Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

   
 
 

 

Fast algorithms for computing the Boltzmann collision operator


Authors: Clément Mouhot and Lorenzo Pareschi
Journal: Math. Comp. 75 (2006), 1833-1852
MSC (2000): Primary 65T50, 68Q25, 74S25, 76P05
DOI: https://doi.org/10.1090/S0025-5718-06-01874-6
Published electronically: July 12, 2006
MathSciNet review: 2240637
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The development of accurate and fast numerical schemes for the five-fold Boltzmann collision integral represents a challenging problem in scientific computing. For a particular class of interactions, including the so-called hard spheres model in dimension three, we are able to derive spectral methods that can be evaluated through fast algorithms. These algorithms are based on a suitable representation and approximation of the collision operator. Explicit expressions for the errors in the schemes are given and spectral accuracy is proved. Parallelization properties and adaptivity of the algorithms are also discussed.


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

  • 1. L. Arkeryd, On the Boltzmann equation, Arch. Rational Mech. Anal. 45 (1972), 1-34. MR 0339665 (49:4423); MR 0339666 (49:4424)
  • 2. Dario Benedetto, Emanuele Caglioti, and Mario Pulvirenti, A kinetic equation for granular media, M2AN Math. Model. Numer. Anal. 31 (1997), 615-641.MR 1471181 (98k:82145)
  • 3. -, Erratum: ``A kinetic equation for granular media'', M2AN Math. Model. Numer. Anal. 33 (1999), no. 2, 439-441.MR 1700043 (2000f:82070)
  • 4. G. A. Bird, Molecular gas dynamics and the direct simulation of gas flows, Oxford Engineering Science Series, vol. 42, The Clarendon Press Oxford University Press, New York, 1994, 1994. MR 1352466 (97e:76078)
  • 5. A. Bobylev and S. Rjasanow, Difference scheme for the Boltzmann equation based on the fast Fourier transform, European J. Mech. B Fluids 16 (1997), no. 2, 293-306. MR 1439069 (98c:82057)
  • 6. A. V. Bobylëv, The theory of the nonlinear spatially uniform Boltzmann equation for Maxwell molecules, Mathematical physics reviews, Vol. 7, Harwood Academic Publ., Chur, 1988, pp. 111-233. MR 1128328 (92m:82112) 15pt
  • 7. A. V. Bobylev and S. Rjasanow, Fast deterministic method of solving the Boltzmann equation for hard spheres, Eur. J. Mech. B Fluids 18 (1999), no. 5, 869-887. MR 1728639 (2001c:76109)
  • 8. -, Numerical solution of the Boltzmann equation using a fully conservative difference scheme based on the fast Fourier transform, Transport Theory Statist. Phys. 29 (2000), no. 3-5, 289-310.MR 1770434 (2001g:82085)
  • 9. Alexandre Vasiljévitch Bobylev, Andrzej Palczewski, and Jacques Schneider, On approximation of the Boltzmann equation by discrete velocity models, C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), no. 5, 639-644.MR 1322351 (96g:82045)
  • 10. C Buet, A discrete velocity scheme for the Boltzmann operator of rarefied gas dynamics, Transport Theory Statist. Phys. 25 (1996), 33-60.MR 1380030 (96m:82061)
  • 11. Claudio Canuto, M. Yousuff Hussaini, Alfio Quarteroni, and Thomas A. Zang, Spectral methods in fluid dynamics, Springer Series in Computational Physics, Springer-Verlag, New York, 1988. MR 0917480 (89m:76004)
  • 12. Torsten Carleman, Sur la théorie de l'équation intégrodifférentielle de Boltzmann, Acta Math. 60 (1932).
  • 13. Carlo Cercignani, Theory and application of the Boltzmann equation, Elsevier, New York, 1975. MR 0406273 (53:10064)
  • 14. -, The Boltzmann equation and its applications, Applied Mathematical Sciences, vol. 67, Springer-Verlag, New York, 1988. MR 1313028 (95i:82082)
