On the convergence of numerical schemes for the Boltzmann equation
Annales de l'I.H.P. Analyse non linéaire, Tome 20 (2003) no. 5, pp. 731-758.
@article{AIHPC_2003__20_5_731_0,
     author = {Horsin, T. and Mischler, S. and Vasseur, A.},
     title = {On the convergence of numerical schemes for the {Boltzmann} equation},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {731--758},
     publisher = {Elsevier},
     volume = {20},
     number = {5},
     year = {2003},
     doi = {10.1016/S0294-1449(02)00029-X},
     mrnumber = {1995500},
     zbl = {1038.82082},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/S0294-1449(02)00029-X/}
}
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
SP  - 731
EP  - 758
VL  - 20
IS  - 5
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/S0294-1449(02)00029-X/
DO  - 10.1016/S0294-1449(02)00029-X
LA  - en
ID  - AIHPC_2003__20_5_731_0
ER  - 
%0 Journal Article
%A Horsin, T.
%A Mischler, S.
%A Vasseur, A.
%T On the convergence of numerical schemes for the Boltzmann equation
%J Annales de l'I.H.P. Analyse non linéaire
%D 2003
%P 731-758
%V 20
%N 5
%I Elsevier
%U http://www.numdam.org/articles/10.1016/S0294-1449(02)00029-X/
%R 10.1016/S0294-1449(02)00029-X
%G en
%F AIHPC_2003__20_5_731_0
Horsin, T.; Mischler, S.; Vasseur, A. On the convergence of numerical schemes for the Boltzmann equation. Annales de l'I.H.P. Analyse non linéaire, Tome 20 (2003) no. 5, pp. 731-758. doi : 10.1016/S0294-1449(02)00029-X. http://www.numdam.org/articles/10.1016/S0294-1449(02)00029-X/

[1] Agoshkov V.I., Spaces of functions with differential-difference characteristics and the smoothness of solutions of the transport equation, Dokl. Akad. Nauk SSSR 276 (6) (1984) 1289-1293. | MR | Zbl

[2] Bouchut F., Desvillettes L., Averaging lemmas without time Fourier transform and application to discretized kinetic equation, Proc. Roy. Soc. Edinburgh Sect. A 129 (1) (1999) 19-36. | MR | Zbl

[3] Cercignani C., The Boltzmann Equation and its Application, Springer-Verlag, Berlin, 1988. | MR

[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. | MR | Zbl

[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. | MR | Zbl

[6] Diperna R.J., Lions P.-L., Global weak solutions of Vlasov-Maxwell systems, Comm. Pure Appl. Math. 42 (1989) 729-757. | MR | Zbl

[7] Diperna R.J., Lions P.-L., Global solutions of Boltzmann equation and the entropy inequality, Arch. Rat. Mech. Anal. 114 (1991) 47-55. | MR | Zbl

[8] Diperna R.J., Lions P.-L., Meyer Y., Lp regularity of velocity averages, Ann. Inst. H. Poincaré Anal. Non Linéaire 8 (1991) 271-287. | Numdam | MR | Zbl

[9] Gabetta E., Pareschi L., Toscani G., Relaxation schemes for nonlinear kinetic equations, SIAM J. Numer. Anal. 34 (6) (1997) 2168-2194. | MR | Zbl

[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] 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. | MR | Zbl

[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. | MR | Zbl

[13] Lions P.-L., Régularité optimale des moyennes en vitesses, Note C. R. Acad. Sci. Paris, Série I 320 (1995) 911-915. | MR | Zbl

[14] Lions P.-L., Régularité optimale des moyennes en vitesses II, C. R. Acad. Sci. Paris, Série I 326 (1998) 945-948. | MR | Zbl

[15] Martin Y.L., Rogier F., Schneider J., Une méthode déterministe pour la résolution de l'équation de Boltzmann inhomogène, C. R. Acad. Sci. Paris 314 (1992) 483-487. | MR | Zbl

[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 I 330 (2000) 857-862. | MR | Zbl

[17] Mischler S., Convergence of discrete velocities schemes for the Boltzmann equation, Arch. Rat. Mech. Anal. 140 (1997) 53-77. | MR | Zbl

[18] Mischler S., Wennberg B., On the homogeneous spatially Boltzmann equation, Annales de l'Institut Henri Poincaré 16 (4) (1999) 467-501. | Numdam | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[21] V.A. Panferov, A.G. Heintz, A new consistent discrete-velocity model for the Boltzmann equation, Preprint, University of Goteborg, 1999.

[22] Perthame B., Souganidis P.E., A limiting case for velocity averaging, Ann. Sci. Ecole Norm. Sup. (4) 31 (4) (1998) 591-598. | Numdam | MR | Zbl

[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). | MR | Zbl

[24] J. Schneider, Une méthode déterministe pour la résolution de l'équation de Boltzmann, Thesis, University Paris 6, France, 1993.

[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. | Zbl

[26] Vasseur A., Time regularity for the system of isentropic gas dynamics with γ=3, Comm. Partial Differential Equations 24 (1999) 1987-1997. | Zbl

[27] C. Villani, A review of mathematical topics in collisionnal kinetic theory, to appear. | MR | Zbl

Cité par Sources :