On the convergence of numerical schemes for the Boltzmann equation
Horsin, T. ; Mischler, S. ; Vasseur, A.
Annales de l'I.H.P. Analyse non linéaire, Tome 20 (2003), p. 731-758 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU) 
@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},
     volume = {20},
     year = {2003},
     pages = {731-758},
     doi = {10.1016/S0294-1449(02)00029-X},
     mrnumber = {1995500},
     zbl = {1038.82082},
     language = {en},
     url = {http://dml.mathdoc.fr/item/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) pp. 731-758. doi : 10.1016/S0294-1449(02)00029-X. http://gdmltest.u-ga.fr/item/AIHPC_2003__20_5_731_0/

[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 753365 | Zbl 0599.35009

[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 1669221 | Zbl 0933.35159

[3] Cercignani C., The Boltzmann Equation and its Application, Springer-Verlag, Berlin, 1988. | MR 1313028

[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 1428146 | Zbl 0876.35088

[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 1014927 | Zbl 0698.45010

[6] Diperna R.J., Lions P.-L., Global weak solutions of Vlasov-Maxwell systems, Comm. Pure Appl. Math. 42 (1989) 729-757. | MR 1003433 | Zbl 0698.35128

[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 1088276 | Zbl 0724.45011

[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 1127927 | Zbl 0763.35014

[9] Gabetta E., Pareschi L., Toscani G., Relaxation schemes for nonlinear kinetic equations, SIAM J. Numer. Anal. 34 (6) (1997) 2168-2194. | MR 1480374 | Zbl 0897.76071

[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 923047 | Zbl 0652.47031

[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 808622 | Zbl 0591.45007

[13] Lions P.-L., Régularité optimale des moyennes en vitesses, Note C. R. Acad. Sci. Paris, Série I 320 (1995) 911-915. | MR 1328710 | Zbl 0827.35110

[14] Lions P.-L., Régularité optimale des moyennes en vitesses II, C. R. Acad. Sci. Paris, Série I 326 (1998) 945-948. | MR 1649933 | Zbl 0922.35135

[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 1154392 | Zbl 0747.65098

[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 1769961 | Zbl 0960.65145

[17] Mischler S., Convergence of discrete velocities schemes for the Boltzmann equation, Arch. Rat. Mech. Anal. 140 (1997) 53-77. | MR 1482928 | Zbl 0898.76089

[18] Mischler S., Wennberg B., On the homogeneous spatially Boltzmann equation, Annales de l'Institut Henri Poincaré 16 (4) (1999) 467-501. | Numdam | MR 1697562 | Zbl 0946.35075

[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 1632506 | Zbl 0918.76048

[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 1472201 | Zbl 0895.76083

[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 1634024 | Zbl 0956.45010

[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 1257657 | Zbl 0811.76050

[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 1020.65054

[26] Vasseur A., Time regularity for the system of isentropic gas dynamics with γ=3, Comm. Partial Differential Equations 24 (1999) 1987-1997. | Zbl 0940.35169

[27] C. Villani, A review of mathematical topics in collisionnal kinetic theory, to appear. | MR 1942465 | Zbl 1170.82369