Hostname: page-component-8448b6f56d-m8qmq Total loading time: 0 Render date: 2024-04-19T21:16:45.724Z Has data issue: false hasContentIssue false
  • English
  • Français

Diffusion Limit of the Lorentz Model: Asymptotic Preserving Schemes

Published online by Cambridge University Press:  15 September 2002

Christophe Buet ,
Stéphane Cordier ,
Brigitte Lucquin-Desreux  and
Simona Mancini
Christophe Buet
Affiliation:
CEA/DAM Ile de France, BP 12, 91680 Bruyères-Le-Châtel, France. Christophe.Buet@cea.fr.
Stéphane Cordier
Affiliation:
Laboratoire MAPMO, UMR 6628, Université d'Orléans, 45067 Orléans, France. Stephane.Cordier@univ-orleans.fr.
Brigitte Lucquin-Desreux
Affiliation:
Laboratoire d'Analyse Numérique, UMR 7598, Université Pierre et Marie Curie, BP 187, 75252 Paris Cedex 05, France. lucquin@ann.jussieu.fr. smancini@ann.jussieu.fr.
Simona Mancini
Affiliation:
Laboratoire d'Analyse Numérique, UMR 7598, Université Pierre et Marie Curie, BP 187, 75252 Paris Cedex 05, France. lucquin@ann.jussieu.fr. smancini@ann.jussieu.fr.
Get access

Abstract

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.

Keywords

Hilbert expansiondiffusion limit.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 36 , Issue 4 , July 2002 , pp. 631 - 655
Copyright
© EDP Sciences, SMAI, 2002

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Adams, M.L., Subcell balance methods for radiative transfer on arbitrary grids. Transport Theory Statist. Phys. 27 (1997) 385-431. CrossRef
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
Buet, C., Cordier, S. and Lucquin-Desreux, B., The grazing collision limit for the Boltzmann-Lorentz model. Asymptot. Anal. 25 (2001) 93-107.
Caflisch, R.E., Jin, S. and Russo, G., Uniformly accurate schemes for hyperbolic systems with relaxation. SIAM J. Numer. Anal. 34 (1997) 246-281. CrossRef
Chen, G.Q., Levermore, C.D. and Liu, T.P., Hyperbolic conservation laws with stiff relaxation terms and entropy. Comm. Pure Appl. Math. 47 (1994) 187-830. CrossRef
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
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.
Degond, P. and Lucquin-Desreux, B., The asymptotics of collision operators for two species of particles of disparate masses. Math. Models Methods Appl. Sci. 6 (1996) 405-436. CrossRef
Desvillettes, L., On asymptotics of the Boltzmann equation when the collisions become grazing. Transport Theory Statist. Phys. 21 (1992) 259-276. CrossRef
Glimm, J., Marshall, G. and Plohr, B.J., A generalized Riemann problem for quasi one dimensional gas flows. Adv. in Appl. Math. 5 (1984) 1-30. CrossRef
E. Godlewski and P.A. Raviart, Numerical approximations of hyperbolic systems of conservation laws. Springer-Verlag, New York, Appl. Math. Sci. 118 (1996).
Goldstein, S., On diffusion by discontinuous movements, and on the telegraph equation. Quart. J. Mech. Appl. Math. 4 (1951) 129-156. CrossRef
Golse, F., Jin, S. and Levermore, C.D., The convergence of numerical transfer schemes in diffusive regimes I: discrete-ordinate method. SIAM J. Numer. Anal. 36 (1999) 1333-1369. CrossRef
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.
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.
Gosse, L. and Leroux, A.Y., A well-balanced scheme designed for inhomogeneous scalar conservation laws. C. R. Acad. Sci. Paris Sér. I Math. I323 (1996) 543-546.
Greenberg, J.M. and Leroux, A.Y., A well balanced scheme for the numerical processing of source terms in hyperbolic equations. SIAM J. Numer. Anal. 33 (1996) 1-16. CrossRef
Hermeline, F., A finite volume method for the approximation of diffusion operators on distorted meshes. J. Comput. Phys 160 (2000) 481-499. CrossRef
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).
Jin, S., Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations. SIAM J. Sci. Comput. 21 (1999) 441-454. CrossRef
Jin, S., Numerical integrations of systems of conservation laws of mixed type. SIAM J. Appl. Math. 55 (1995) 1536-1551. CrossRef
Jin, S. and Levermore, C.D., The discrete-ordinate method in diffusive regimes. Transport Theory Statist. Phys. 20 (1991) 413-439. CrossRef
Jin, S. and Levermore, C.D., Fully-discrete numerical transfer in diffusive regimes. Transport Theory Statist. Phys. 22 (1993) 739-791. CrossRef
S. Jin and C.D. Levermore, Numerical schemes for hyperbolic conservation laws with stiff relaxation terms. J. Comput. Phys. 126 (1996) 449-467.
S. Jin and L. Pareschi, Discretization of the multiscale semiconductor Boltzmann equation by diffusive relaxation schemes. J. Comput. Phys. 161 (2000) 312-330.
Jin, S., Pareschi, L. and Toscani, G., Diffusive relaxation schemes for multiscale discrete-velocity kinetic equations. SIAM J. Numer. Anal. 35 (1998) 2405-2439. CrossRef
S. Jin, L. Pareschi and G. Toscani, Uniformly accurate diffusive relaxation schemes for multiscale transport equations. SIAM J. Numer. Anal. (2000).
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.
Klar, A., An asymptotic-induced scheme for non stationary transport equations in the diffusive limit. SIAM J. Numer. Anal 35 (1998) 1073-1094. CrossRef
Larsen, E.W., The asymptotic diffusion limit of discretized transport problems. Nuclear Sci. Eng. 112 (1992) 336-346. CrossRef
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.
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.
LeVeque, R.J., Balancing source terms and flux gradients in high-resolution Godunov methods: the quasi-steady wave-propagation algorithm. J. Comput. Phys. 146 (1998) 346-365. CrossRef
Lions, P.L., Perthame, B. and Souganidis, P.E., 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. 3.0.CO;2-5>CrossRef
Lucquin-Desreux, B., Diffusion of electrons by multicharged ions. Math. Models Methods Appl. Sci. 10 (2000) 409-440. CrossRef
B. Lucquin-Desreux and S. Mancini, A finite element approximation of grazing collisions (submitted).
P.A. Markowich, C. Ringhoffer and C. Schmeiser, Semiconductor equations. Springer-Verlag (1994).
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.
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.
Naldi, G. and Pareschi, L., Numerical schemes for kinetic equations in diffusive regimes. Appl. Math. Lett. 11 (1998) 29-55. CrossRef
L. Pareschi, Central differencing based numerical schemes for hyperbolic conservation laws with relaxation terms. J. Num. Anal. (to appear).
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).
Prendergast, K.H. and Numerical, K. Xu hydrodynamics for gas-kinetic theory. J. Comput. Phys. 109 (1993) 53-66. CrossRef
Prendergast, K.H. and Numerical Navier-Stokes, K. Xu solutions from gas kinetic theory. J. Comput. Phys. 114 (1994) 9-17.
G. Samba, Limite asymptotique d'un schéma d'éléments finis linéaires discontinus lumpés en régime diffusion. Rapport CEA (to appear).
G.I. Taylor, Diffusion by continuous movements. Proc. London Math. Soc. 20 (1921) 196-212.
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.