  • 15. Carlo Cercignani, Reinhard Illner, and Mario Pulvirenti, The mathematical theory of dilute gases, Applied Mathematical Sciences, vol. 106, Springer-Verlag, New York, 1994. MR 1307620 (96g:82046)
  • 16. James W. Cooley and John W. Tukey, An algorithm for the machine calculation of complex Fourier series, Math. Comput. 19 (1965), 297-301.MR 0178586 (31:2843)
  • 17. F. Coquel, F. Rogier, and J. Schneider, A deterministic method for solving the homogeneous Boltzmann equation, Rech. Aérospat. (1992), no. 3, 1-10.MR 1192070 (93k:76099)
  • 18. Pierre Degond and Brigitte Lucquin-Desreux, The Fokker-Planck asymptotics of the Boltzmann collision operator in the Coulomb case, M3AS (1992), no. 2, 167-182.MR 1167768 (93g:82083)
  • 19. Pierre Degond, Lorenzo Pareschi, and Giovanni Russo, Modeling and computational methods for kinetic equations, Modeling and Simulation in Science, Engineering and Technology, 2004. MR 2069221 (2005a:82001)
  • 20. M. Escobedo and S. Mischler, On a quantum Boltzmann equation for a gas of photons, J. Math. Pures Appl. (2001), no. 9, 417-515. MR 1831432 (2002j:82071)
  • 21. Francis Filbet, Clément Mouhot, and Lorenzo Pareschi, Solving the Boltzmann equation in $ {N}\log_2 {N}$, SIAM J. Sci. Comp. (submitted).
  • 22. Francis Filbet and Lorenzo Pareschi, A numerical method for the accurate solution of the Fokker-Planck-Landau equation in the non homogeneous case, J. Comput. Phys. 179 (2002), no. 1, 1-26. MR 1906573 (2003c:82085)
  • 23. -, Accurate numerical methods for the collisional motion of (heated) granular flows, J. Comput. Phys. 202 (2005), 216-235.MR 2102883 (2005g:82122)
  • 24. Francis Filbet and Giovanni Russo, High order numerical methods for the space nonhomogeneous Boltzmann equation, J. Comput. Phys. 186 (2003), no. 2, 457-480. MR 1973198 (2004c:82116)
  • 25. E. Gabetta, L. Pareschi, and G. Toscani, Relaxation schemes for nonlinear kinetic equations, SIAM J. Numer. Anal. 34 (1997), no. 6, 2168-2194.MR 1480374 (99c:76096)
  • 26. Tommy Gustafsson, $ {L}\sp p$-estimates for the nonlinear spatially homogeneous Boltzmann equation, Arch. Rational Mech. Anal. 92 (1986), no. 1, 23-57.MR 0816620 (87h:82054)
  • 27. -, Global $ {L}\sp p$-properties for the spatially homogeneous Boltzmann equation, Arch. Rational Mech. Anal. 103 (1988), no. 1, 1-38.MR 0946968 (89h:82019)
  • 28. I. Ibragimov and S. Rjasanow, Numerical solution of the Boltzmann equation on the uniform grid, Computing 69 (2002), no. 2, 163-186.MR 1954793 (2004i:82060)
  • 29. P. A. Markowich, C. A. Ringhofer, and C. Schmeiser, Semiconductor equations, Springer-Verlag, Vienna, 1990. MR 1063852 (91j:78011)
  • 30. Yves-Loïc Martin, François Rogier, and Jacques Schneider, Une méthode déterministe pour la résolution de l'équation de Boltzmann inhomogène, C. R. Acad. Sci. Paris Sér. I Math. 314 (1992), no. 6, 483-487. MR 1154392 (93b:65198)
  • 31. Clément Mouhot, Quantitative lower bounds for the full Boltzmann equation, Part I: Periodic boundary conditions, Comm. Part. Diff. Equations 30 (2005), no. 5-6, 881-917. MR 2153518 (2006a:76096)
  • 32. Clément Mouhot and Lorenzo Pareschi, An $ {O}({N}\log_2 {N})$ algorithm for computing discrete velocity models, (Work in progress).
  • 33. Clément Mouhot and Cédric Villani, Regularity theory for the spatially homogeneous Boltzmann equation with cut-off, Arch. Rational Mech. Anal. 173 (2004), no. 2, 169-212. MR 2081030
  • 34. Giovanni Naldi, Lorenzo Pareschi, and Giuseppe Toscani, Spectral methods for one-dimensional kinetic models of granular flows and numerical quasi elastic limit, M2AN Math. Model. Numer. Anal. 37 (2003), no. 1, 73-90. MR 1972651 (2004d:65117) 20pt
