Abstract
In this work we discuss the development of fast algorithms for the inelastic Boltzmann equation describing the collisional motion of a granular gas. In such systems the collisions between particles occur in an inelastic way and are characterized by a coefficient of restitution which in the general case depends on the relative velocity of the collision. In the quasi-elastic approximation the granular operator is replaced by the sum of an elastic Boltzmann operator and a nonlinear friction term. Fast numerical methods based on a suitable spectral representation of the approximated model are then presented.
The financial support of the European network HYKE, funded by the EC as contract HPRN-CT-2002-00282 and of the project NUMSTAT 2005, Comitato dei Sostenitori, funded by the University of Ferrara is acknowledged.
Chapter PDF
Similar content being viewed by others
References
N. Bellomo, M. Esteban, M. Lachowicz. Nonlinear kinetic equations with dissipative collisions. Appl. Math. Letters 8:46–52, 1995.
D. Benedetto, E. Caglioti, M. Pulvirenti. A kinetic equation for granular media. M2AN Math. Model. Numer. Anal. 31:615–641, 1997.
D. Benedetto, E. Caglioti, J.A. Carrillo, M. Pulvirenti. A non-maxwellian steady distribution for one-dimensional granular media. J. Statist. Phys. 91:979–990, 1998.
A.V. Bobylev, J.A. Carrillo, I.M. Gamba. On some properties of kinetic and hydrodynamic equations for inelastic interactions J. Statist. Phys. 98:743–773, 2000.
N.V. Brilliantov, T. Pöschel. Granular gases with impact-velocity dependent restitution coefficient. In Granular Gases: 100–124, Lecture Notes in Physics, Vol. 564, Springer-Verlag, Berlin, 2000.
J.A. Carrillo, C. Cercignani, I.M. Gamba. Steady states of a Boltzmann equation for driven granular media. Phys. Rev. E (3) 62:7700–7707, 2000.
C. Cercignani. Recent developments in the mechanism of granular materials. Fisica Matematica e ingegneria delle strutture, Pitagora Editrice, Bologna, 1995.
C. Cercignani, R. Illner, M. Pulvirenti. The mathematical theory of dilute gases. Applied Mathematical Sciences, Vol. 106, Springer-Verlag, New-York, 1994.
F. Filbet, C. Mouhot, L. Pareschi. Solving the Boltzmann equation in N log2 N. SISC (to appear), 2005.
F. Filbet, C. Mouhot, L. Pareschi. Work in progress.
I. Goldhirsch. Scales and kinetics of granular flows. Chaos 9:659–672, 1999.
S. McNamara, W.R. Young. Kinetics of a one-dimensional granular medium in the quasielastic limit. Phys. Fluids A 5:34–45, 1993.
C. Mouhot, L. Pareschi. Fast algorithms for computing the Boltzmann collision operator. Math. Comp. (to appear), 2005.
G. Naldi, L. Pareschi, G. Toscani. Spectral methods for one-dimensional kinetic models of granular flows and numerical quasi elastic limit. M2AN Math. Model. Numer. Anal. 37:73–90,2003.
L. Pareschi, G. Russo. Numerical solution of the Boltzmann equation. Spectrally accurate approximation of the collision operator. SIAM J. Numer. Anal. 37:1217–1245, 2000.
L. Pareschi, G. Toscani. Modelling and numerics of granular gases. In Modeling and computational methods for kinetic equations: 259–285, Series Model. Simul. Sci. Eng. Technol., Birkhauser, Boston, 2004.
G. Toscani. One-dimensional kinetic models of granular flows, M2AN Math. Model. Numer. Anal. 34:1277–1292, 2000.
G. Toscani. Kinetic and hydrodinamic models of nearly elastic granular flows, Monatsch. Math. 142:179–192, 2004.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 International Federation for Information Processing
About this paper
Cite this paper
Ferrari, E., Naldi, G., Toscani, G. (2006). Modelling and Fast Numerical Methods for Granular Flows. In: Ceragioli, F., Dontchev, A., Furuta, H., Marti, K., Pandolfi, L. (eds) Systems, Control, Modeling and Optimization. CSMO 2005. IFIP International Federation for Information Processing, vol 202. Springer, Boston, MA . https://doi.org/10.1007/0-387-33882-9_14
Download citation
DOI: https://doi.org/10.1007/0-387-33882-9_14
Publisher Name: Springer, Boston, MA
Print ISBN: 978-0-387-33881-1
Online ISBN: 978-0-387-33882-8
eBook Packages: Computer ScienceComputer Science (R0)