Naldi, Giovanni; Pareschi, Lorenzo; Toscani, Giuseppe
Spectral methods for one-dimensional kinetic models of granular flows and numerical quasi elastic limit
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 37 (2003) no. 1 , p. 73-90
Zbl 1046.76034 | MR 1972651
doi : 10.1051/m2an:2003019
URL stable :

Classification:  65L60,  65R20,  76P05,  82C40
In this paper we introduce numerical schemes for a one-dimensional kinetic model of the Boltzmann equation with dissipative collisions and variable coefficient of restitution. In particular, we study the numerical passage of the Boltzmann equation with singular kernel to nonlinear friction equations in the so-called quasi elastic limit. To this aim we introduce a Fourier spectral method for the Boltzmann equation [25, 26] and show that the kernel modes that define the spectral method have the correct quasi elastic limit providing a consistent spectral method for the limiting nonlinear friction equation.


[1] D. Benedetto, E. Caglioti and M. Pulvirenti, A kinetic equation for granular media. Math. Mod. Numer. Anal. 31 (1997) 615-641. Numdam | Zbl 0888.73006

[2] D. Benedetto, E. Caglioti, J.A. Carrillo and M. Pulvirenti, A non maxwellian steady distribution for one-dimensional granular media. J. Statist. Phys. 91 (1998) 979-990. Zbl 0921.60057

[3] G.A. Bird, Molecular gas dynamics and direct simulation of gas flows. Clarendon Press, Oxford, UK (1994). MR 1352466

[4] C. Bizon, M.D. Shattuck, J.B. Swift and H.L. Swinney, Transport coefficients from granular media from molecular dynamics simulations. Phys. Rev. E 60 (1999) 4340-4351.

[5] A.V. Bobylev, J.A. Carrillo and I. Gamba, On some properties of kinetic and hydrodynamics equations for inelastic interactions. J. Statist. Phys. 98 (2000) 743-773. Zbl 1056.76071

[6] A.V. Bobylev and K. Nanbu, Theory of collision algorithms for gases and plasmas based on the Boltzmann equation and the Landau-Fokker-Planck equation. Phys. Rev. E 61 (2000) 4576-4586.

[7] N.V. Brilliantov and T. Pöschel, Granular gases the early stage, in Coherent Structures in Classical Systems, Miguel Rubi Ed., Springer (in press). MR 1995113 | Zbl 0992.76080

[8] C. Buet, A discrete-velocity scheme for the Boltzmann operator of rarefied gas dynamics. Transport Theory Statist. Phys. 25 (1996) 33-60. Zbl 0857.76079

[9] J.A. Carrillo, C. Cercignani and I.M. Gamba, Steady states of a Boltzmann equation for driven granular media. Phys. Rev. E 62 (2000) 7700-7707.

[10] J.A. Carrillo, R.J. Mccann and C. Villani, Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates. Rev. Mat. Iberoamericana (to appear). MR 2053570 | Zbl 1073.35127

[11] C. Canuto, M.Y. Hussaini, A. Quarteroni and T.A. Zang, Spectral methods in fluid dynamics. Springer Verlag, New York (1988). MR 917480 | Zbl 0658.76001

[12] 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. Zbl 0755.35091

[13] L. Desvillettes, On asymptotics of the Boltzmann equation when the collisions become grazing. Transport Theory Statist. Phys. 21 (1992) 259-276. Zbl 0769.76059

[14] L. Desvillettes, C. Graham and S. Melehard, Probabilistic interpretation and numerical approximation of a Kac equation without cutoff. Stochastic Process. Appl. 84 (1999) 115-135. Zbl 1009.76081

[15] Y. Du, H. Li and L.P. Kadanoff, Breakdown of hydrodynamics in a one-dimensional system of inelastic particles. Phys. Rev. Lett. 74 (1995) 1268-1271.

[16] F. Filbet and L. Pareschi, A numerical method for the accurate solution of the Fokker-Planck-landau equation in the nonhomogeneous case. J. Comput. Phys 179 (2002) 1-26. Zbl 1003.82011

[17] I. Goldhirsch, Scales and kinetics of granular flows. Chaos 9 (1999) 659-672. Zbl 1055.76569

[18] H. Guérin and S. Méléard, Convergence from Boltzmann to Landau process with soft potential and particle approximations. Preprint PMA 698, Paris VI (2001). MR 1972130 | Zbl 1031.82035

[19] L. Kantorovich, On translation of mass (in Russian). Dokl. AN SSSR 37 (1942) 227-229.

[20] H. Li and G. Toscani, Long-time asymptotics of kinetic models of granular flows. Preprint (2002). Zbl 1116.82025

[21] B. Lucquin-Desreux and S. Mancini, A finite element approximation of grazing collisions. Preprint N. 1034, Laboratoire d'Analyse Numérique, Paris VI (2001). Zbl 1053.82029

[22] S. Mcnamara and W.R. Young, Kinetics of a one-dimensional granular medium in the quasi-elastic limit. Phys. Fluids A 5 (1993) 34-45.

[23] K. Nanbu, Direct simulation scheme derived from the Boltzmann equation. J. Phys. Soc. Japan 49 (1980) 2042-2049.

[24] L. Pareschi, On the fast evaluation of kinetic equations for driven granular flows. Proceedings ENUMATH 2001 (to appear). MR 2360746 | Zbl pre02064924

[25] L. Pareschi and B. Perthame, A Fourier spectral method for homogeneous Boltzmann equations. Transport Theory Statist. Phys. 25 (1996) 369-383. Zbl 0870.76074

[26] L. Pareschi and G. Russo, Numerical solution of the Boltzmann equation. I. Spectrally accurate approximation of the collision operator. SIAM J. Numer. Anal. 37 (2000) 1217-1245. Zbl 1049.76055

[27] L. Pareschi, G. Toscani and C. Villani, Spectral methods for the non cut-off Boltzmann equation and numerical grazing collision limit. Numer. Math. 93 (2003) 527-548. Electronic DOI 10.1007/s002110100384. Zbl 1106.65324

[28] R. Ramírez, T. Pöschel, N.V. Brilliantov and T. Schwager, Coefficient of restitution of colliding viscoelastic spheres. Phys. Rev. E 60 (1999) 4465-4472.

[29] F. Rogier and J. Schneider, A direct method for solving the Boltzmann equation. Transport Theory Statist. Phys. 23 (1994) 313-338. Zbl 0811.76050

[30] G. Toscani, One-dimensional kinetic models of granular flows. ESAIM: M2AN 34 (2000) 1277-1291. Numdam | Zbl 0981.76098

[31] G. Toscani and C. Villani, Probability metrics and uniqueness of the solution to the Boltzmann equation for a Maxwell gas. J. Statist. Phys. 94 (1999) 619-637. Zbl 0958.82044

[32] L.N. Vasershtein, Markov processes on countable product space describing large systems of automata (in Russian). Problemy Peredachi Informatsii 5 (1969) 64-73. Zbl 0273.60054

[33] C. Villani, On a new class of weak solutions for the spatially homogeneous Boltzmann and Landau equations. Arch. Ration. Mech. Anal. 143 (1998) 273-307. Zbl 0912.45011

[34] C. Villani, On the spatially homogeneous Landau equation for Maxwellian molecules. Math. Models Methods Appl. Sci. 8 (1998) 957-983. Zbl 0957.82029