Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Analysis of spectral methods for the homogeneous Boltzmann equation


Authors: Francis Filbet and Clément Mouhot
Journal: Trans. Amer. Math. Soc. 363 (2011), 1947-1980
MSC (2000): Primary 82C40, 65M70, 76P05
DOI: https://doi.org/10.1090/S0002-9947-2010-05303-6
Published electronically: November 15, 2010
MathSciNet review: 2746671
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The development of accurate and fast algorithms for the Boltzmann collision integral and their analysis represent a challenging problem in scientific computing and numerical analysis. Recently, several works were devoted to the derivation of spectrally accurate schemes for the Boltzmann equation, but very few of them were concerned with the stability analysis of the method. In particular there was no result of stability except when the method was modified in order to enforce the positivity preservation, which destroys the spectral accuracy. In this paper we propose a new method to study the stability of homogeneous Boltzmann equations perturbed by smoothed balanced operators which do not preserve positivity of the distribution. This method takes advantage of the ``spreading'' property of the collision, together with estimates on regularity and entropy production. As an application we prove stability and convergence of spectral methods for the Boltzmann equation, when the discretization parameter is large enough (with explicit bound).


References [Enhancements On Off] (What's this?)

  • 1. Bird, G. A.:
    Molecular gas dynamics.
    Clarendon Press, Oxford (1994).
  • 2. Bobylev, A. V.:
    The theory of the nonlinear spatially uniform Boltzmann equation for Maxwell molecules.
    Math. Phys. Reviews, vol. 7 (1988) pp. 111-233. MR 1128328 (92m:82112)
  • 3. Bobylev, A. V. and Rjasanow, S.:
    Difference scheme for the Boltzmann equation based on the fast Fourier transform.
    European J. Mech. B Fluids 16 (1997) pp. 293-306. MR 1439069 (98c:82057)
  • 4. Bobylev, A. V. and Rjasanow, S.:
    Fast deterministic method of solving the Boltzmann equation for hard spheres.
    Eur. J. Mech. B Fluids 18 (1999) pp. 869-887. MR 1728639 (2001c:76109)
  • 5. Bolley, F., Villani, C.:
    Weighted Csiszár-Kullback-Pinsker inequalities and applications to transportation inequalities.
    Ann. Fac. Sci. Toulouse Math. (6) 14 (2005) pp. 331-352. MR 2172583 (2006j:60018)
  • 6. Canuto, C., Hussaini, M. Y., Quarteroni, A. and Zang, T. A.:
    Spectral methods in fluid dynamics.
    Springer Series in Computational Physics, Springer-Verlag, New York 1988). MR 917480 (89m:76004)
  • 7. Carleman T.:
    Sur la théorie de l'équation intégrodifférentielle de Boltzmann.
    Acta Math. 60 (1932).
  • 8. Cercignani, C., Illner, R. and Pulvirenti, M.:
    The Mathematical Theory of Dilute Gases.
    Appl. Math. Sci. 106, Springer-Verlag, New York (1994). MR 1307620 (96g:82046)
  • 9. Fainsilber, L., Kurlberg, P. and Wennberg, B.;
    Lattice points on circles and discrete velocity models for the Boltzmann equation.
    SIAM J. Math. Anal. 37 (2006) pp. 1903-1922. MR 2213399 (2007f:11116)
  • 10. Filbet, F. and Pareschi, L.:
    A numerical method for the accurate solution of the Fokker-Planck-Landau equation in the non homogeneous case.
    J. Comput. Phys. 179 (2002) pp. 1-26. MR 1906573 (2003c:82085)
  • 11. Filbet, F. and Russo, G.:
    High order numerical methods for the space non-homogeneous Boltzmann equation.
    J. Comput. Phys. 186 (2003) pp. 457-480. MR 1973198 (2004c:82116)
  • 12. Filbet, F. and Russo, G.:
    Accurate numerical methods for the Boltzmann equation.
    Modeling and computational methods for kinetic equations, pp. 117-145, Model. Simul. Sci. Eng. Technol., Birkhäuser Boston, Boston, MA 2004. MR 2068582 (2005d:82117)
  • 13. Filbet, F., Pareschi, L. and Toscani, G.:
    Accurate numerical methods for the collisional motion of (heated) granular flows.
    J. Comput. Phys. 202 (2005) pp. 216-235. MR 2102883 (2005g:82122)
  • 14. Filbet, F., Mouhot, C. and Pareschi, L.:
    Solving the Boltzmann equation in $ N\log\sb 2N$.
    SIAM J. Sci. Comput. 28 (2006) pp. 1029-1053 MR 2240802 (2007c:82076)
  • 15. Ibragimov, I. and Rjasanow, S.:
    Numerical solution of the Boltzmann equation on the uniform grid.
    Computing 69 (2002) pp. 163-186. MR 1954793 (2004i:82060)
  • 16. Lions, P.-L.:
    Compactness in Boltzmann's equation via Fourier integral operators and applications, I.
    J. Math. Kyoto Univ. 34 (1994) pp. 391-427. MR 1284432 (96f:35133)
  • 17. Lions, P.-L.:
    Compactness in Boltzmann's equation via Fourier integral operators and applications, II.
    J. Math. Kyoto Univ. 34 (1994) pp. 429-461. MR 1284432 (96f:35133)
  • 18. Mischler, S. and Mouhot, C.:
    Cooling process for inelastic Boltzmann equations for hard spheres. II. Self-similar solutions and tail behavior.
    J. Stat. Phys. 124 (2006) pp. 703-746. MR 2264623 (2007g:82054)
  • 19. Mouhot, C.:
    Quantitative lower bounds for the full Boltzmann equation. I. Periodic boundary conditions.
    Comm. Partial Differential Equations 30 (2005) pp. 881-917. MR 2153518 (2006a:76096)
  • 20. Mouhot, C. and Villani, C.:
    Regularity theory for the spatially homogeneous Boltzmann equation with cut-off.
    Arch. Rational Mech. Anal. 173 (2004) pp. 169-212. MR 2081030 (2006f:82073)
  • 21. Mouhot, C. and Pareschi, L.:
    Fast algorithms for computing the Boltzmann collision operator.
    Math. Comp. 75 (2006) pp. 1833-1852. MR 2240637 (2007d:65095)
  • 22. Nanbu, K.:
    Direct simulation scheme derived from the Boltzmann equation. I. Monocomponent Gases.
    J. Phys. Soc. Japan 52 (1983) pp. 2042-2049.
  • 23. Palczewski, A., Schneider, J. and Bobylev, A. V.:
    A consistency result for a discrete-velocity model of the Boltzmann equation.
    SIAM J. Numer. Anal. 34 (1997) pp. 1865-1883. MR 1472201 (99d:82068)
  • 24. Palczewski, A. and Schneider, J.:
    Existence, stability, and convergence of solutions of discrete velocity models to the Boltzmann equation.
    J. Statist. Phys. 91 (1998) pp. 307-326. MR 1632506 (99g:82072)
  • 25. Panferov, V. A. and Heintz, A. G.:
    A new consistent discrete-velocity model for the Boltzmann equation.
    Math. Methods Appl. Sci. 25 (2002) pp. 571-593. MR 1895119 (2003d:82100)
  • 26. Pareschi, L. and Perthame, B.:
    A Fourier spectral method for homogeneous Boltzmann equations.
    Trans. Theo. Stat. Phys. 25 (1996) pp. 369-382. MR 1407541 (97j:82133)
  • 27. Pareschi, L. and Russo, G.:
    Numerical solution of the Boltzmann equation I. Spectrally accurate approximation of the collision operator.
    SIAM J. Numer. Anal. 37 (2000) pp. 1217-1245. MR 1756425 (2001g:65175)
  • 28. Pareschi, L., Russo, G. and Toscani, G.:
    Fast spectral methods for the Fokker-Planck-Landau collision operator.
    J. Comput. Phys. 165 (2000) pp. 216-236. MR 1795398 (2001i:65112)
  • 29. Pareschi, L. and Russo, G.:
    On the stability of spectral methods for the homogeneous Boltzmann equation.
    Trans. Theo. Stat. Phys. 29 (2000) pp. 431-447. MR 1770438 (2001c:82070)
  • 30. Pareschi, L., Toscani, G. and Villani, C.:
    Spectral methods for the non cut-off Boltzmann equation and numerical grazing collision limit, Numer. Math. 93 (2003) pp. 527-548. MR 1953752 (2003k:65128)
  • 31. Pulvirenti, A. and Wennberg, B.:
    A Maxwellian lower bound for solutions to the Boltzmann equation.
    Comm. Math. Phys. 183 (1997) pp. 145-160. MR 1461954 (99f:82056)
  • 32. Toscani, G. and Villani, C.:
    Sharp entropy dissipation bounds and explicit rate of trend to equilibrium for the spatially homogeneous Boltzmann equation,
    Commun. Math. Phys. 203 (1999) pp. 667-706. MR 1700142 (2000e:82039)
  • 33. Villani, C.:
    A survey of mathematical topics in kinetic theory.
    Handbook of fluid mechanics, S. Friedlander and D. Serre, Eds. Elsevier Publ. (2002).
  • 34. Villani, C.:
    Cercignani's conjecture is sometimes true and always almost true.
    Comm. Math. Phys. 234 (2003) pp. 455-490. MR 1964379 (2004b:82048)
  • 35. Wennberg, B.;
    Regularity in the Boltzmann equation and the Radon transform.
    Comm. Partial Differential Equations 19 (1994), pp. 2057-2074. MR 1301182 (95k:76103)
  • 36. Wennberg, B.:
    The geometry of binary collisions and generalized Radon transforms.
    Arch. Rational Mech. Anal. 139 (1997) pp. 291-302. MR 1480243 (98k:82166)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 82C40, 65M70, 76P05