  • 35. Kenichi Nanbu, Direct simulation scheme derived from the Boltzmann equation. I. Monocomponent gases, J. Phys. Soc. Japan 52 (1983), 2042-2049.
  • 36. Andrzej Palczewski and Jacques Schneider, Existence, stability, and convergence of solutions of discrete velocity models to the Boltzmann equation, J. Statist. Phys. 91 (1998), no. 1-2, 307-326. MR 1632506 (99g:82072)
  • 37. Andrzej Palczewski, Jacques Schneider, and Alexandre V. Bobylev, A consistency result for a discrete-velocity model of the Boltzmann equation, SIAM J. Numer. Anal. 34 (1997), no. 5, 1865-1883.MR 1472201 (99d:82068)
  • 38. Vladislav A. Panferov and Alexei G. Heintz, A new consistent discrete-velocity model for the Boltzmann equation, Math. Methods Appl. Sci. 25 (2002), no. 7, 571-593. MR 1895119 (2003d:82100)
  • 39. L. Pareschi, Computational methods and fast algorithms for Boltzmann equations, Chapter 7 Lecture Notes on the discretization of the Boltzmann equation, 2003, pp. 527-548.
  • 40. L. Pareschi, G. Russo, and G. Toscani, Fast spectral methods for the Fokker-Planck-Landau collision operator, J. Comput. Phys. 165 (2000), no. 1, 216-236. MR 1795398 (2001i:65112)
  • 41. 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), no. 3, 527-548. MR 1953752 (2003k:65128)
  • 42. Lorenzo Pareschi and Benoit Perthame, A Fourier spectral method for homogeneous Boltzmann equations, Transport Theory Statist. Phys. 25 (1996), no. 3-5, 369-382. MR 1407541 (97j:82133)
  • 43. Lorenzo Pareschi and Giovanni Russo, Numerical solution of the Boltzmann equation. I. Spectrally accurate approximation of the collision operator, SIAM J. Numer. Anal. 37 (2000), no. 4, 1217-1245.MR 1756425 (2001g:65175)
  • 44. -, On the stability of spectral methods for the homogeneous Boltzmann equation, Transport Theory Statist. Phys. 29 (2000), no. 3-5, 431-447.MR 1770438 (2001c:82070)
  • 45. Lorenzo Pareschi, Giovanni Russo, and Giuseppe Toscani, Méthode spectrale rapide pour l'équation de Fokker-Planck-Landau, C. R. Acad. Sci. Paris Sér. I Math. 330 (2000), no. 6, 517-522.MR 1756970 (2001a:65125)
  • 46. Ada Pulvirenti and Bernt Wennberg, A Maxwellian lower bound for solutions to the Boltzmann equation, Comm. Math. Phys. 183 (1997), no. 1, 145-160.MR 1461954 (99f:82056)
  • 47. François Rogier and Jacques Schneider, A direct method for solving the Boltzmann equation, Transport Theory Statist. Phys. 23 (1994), no. 1-3, 313-338.MR 1257657 (94k:82100)
  • 48. Michelle Schatzman, Analyse numérique, InterEditions, Paris, 1991.MR 1200897 (93i:65002)
  • 49. C. Villani, A survey of mathematical topics in kinetic theory, Handbook of fluid mechanics, S. Friedlander and D. Serre, Eds. Elsevier Publ., 2002.MR 1942465 (2003k:82087)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 65T50, 68Q25, 74S25, 76P05

Retrieve articles in all journals with MSC (2000): 65T50, 68Q25, 74S25, 76P05


Additional Information

Clément Mouhot
Affiliation: UMPA, ENS Lyon, 46 allée d’Italie, 69364 Lyon Cedex 07, France
Email: cmouhot@umpa.ens-lyon.fr

Lorenzo Pareschi
Affiliation: Dipartimento di Matematica, Università di Ferrara, Via Machiavelli 35, 44100 Ferrara, Italy
Email: lorenzo.pareschi@unife.it

DOI: https://doi.org/10.1090/S0025-5718-06-01874-6
Keywords: Boltzmann equation, spectral methods, fast Fourier transform, fast algorithms
Received by editor(s): February 7, 2004
Received by editor(s) in revised form: March 13, 2005
Published electronically: July 12, 2006
Additional Notes: The first author was supported by the European network HYKE, funded by the EC as contract HPRN-CT-2002-00282
Article copyright: © Copyright 2006 by the authors

American Mathematical Society