# Spectral methods for one-dimensional kinetic models of granular flows and numerical quasi elastic limit

Giovanni Naldi; Lorenzo Pareschi; Giuseppe Toscani

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

- Volume: 37, Issue: 1, page 73-90
- ISSN: 0764-583X

## Access Full Article

top## Abstract

top## How to cite

topNaldi, Giovanni, Pareschi, Lorenzo, and Toscani, Giuseppe. "Spectral methods for one-dimensional kinetic models of granular flows and numerical quasi elastic limit." ESAIM: Mathematical Modelling and Numerical Analysis 37.1 (2010): 73-90. <http://eudml.org/doc/194157>.

@article{Naldi2010,

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.
},

author = {Naldi, Giovanni, Pareschi, Lorenzo, Toscani, Giuseppe},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Boltzmann equation;
granular media; spectral methods; singular integrals; nonlinear
friction equation; quasi elastic limit.; Fourier method; nonlinear friction equation},

language = {eng},

month = {3},

number = {1},

pages = {73-90},

publisher = {EDP Sciences},

title = {Spectral methods for one-dimensional kinetic models of granular flows and numerical quasi elastic limit},

url = {http://eudml.org/doc/194157},

volume = {37},

year = {2010},

}

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

DA - 2010/3//

PB - EDP Sciences

VL - 37

IS - 1

SP - 73

EP - 90

AB -
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.

LA - eng

KW - Boltzmann equation;
granular media; spectral methods; singular integrals; nonlinear
friction equation; quasi elastic limit.; Fourier method; nonlinear friction equation

UR - http://eudml.org/doc/194157

ER -

## References

top- D. Benedetto, E. Caglioti and M. Pulvirenti, A kinetic equation for granular media. Math. Mod. Numer. Anal.31 (1997) 615-641. Zbl0888.73006
- 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. Zbl0921.60057
- G.A. Bird, Molecular gas dynamics and direct simulation of gas flows. Clarendon Press, Oxford, UK (1994).
- C. Bizon, M.D. Shattuck, J.B. Swift and H.L. Swinney, Transport coefficients from granular media from molecular dynamics simulations. Phys. Rev. E60 (1999) 4340-4351.
- 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. Zbl1056.76071
- 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. E61 (2000) 4576-4586.
- N.V. Brilliantov and T. Pöschel, Granular gases the early stage, in Coherent Structures in Classical Systems, Miguel Rubi Ed., Springer (in press). Zbl0992.76080
- C. Buet, A discrete-velocity scheme for the Boltzmann operator of rarefied gas dynamics. Transport Theory Statist. Phys.25 (1996) 33-60. Zbl0857.76079
- J.A. Carrillo, C. Cercignani and I.M. Gamba, Steady states of a Boltzmann equation for driven granular media. Phys. Rev. E62 (2000) 7700-7707.
- 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). Zbl1073.35127
- C. Canuto, M.Y. Hussaini, A. Quarteroni and T.A. Zang, Spectral methods in fluid dynamics. Springer Verlag, New York (1988). Zbl0658.76001
- 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. Zbl0755.35091
- L. Desvillettes, On asymptotics of the Boltzmann equation when the collisions become grazing. Transport Theory Statist. Phys.21 (1992) 259-276. Zbl0769.76059
- 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. Zbl1009.76081
- 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.
- F. Filbet and L. Pareschi, A numerical method for the accurate solution of the Fokker-Planck-landau equation in the nonhomogeneous case. J. Comput. Phys179 (2002) 1-26. Zbl1003.82011
- I. Goldhirsch, Scales and kinetics of granular flows. Chaos9 (1999) 659-672. Zbl1055.76569
- H. Guérin and S. Méléard, Convergence from Boltzmann to Landau process with soft potential and particle approximations. Preprint PMA 698, Paris VI (2001). Zbl1031.82035
- L. Kantorovich, On translation of mass (in Russian). Dokl. AN SSSR37 (1942) 227-229.
- H. Li and G. Toscani, Long-time asymptotics of kinetic models of granular flows. Preprint (2002). Zbl1116.82025
- B. Lucquin-Desreux and S. Mancini, A finite element approximation of grazing collisions. Preprint N. 1034, Laboratoire d'Analyse Numérique, Paris VI (2001). Zbl1053.82029
- S. McNamara and W.R. Young, Kinetics of a one-dimensional granular medium in the quasi-elastic limit. Phys. Fluids A5 (1993) 34-45.
- K. Nanbu, Direct simulation scheme derived from the Boltzmann equation. J. Phys. Soc. Japan49 (1980) 2042-2049.
- L. Pareschi, On the fast evaluation of kinetic equations for driven granular flows. Proceedings ENUMATH 2001 (to appear). Zbl1254.76157
- L. Pareschi and B. Perthame, A Fourier spectral method for homogeneous Boltzmann equations. Transport Theory Statist. Phys.25 (1996) 369-383. Zbl0870.76074
- 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. Zbl1049.76055
- 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. Zbl1106.65324
- R. Ramírez, T. Pöschel, N.V. Brilliantov and T. Schwager, Coefficient of restitution of colliding viscoelastic spheres. Phys. Rev. E60 (1999) 4465-4472.
- F. Rogier and J. Schneider, A direct method for solving the Boltzmann equation. Transport Theory Statist. Phys.23 (1994) 313-338. Zbl0811.76050
- G. Toscani, One-dimensional kinetic models of granular flows. ESAIM: M2AN34 (2000) 1277-1291. Zbl0981.76098
- 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. Zbl0958.82044
- L.N. Vasershtein, Markov processes on countable product space describing large systems of automata (in Russian). Problemy Peredachi Informatsii5 (1969) 64-73.
- 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. Zbl0912.45011
- C. Villani, On the spatially homogeneous Landau equation for Maxwellian molecules. Math. Models Methods Appl. Sci.8 (1998) 957-983. Zbl0957.82029

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.