On the convergence of numerical schemes for the Boltzmann equation

T. Horsin; S. Mischler; A. Vasseur

Annales de l'I.H.P. Analyse non linéaire (2003)

  • Volume: 20, Issue: 5, page 731-758
  • ISSN: 0294-1449

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Horsin, T., Mischler, S., and Vasseur, A.. "On the convergence of numerical schemes for the Boltzmann equation." Annales de l'I.H.P. Analyse non linéaire 20.5 (2003): 731-758. <http://eudml.org/doc/78595>.

@article{Horsin2003,
author = {Horsin, T., Mischler, S., Vasseur, A.},
journal = {Annales de l'I.H.P. Analyse non linéaire},
keywords = {explicit discretization schemes; Maxwellian bounds; time-discrete averaging lemma},
language = {eng},
number = {5},
pages = {731-758},
publisher = {Elsevier},
title = {On the convergence of numerical schemes for the Boltzmann equation},
url = {http://eudml.org/doc/78595},
volume = {20},
year = {2003},
}

TY - JOUR
AU - Horsin, T.
AU - Mischler, S.
AU - Vasseur, A.
TI - On the convergence of numerical schemes for the Boltzmann equation
JO - Annales de l'I.H.P. Analyse non linéaire
PY - 2003
PB - Elsevier
VL - 20
IS - 5
SP - 731
EP - 758
LA - eng
KW - explicit discretization schemes; Maxwellian bounds; time-discrete averaging lemma
UR - http://eudml.org/doc/78595
ER -

References

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  1. [1] Agoshkov V.I., Spaces of functions with differential-difference characteristics and the smoothness of solutions of the transport equation, Dokl. Akad. Nauk SSSR276 (6) (1984) 1289-1293. Zbl0599.35009MR753365
  2. [2] Bouchut F., Desvillettes L., Averaging lemmas without time Fourier transform and application to discretized kinetic equation, Proc. Roy. Soc. Edinburgh Sect. A129 (1) (1999) 19-36. Zbl0933.35159MR1669221
  3. [3] Cercignani C., The Boltzmann Equation and its Application, Springer-Verlag, Berlin, 1988. Zbl0646.76001MR1313028
  4. [4] Desvillettes L., Mischler S., About the splitting algorithm for Boltzmann and B.G.K. equations, Math. Mod. Meth. Appl. Sci.6 (8) (1996) 1079-1101. Zbl0876.35088MR1428146
  5. [5] DiPerna R.J., Lions P.-L., On the Cauchy problem for Boltzmann equations: global existence and weak stability, Ann. Math.130 (1989) 321-366. Zbl0698.45010MR1014927
  6. [6] DiPerna R.J., Lions P.-L., Global weak solutions of Vlasov–Maxwell systems, Comm. Pure Appl. Math.42 (1989) 729-757. Zbl0698.35128MR1003433
  7. [7] DiPerna R.J., Lions P.-L., Global solutions of Boltzmann equation and the entropy inequality, Arch. Rat. Mech. Anal.114 (1991) 47-55. Zbl0724.45011MR1088276
  8. [8] DiPerna R.J., Lions P.-L., Meyer Y., Lp regularity of velocity averages, Ann. Inst. H. Poincaré Anal. Non Linéaire8 (1991) 271-287. Zbl0763.35014MR1127927
  9. [9] Gabetta E., Pareschi L., Toscani G., Relaxation schemes for nonlinear kinetic equations, SIAM J. Numer. Anal.34 (6) (1997) 2168-2194. Zbl0897.76071MR1480374
  10. [10] Goldstein D., Sturtevant B., Broadwell J.E., Investigation of the motion of discrete-velocity gases, in: Muntz E.P., Weaver D.P., Campbell D.H. (Eds.), Rarefied Gas Dynamics: Theoretical and Computational Techniques, Progress in Astronautics and Aeronautics, 118, AIAA, Washington, DC, 1989. 
  11. [11] Golse F., Lions P.-L., Perthame B., Sentis R., Regularity of the moments of the solution of a transport equation, J. Funct. Anal.76 (1988) 110-125. Zbl0652.47031MR923047
  12. [12] Golse F., Perthame B., Sentis R., Un résultat de compacité pour l'équation de transport et application au calcul de la valeur propre principale d'un opérateur de transport, C. R. Acad. Sci.301 (1985) 341-344. Zbl0591.45007MR808622
  13. [13] Lions P.-L., Régularité optimale des moyennes en vitesses, Note C. R. Acad. Sci. Paris, Série I320 (1995) 911-915. Zbl0827.35110MR1328710
  14. [14] Lions P.-L., Régularité optimale des moyennes en vitesses II, C. R. Acad. Sci. Paris, Série I326 (1998) 945-948. Zbl0922.35135MR1649933
  15. [15] Martin Y.L., Rogier F., Schneider J., Une méthode déterministe pour la résolution de l'équation de Boltzmann inhomogène, C. R. Acad. Sci. Paris314 (1992) 483-487. Zbl0747.65098MR1154392
  16. [16] Michel P., Schneider J., Approximation simultanée de réels par des nombres rationnels et noyau de collision de l'équation de Boltzmann, C. R. Acad. Sci. Paris, Série I330 (2000) 857-862. Zbl0960.65145MR1769961
  17. [17] Mischler S., Convergence of discrete velocities schemes for the Boltzmann equation, Arch. Rat. Mech. Anal.140 (1997) 53-77. Zbl0898.76089MR1482928
  18. [18] Mischler S., Wennberg B., On the homogeneous spatially Boltzmann equation, Annales de l'Institut Henri Poincaré16 (4) (1999) 467-501. Zbl0946.35075MR1697562
  19. [19] Palczewski A., Schneider J., Existence, stability, and convergence of solutions of discrete velocity models to the Boltzmann equation, J. Statist. Phys.91 (1998) 307-326. Zbl0918.76048MR1632506
  20. [20] Palczewski A., Schneider J., Bobylev A., Consistency result for a discrete-velocity model of the Boltzmann equation, SIAM J. Numer. Anal.34 (5) (1997) 1865-1883. Zbl0895.76083MR1472201
  21. [21] V.A. Panferov, A.G. Heintz, A new consistent discrete-velocity model for the Boltzmann equation, Preprint, University of Goteborg, 1999. Zbl0997.82036
  22. [22] Perthame B., Souganidis P.E., A limiting case for velocity averaging, Ann. Sci. Ecole Norm. Sup. (4)31 (4) (1998) 591-598. Zbl0956.45010MR1634024
  23. [23] Rogier F., Schneider J., A direct method for solving the Boltzmann equation, Proc. du Colloque Eromech 287, Discrete Models in Fluid Dynamics, Transport Theory Statis. Phys. (1–3) (1994). Zbl0811.76050MR1257657
  24. [24] J. Schneider, Une méthode déterministe pour la résolution de l'équation de Boltzmann, Thesis, University Paris 6, France, 1993. 
  25. [25] Vasseur A., Convergence of a semi-discrete kinetic scheme for the system of isentropic gas dynamics with γ=3, Indiana Univ. Math. J.48 (1999) 347-364. Zbl1020.65054
  26. [26] Vasseur A., Time regularity for the system of isentropic gas dynamics with γ=3, Comm. Partial Differential Equations24 (1999) 1987-1997. Zbl0940.35169
  27. [27] C. Villani, A review of mathematical topics in collisionnal kinetic theory, to appear. Zbl1170.82369MR1942465

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