Retrieve articles in all journals with MSC (2000): 82C40, 65M70, 76P05


Additional Information

Francis Filbet
Affiliation: Université de Lyon, Institut Camille Jordan, CNRS UMR 5208, Université de Lyon 1, INSAL, ECL, 43 boulevard 11 novembre 1918, F-69622 Villeurbanne cedex, France
Email: filbet@math.univ-lyon1.fr

Clément Mouhot
Affiliation: Centre for Mathematical Sciences, University of Cambridge, DAMTP, Wilberforce Road, Cambridge CB3 0WA, England
Address at time of publication: CNRS & École Normale Supérieure, DMA, UMR CNRS 8553, 45, rue d’Ulm, F-75230 Paris cedex 05, FRANCE
Email: Clement.Mouhot@ens.fr

DOI: https://doi.org/10.1090/S0002-9947-2010-05303-6
Keywords: Boltzmann equation, spectral methods, numerical stability, asymptotic stability, Fourier-Galerkin method
Received by editor(s): December 2, 2008
Published electronically: November 15, 2010
Additional Notes: The first author would like to express his gratitude to the ANR JCJC-0136 MNEC (Méthode Numérique pour les Equations Cinétiques) and to the ERC StG #239983 (NuSiKiMo) for funding.
The second author would like to thank Cambridge University who provided repeated hospitality in 2009 thanks to the Award No. KUK-I1-007-43, funded by the King Abdullah University of Science and Technology (KAUST)
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society