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

Abstract

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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.

How to cite

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Naldi, 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

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  1. D. Benedetto, E. Caglioti and M. Pulvirenti, A kinetic equation for granular media. Math. Mod. Numer. Anal.31 (1997) 615-641.  Zbl0888.73006
  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.  Zbl0921.60057
  3. G.A. Bird, Molecular gas dynamics and direct simulation of gas flows. Clarendon Press, Oxford, UK (1994).  
  4. 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.  
  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.  Zbl1056.76071
  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. E61 (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).  Zbl0992.76080
  8. C. Buet, A discrete-velocity scheme for the Boltzmann operator of rarefied gas dynamics. Transport Theory Statist. Phys.25 (1996) 33-60.  Zbl0857.76079
  9. 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.  
  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).  Zbl1073.35127
  11. C. Canuto, M.Y. Hussaini, A. Quarteroni and T.A. Zang, Spectral methods in fluid dynamics. Springer Verlag, New York (1988).  Zbl0658.76001
  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.  Zbl0755.35091
  13. L. Desvillettes, On asymptotics of the Boltzmann equation when the collisions become grazing. Transport Theory Statist. Phys.21 (1992) 259-276.  Zbl0769.76059
  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.  Zbl1009.76081
  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. Phys179 (2002) 1-26.  Zbl1003.82011
  17. I. Goldhirsch, Scales and kinetics of granular flows. Chaos9 (1999) 659-672.  Zbl1055.76569
  18. H. Guérin and S. Méléard, Convergence from Boltzmann to Landau process with soft potential and particle approximations. Preprint PMA 698, Paris VI (2001).  Zbl1031.82035
  19. L. Kantorovich, On translation of mass (in Russian). Dokl. AN SSSR37 (1942) 227-229.  
  20. H. Li and G. Toscani, Long-time asymptotics of kinetic models of granular flows. Preprint (2002).  Zbl1116.82025
  21. B. Lucquin-Desreux and S. Mancini, A finite element approximation of grazing collisions. Preprint N. 1034, Laboratoire d'Analyse Numérique, Paris VI (2001).  Zbl1053.82029
  22. S. McNamara and W.R. Young, Kinetics of a one-dimensional granular medium in the quasi-elastic limit. Phys. Fluids A5 (1993) 34-45.  
  23. K. Nanbu, Direct simulation scheme derived from the Boltzmann equation. J. Phys. Soc. Japan49 (1980) 2042-2049.  
  24. L. Pareschi, On the fast evaluation of kinetic equations for driven granular flows. Proceedings ENUMATH 2001 (to appear).  Zbl1254.76157
  25. L. Pareschi and B. Perthame, A Fourier spectral method for homogeneous Boltzmann equations. Transport Theory Statist. Phys.25 (1996) 369-383.  Zbl0870.76074
  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.  Zbl1049.76055
  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.  Zbl1106.65324
  28. R. Ramírez, T. Pöschel, N.V. Brilliantov and T. Schwager, Coefficient of restitution of colliding viscoelastic spheres. Phys. Rev. E60 (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.  Zbl0811.76050
  30. G. Toscani, One-dimensional kinetic models of granular flows. ESAIM: M2AN34 (2000) 1277-1291.  Zbl0981.76098
  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.  Zbl0958.82044
  32. L.N. Vasershtein, Markov processes on countable product space describing large systems of automata (in Russian). Problemy Peredachi Informatsii5 (1969) 64-73.  
  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.  Zbl0912.45011
  34. C. Villani, On the spatially homogeneous Landau equation for Maxwellian molecules. Math. Models Methods Appl. Sci.8 (1998) 957-983.  Zbl0957.82029

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