Hostname: page-component-77c89778f8-7drxs Total loading time: 0 Render date: 2024-07-21T19:48:33.555Z Has data issue: false hasContentIssue false
  • English
  • Français

Spectral methods for one-dimensional kinetic models ofgranular flows and numerical quasi elastic limit

Published online by Cambridge University Press:  15 March 2003

Giovanni Naldi ,
Lorenzo Pareschi  and
Giuseppe Toscani
Giovanni Naldi
Affiliation:
Department of Mathematics and Applications, University of Milano-Bicocca, Milano, Italy. naldi@matapp.unimib.it.
Lorenzo Pareschi
Affiliation:
Department of Mathematics, University of Ferrara, Via Machiavelli 35, 44100 Ferrara, Italy. pareschi@dm.unife.it.
Giuseppe Toscani
Affiliation:
Department of Mathematics, University of Pavia, Via Ferrata 1, 27100 Pavia, Italy. toscani@dimat.unipv.it.
Get access

Abstract

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 [CITE] 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.

Keywords

Boltzmann equationgranular mediaspectral methodssingular integralsnonlinear friction equationquasi elastic limit.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 37 , Issue 1 , January 2003 , pp. 73 - 90
Copyright
© EDP Sciences, SMAI, 2003

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Benedetto, D., Caglioti, E. and Pulvirenti, M., A kinetic equation for granular media. Math. Mod. Numer. Anal. 31 (1997) 615-641. CrossRef
Benedetto, D., Caglioti, E., Carrillo, J.A. and Pulvirenti, M., A non maxwellian steady distribution for one-dimensional granular media. J. Statist. Phys. 91 (1998) 979-990. CrossRef
G.A. Bird, Molecular gas dynamics and direct simulation of gas flows. Clarendon Press, Oxford, UK (1994).
Bizon, C., Shattuck, M.D., Swift, J.B. and Swinney, H.L., Transport coefficients from granular media from molecular dynamics simulations. Phys. Rev. E 60 (1999) 4340-4351. CrossRef
Bobylev, A.V., Carrillo, J.A. and Gamba, I., On some properties of kinetic and hydrodynamics equations for inelastic interactions. J. Statist. Phys. 98 (2000) 743-773. CrossRef
Bobylev, A.V. and Nanbu, K., 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. CrossRef
N.V. Brilliantov and T. Pöschel, Granular gases the early stage, in Coherent Structures in Classical Systems, Miguel Rubi Ed., Springer (in press).
Buet, C., A discrete-velocity scheme for the Boltzmann operator of rarefied gas dynamics. Transport Theory Statist. Phys. 25 (1996) 33-60. CrossRef
Carrillo, J.A., Cercignani, C. and Gamba, I.M., Steady states of a Boltzmann equation for driven granular media. Phys. Rev. E 62 (2000) 7700-7707. CrossRef
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).
C. Canuto, M.Y. Hussaini, A. Quarteroni and T.A. Zang, Spectral methods in fluid dynamics. Springer Verlag, New York (1988).
Degond, P. and Lucquin-Desreux, B., The Fokker-Planck asymptotics of the Boltzmann collision operator in the Coulomb case. Math. Models Methods Appl. Sci. 2 (1992) 167-182. CrossRef
Desvillettes, L., On asymptotics of the Boltzmann equation when the collisions become grazing. Transport Theory Statist. Phys. 21 (1992) 259-276. CrossRef
Desvillettes, L., Graham, C. and Melehard, S., Probabilistic interpretation and numerical approximation of a Kac equation without cutoff. Stochastic Process. Appl. 84 (1999) 115-135. CrossRef
Du, Y., Li, H. and Kadanoff, L.P., Breakdown of hydrodynamics in a one-dimensional system of inelastic particles. Phys. Rev. Lett. 74 (1995) 1268-1271. CrossRef
Filbet, F. and Pareschi, L., A numerical method for the accurate solution of the Fokker-Planck-landau equation in the nonhomogeneous case. J. Comput. Phys 179 (2002) 1-26. CrossRef
Goldhirsch, I., Scales and kinetics of granular flows. Chaos 9 (1999) 659-672. CrossRef
H. Guérin and S. Méléard, Convergence from Boltzmann to Landau process with soft potential and particle approximations. Preprint PMA 698, Paris VI (2001).
Kantorovich, L., On translation of mass (in Russian). Dokl. AN SSSR 37 (1942) 227-229.
H. Li and G. Toscani, Long-time asymptotics of kinetic models of granular flows. Preprint (2002).
B. Lucquin-Desreux and S. Mancini, A finite element approximation of grazing collisions. Preprint N. 1034, Laboratoire d'Analyse Numérique, Paris VI (2001).
McNamara, S. and Young, W.R., Kinetics of a one-dimensional granular medium in the quasi-elastic limit. Phys. Fluids A 5 (1993) 34-45. CrossRef
Nanbu, K., Direct simulation scheme derived from the Boltzmann equation. J. Phys. Soc. Japan 49 (1980) 2042-2049. CrossRef
L. Pareschi, On the fast evaluation of kinetic equations for driven granular flows. Proceedings ENUMATH 2001 (to appear).
Pareschi, L. and Perthame, B., Fourier, A spectral method for homogeneous Boltzmann equations. Transport Theory Statist. Phys. 25 (1996) 369-383. CrossRef
Pareschi, L. and Russo, G., Numerical solution of the Boltzmann equation. I. Spectrally accurate approximation of the collision operator. SIAM J. Numer. Anal. 37 (2000) 1217-1245. CrossRef
Pareschi, L., Toscani, G. and Villani, C., 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. CrossRef
Ramírez, R., Pöschel, T., Brilliantov, N.V. and Schwager, T., Coefficient of restitution of colliding viscoelastic spheres. Phys. Rev. E 60 (1999) 4465-4472. CrossRef
Rogier, F. and Schneider, J., A direct method for solving the Boltzmann equation. Transport Theory Statist. Phys. 23 (1994) 313-338. CrossRef
Toscani, G., One-dimensional kinetic models of granular flows. ESAIM: M2AN 34 (2000) 1277-1291. CrossRef
Toscani, G. and Villani, C., Probability metrics and uniqueness of the solution to the Boltzmann equation for a Maxwell gas. J. Statist. Phys. 94 (1999) 619-637. CrossRef
Vasershtein, L.N., Markov processes on countable product space describing large systems of automata (in Russian). Problemy Peredachi Informatsii 5 (1969) 64-73.
Villani, C., On a new class of weak solutions for the spatially homogeneous Boltzmann and Landau equations. Arch. Ration. Mech. Anal. 143 (1998) 273-307. CrossRef
Villani, C., On the spatially homogeneous Landau equation for Maxwellian molecules. Math. Models Methods Appl. Sci. 8 (1998) 957-983. CrossRef