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.
@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},
volume = {37},
year = {2003},
pages = {73-90},
doi = {10.1051/m2an:2003019},
mrnumber = {1972651},
zbl = {1046.76034},
language = {en},
url = {http://dml.mathdoc.fr/item/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, Tome 37 (2003) pp. 73-90. doi : 10.1051/m2an:2003019. http://gdmltest.u-ga.fr/item/M2AN_2003__37_1_73_0/
[1] , and, A kinetic equation for granular media. Math. Mod. Numer. Anal. 31 (1997) 615-641. | Numdam | Zbl 0888.73006
[2] ,, and, A non maxwellian steady distribution for one-dimensional granular media. J. Statist. Phys. 91 (1998) 979-990. | Zbl 0921.60057
[3] , Molecular gas dynamics and direct simulation of gas flows. Clarendon Press, Oxford, UK (1994). | MR 1352466
[4] ,, and, Transport coefficients from granular media from molecular dynamics simulations. Phys. Rev. E 60 (1999) 4340-4351.
[5] , and, On some properties of kinetic and hydrodynamics equations for inelastic interactions. J. Statist. Phys. 98 (2000) 743-773. | Zbl 1056.76071
[6] and, 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] and, Granular gases the early stage, in Coherent Structures in Classical Systems, Miguel Rubi Ed., Springer (in press). | MR 1995113 | Zbl 0992.76080
[8] , A discrete-velocity scheme for the Boltzmann operator of rarefied gas dynamics. Transport Theory Statist. Phys. 25 (1996) 33-60. | Zbl 0857.76079
[9] , and, Steady states of a Boltzmann equation for driven granular media. Phys. Rev. E 62 (2000) 7700-7707.
[10] , and, 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] ,, and, Spectral methods in fluid dynamics. Springer Verlag, New York (1988). | MR 917480 | Zbl 0658.76001
[12] and, 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] , On asymptotics of the Boltzmann equation when the collisions become grazing. Transport Theory Statist. Phys. 21 (1992) 259-276. | Zbl 0769.76059
[14] , and, Probabilistic interpretation and numerical approximation of a Kac equation without cutoff. Stochastic Process. Appl. 84 (1999) 115-135. | Zbl 1009.76081
[15] , and, Breakdown of hydrodynamics in a one-dimensional system of inelastic particles. Phys. Rev. Lett. 74 (1995) 1268-1271.
[16] and, 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] , Scales and kinetics of granular flows. Chaos 9 (1999) 659-672. | Zbl 1055.76569
[18] and, Convergence from Boltzmann to Landau process with soft potential and particle approximations. Preprint PMA 698, Paris VI (2001). | MR 1972130 | Zbl 1031.82035
[19] , On translation of mass (in Russian). Dokl. AN SSSR 37 (1942) 227-229.
[20] and, Long-time asymptotics of kinetic models of granular flows. Preprint (2002). | Zbl 1116.82025
[21] and, A finite element approximation of grazing collisions. Preprint N. 1034, Laboratoire d'Analyse Numérique, Paris VI (2001). | Zbl 1053.82029
[22] and, Kinetics of a one-dimensional granular medium in the quasi-elastic limit. Phys. Fluids A 5 (1993) 34-45.
[23] , Direct simulation scheme derived from the Boltzmann equation. J. Phys. Soc. Japan 49 (1980) 2042-2049.
[24] , On the fast evaluation of kinetic equations for driven granular flows. Proceedings ENUMATH 2001 (to appear). | MR 2360746 | Zbl pre02064924
[25] and, A Fourier spectral method for homogeneous Boltzmann equations. Transport Theory Statist. Phys. 25 (1996) 369-383. | Zbl 0870.76074
[26] and, 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] , and, 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] ,, and, Coefficient of restitution of colliding viscoelastic spheres. Phys. Rev. E 60 (1999) 4465-4472.
[29] and, A direct method for solving the Boltzmann equation. Transport Theory Statist. Phys. 23 (1994) 313-338. | Zbl 0811.76050
[30] , One-dimensional kinetic models of granular flows. ESAIM: M2AN 34 (2000) 1277-1291. | Numdam | Zbl 0981.76098
[31] and, 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] , Markov processes on countable product space describing large systems of automata (in Russian). Problemy Peredachi Informatsii 5 (1969) 64-73. | Zbl 0273.60054
[33] , 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] , On the spatially homogeneous Landau equation for Maxwellian molecules. Math. Models Methods Appl. Sci. 8 (1998) 957-983. | Zbl 0957.82029