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, Volume 37 (2003) no. 1, pp. 73-90.

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.

DOI: 10.1051/m2an:2003019
Classification: 65L60,  65R20,  76P05,  82C40
Keywords: Boltzmann equation, granular media, spectral methods, singular integrals, nonlinear friction equation, quasi elastic limit
@article{M2AN_2003__37_1_73_0,
author = {Naldi, Giovanni and Pareschi, Lorenzo and Toscani, Giuseppe},
title = {Spectral methods for one-dimensional kinetic models of granular flows and numerical quasi elastic limit},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
pages = {73--90},
publisher = {EDP-Sciences},
volume = {37},
number = {1},
year = {2003},
doi = {10.1051/m2an:2003019},
zbl = {1046.76034},
mrnumber = {1972651},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an:2003019/}
}
TY  - JOUR
AU  - Naldi, Giovanni
AU  - Pareschi, Lorenzo
AU  - Toscani, Giuseppe
TI  - Spectral methods for one-dimensional kinetic models of granular flows and numerical quasi elastic limit
JO  - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY  - 2003
DA  - 2003///
SP  - 73
EP  - 90
VL  - 37
IS  - 1
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an:2003019/
UR  - https://zbmath.org/?q=an%3A1046.76034
UR  - https://www.ams.org/mathscinet-getitem?mr=1972651
UR  - https://doi.org/10.1051/m2an:2003019
DO  - 10.1051/m2an:2003019
LA  - en
ID  - M2AN_2003__37_1_73_0
ER  - 
%0 Journal Article
%A Naldi, Giovanni
%A Pareschi, Lorenzo
%A Toscani, Giuseppe
%T Spectral methods for one-dimensional kinetic models of granular flows and numerical quasi elastic limit
%J ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
%D 2003
%P 73-90
%V 37
%N 1
%I EDP-Sciences
%U https://doi.org/10.1051/m2an:2003019
%R 10.1051/m2an:2003019
%G en
%F M2AN_2003__37_1_73_0
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, Volume 37 (2003) no. 1, pp. 73-90. doi : 10.1051/m2an:2003019. http://www.numdam.org/articles/10.1051/m2an:2003019/

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

[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

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

[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

[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 | Zbl

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

[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 | Zbl

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

[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

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

[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

[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

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

[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 | Zbl

[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

[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

[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

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

[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

[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

[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

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

[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

[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

[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

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

Cited by Sources: