Diffusion limit of the Lorentz model : asymptotic preserving schemes

Christophe Buet; Stéphane Cordier; Brigitte Lucquin-Desreux; Simona Mancini

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2002)

  • Volume: 36, Issue: 4, page 631-655
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Buet, Christophe, et al. "Diffusion limit of the Lorentz model : asymptotic preserving schemes." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 36.4 (2002): 631-655. <http://eudml.org/doc/245407>.

@article{Buet2002,
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.},
author = {Buet, Christophe, Cordier, Stéphane, Lucquin-Desreux, Brigitte, Mancini, Simona},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {Hilbert expansion; diffusion limit},
language = {eng},
number = {4},
pages = {631-655},
publisher = {EDP-Sciences},
title = {Diffusion limit of the Lorentz model : asymptotic preserving schemes},
url = {http://eudml.org/doc/245407},
volume = {36},
year = {2002},
}

TY - JOUR
AU - Buet, Christophe
AU - Cordier, Stéphane
AU - Lucquin-Desreux, Brigitte
AU - Mancini, Simona
TI - Diffusion limit of the Lorentz model : asymptotic preserving schemes
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2002
PB - EDP-Sciences
VL - 36
IS - 4
SP - 631
EP - 655
AB - 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.
LA - eng
KW - Hilbert expansion; diffusion limit
UR - http://eudml.org/doc/245407
ER -

References

top
  1. [1] M.L. Adams, Subcell balance methods for radiative transfer on arbitrary grids. Transport Theory Statist. Phys. 27 (1997) 385–431. Zbl0906.65136
  2. [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 Zbl1017.65070
  3. [3] C. Buet, S. Cordier and B. Lucquin-Desreux, The grazing collision limit for the Boltzmann–Lorentz model. Asymptot. Anal. 25 (2001) 93–107. Zbl1067.82051
  4. [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. Zbl0868.35070
  5. [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. Zbl0806.35112
  6. [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 Zbl1036.82021
  7. [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. Zbl0755.35091
  8. [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. Zbl0853.76079
  9. [9] L. Desvillettes, On asymptotics of the Boltzmann equation when the collisions become grazing. Transport Theory Statist. Phys. 21 (1992) 259–276. Zbl0769.76059
  10. [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. Zbl0566.76056
  11. [11] E. Godlewski and P.A. Raviart, Numerical approximations of hyperbolic systems of conservation laws. Springer-Verlag, New York, Appl. Math. Sci. 118 (1996). Zbl0860.65075MR1410987
  12. [12] S. Goldstein, On diffusion by discontinuous movements, and on the telegraph equation. Quart. J. Mech. Appl. Math. 4 (1951) 129–156. Zbl0045.08102
  13. [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. Zbl1053.82030
  14. [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. Zbl0909.65059
  15. [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. Zbl1018.65108
  16. [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. Zbl0858.65091
  17. [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. Zbl0876.65064
  18. [18] F. Hermeline, A finite volume method for the approximation of diffusion operators on distorted meshes. J. Comput. Phys 160 (2000) 481–499. Zbl0949.65101
  19. [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). Zbl0777.65070
  20. [20] S. Jin, Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations. SIAM J. Sci. Comput. 21 (1999) 441–454. Zbl0947.82008
  21. [21] S. Jin, Numerical integrations of systems of conservation laws of mixed type. SIAM J. Appl. Math. 55 (1995) 1536–1551. Zbl0839.65092
  22. [22] S. Jin and C.D. Levermore, The discrete-ordinate method in diffusive regimes. Transport Theory Statist. Phys. 20 (1991) 413–439. Zbl0760.65125
  23. [23] S. Jin and C.D. Levermore, Fully-discrete numerical transfer in diffusive regimes. Transport Theory Statist. Phys. 22 (1993) 739–791. Zbl0818.65141
  24. [24] S. Jin and C.D. Levermore, Numerical schemes for hyperbolic conservation laws with stiff relaxation terms. J. Comput. Phys. 126 (1996) 449–467. Zbl0860.65089
  25. [25] S. Jin and L. Pareschi, Discretization of the multiscale semiconductor Boltzmann equation by diffusive relaxation schemes. J. Comput. Phys. 161 (2000) 312–330. Zbl1156.82408
  26. [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. Zbl0938.35097
  27. [27] S. Jin, L. Pareschi and G. Toscani, Uniformly accurate diffusive relaxation schemes for multiscale transport equations. SIAM J. Numer. Anal. (2000). Zbl0976.65091MR1781209
  28. [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. Zbl0826.65078
  29. [29] A. Klar, An asymptotic-induced scheme for non stationary transport equations in the diffusive limit. SIAM J. Numer. Anal 35 (1998) 1073–1094. Zbl0918.65091
  30. [30] E.W. Larsen, The asymptotic diffusion limit of discretized transport problems. Nuclear Sci. Eng. 112 (1992) 336–346. 
  31. [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. Zbl0684.65118
  32. [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. Zbl0627.65146
  33. [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. Zbl0931.76059
  34. [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. Zbl0853.76077
  35. [35] B. Lucquin-Desreux, Diffusion of electrons by multicharged ions. Math. Models Methods Appl. Sci. 10 (2000) 409–440. Zbl1012.82022
  36. [36] B. Lucquin-Desreux and S. Mancini, A finite element approximation of grazing collisions (submitted). Zbl1053.82029
  37. [37] P.A. Markowich, C. Ringhoffer and C. Schmeiser, Semiconductor equations. Springer-Verlag (1994). Zbl0765.35001
  38. [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. Zbl0801.65137
  39. [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. Zbl0864.65095
  40. [40] G. Naldi and L. Pareschi, Numerical schemes for kinetic equations in diffusive regimes. Appl. Math. Lett. 11 (1998) 29–55. Zbl06587013
  41. [41] L. Pareschi, Central differencing based numerical schemes for hyperbolic conservation laws with relaxation terms. J. Num. Anal. (to appear). Zbl1020.65048MR1870848
  42. [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). Zbl0969.76055MR1731614
  43. [43] K.H. Prendergast and K. Xu, Numerical hydrodynamics for gas-kinetic theory. J. Comput. Phys. 109 (1993) 53–66. Zbl0791.76059
  44. [44] K.H. Prendergast and K. Xu, Numerical Navier-Stokes solutions from gas kinetic theory. J. Comput. Phys. 114 (1994) 9–17. Zbl0810.76059
  45. [45] G. Samba, Limite asymptotique d’un schéma d’éléments finis linéaires discontinus lumpés en régime diffusion. Rapport CEA (to appear). 
  46. [46] G.I. Taylor, Diffusion by continuous movements. Proc. London Math. Soc. 20 (1921) 196–212. Zbl48.0961.01JFM48.0961.01
  47. [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. Zbl0547.65065

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