pareschi, Discretization of the multiscale semiconductor Boltzmann equation by diffusive relaxation schemes. J. Comput. Phys. 161 (2000) 312-330. | Zbl 1156.82408

[26] S. Jin, L. Pareschi and G. Toscani, Diffusive relaxation schemes for multiscale discrete-velocity kinetic equations. SIAM J. Numer. Anal. 35 (1998) 2405-2439. | Zbl 0938.35097

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

[28] S. Jin and Z. Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimensions. Comm. Pure Appl. Math. XLVIII (1995) 235-276. | Zbl 0826.65078

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

[30] E.W. Larsen, The asymptotic diffusion limit of discretized transport problems. Nuclear Sci. Eng. 112 (1992) 336-346.

[31] E.W. Larsen and J.E. Morel, Asymptotic solutions of numerical transport problems in optically thick, diffusive regimes. II. J. Comput. Phys. 83 (1989) 212-236. | Zbl 0684.65118

[32] 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

[33] R.J. Leveque, Balancing source terms and flux gradients in high-resolution Godunov methods: the quasi-steady wave-propagation algorithm. J. Comput. Phys. 146 (1998) 346-365. | Zbl 0931.76059

[34] P.L. Lions, B. Perthame and P.E. Souganidis, Existence of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates. Comm. Pure Appl. Math. 49 (1996) 599-638. | Zbl 0853.76077

[35] B. Lucquin-Desreux, Diffusion of electrons by multicharged ions. Math. Models Methods Appl. Sci. 10 (2000) 409-440. | Zbl 1012.82022

[36] B. Lucquin-Desreux and S. Mancini, A finite element approximation of grazing collisions (submitted). | Zbl 1053.82029

[37] P.A. Markowich, C. Ringhoffer and C. Schmeiser, Semiconductor equations. Springer-Verlag (1994). | Zbl 0765.35001

[38] W.F. Miller Jr. and T. Noh, Finite differences versus finite elements in slab geometry, even-parity transport theory. Transport Theory Statist. Phys. 22 (1993) 247-270. | Zbl 0801.65137

[39] J.E. Morel, T.A. Wareing and K. Smith, A linear-discontinuous spatial differencing scheme for ${S}_{n}$ radiative transfer calculations. J. Comput. Phys. 128 (1996) 445-462. | Zbl 0864.65095

[40] G. Naldi and L. Pareschi, Numerical schemes for kinetic equations in diffusive regimes. Appl. Math. Lett. 11 (1998) 29-55. | Zbl 1337.65118

[41] L. Pareschi, Central differencing based numerical schemes for hyperbolic conservation laws with relaxation terms. J. Num. Anal. (to appear). | MR 1870848 | Zbl 1020.65048

[42] B. Perthame, An introduction to kinetic schemes for gas dynamics. An introduction to recent developments in theory and numerics for conservation laws. L.N. in Computational Sc. and Eng., 5, D. Kroner, M. Ohlberger and C. Rohde Eds., Springer (1998). | MR 1731614 | Zbl 0969.76055

[43] K.H. Prendergast and K. Xu, Numerical hydrodynamics for gas-kinetic theory. J. Comput. Phys. 109 (1993) 53-66. | Zbl 0791.76059

[44] K.H. Prendergast and K. Xu, Numerical Navier-Stokes solutions from gas kinetic theory. J. Comput. Phys. 114 (1994) 9-17. | Zbl 0810.76059

[45] G. Samba, Limite asymptotique d'un schéma d'éléments finis linéaires discontinus lumpés en régime diffusion. Rapport CEA (to appear).

[46] G.I. Taylor, Diffusion by continuous movements. Proc. London Math. Soc. 20 (1921) 196-212. | JFM 48.0961.01

[47] B. Vanleer, On the relation between the upwind differencing schemes of Engquist-Osher, Godunov and Roe. SIAM J. Sci. Stat. Comp. 5 (1984) 1-20. | Zbl 0547.65065