A brief review of numerical methods for collision operators

  • S. Cordier
  • S. Mancini
Conference paper


In this review, we present numerical methods for approx imating collision operators in kinetic theory. We refer to [29] for a similar paper concerning the Vlasov equation. Various collision operators are described (Boltzmann, Fokker-Planck) and the three main numerical methods (Spectral, Discrete Velocity and Monte Carlo) are briefly shown.


Monte Carlo Boltzmann Equation Direct Simulation Monte Carlo Collision Operator Vlasov Equation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Andriès, P., Le Tallec, P., Perlat, J.-P., Perthame, B. (2000): The Gaussian-BGK model of Boltzmann equation with small Prandtl number. Eur. J. Mech. B Fluids 19, 813–830MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    Babovsky, H., Illner, R. (1989): A convergence proof for Nanbu’s simulation method for the full Boltzmann equation, SIAM J. Numer. Anal. 26, 45–65MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    Bellomo, N., Gatignol, R. (eds.) (2003): Lecture notes on the discretization of the Boltzmann equation. World Scientific, SingaporeGoogle Scholar
  4. [4]
    Bird, G.A. (1970): Direct simulation and the Boltzmann equation. Phys. Fluids A 13, 2672–2681Google Scholar
  5. [5]
    Bird, G.A. (1994): Molecular gas dynamics and the direct simulation of gas flows. Oxford Science Publications, OxfordGoogle Scholar
  6. [6]
    Boudin, L., Desvillettes, L. (2000): On the singularities of the global small solutions of the full Boltzmann equation. Monatsh. Math. 131, 91–108MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    Bourgat, J.-E, Le Tallec, P., Perthame, B., Qiu, Y. (1994): Coupling Boltzmann and Euler equations without overlapping. In: Quarteroni, A. et al. (eds.): Domain decomposition methods in science and engineering. (Contemporary Mathematics, vol. 157). American Mathematical Society, Providence, RI, pp. 377–398CrossRefGoogle Scholar
  8. [8]
    Buet, C. (1997): Conservative and entropy schemes for Boltzmann collision operator of polyatomic gases. Math. Models Methods Appl. Sci. 7, 165–192MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    Buet, C., Cordier, S. (1999): Numerical analysis of conservative and entropy schemes for the Fokker-Planck-Landau equation. SIAM J. Numer. Anal. 36, 953–973MathSciNetCrossRefGoogle Scholar
  10. [10]
    Buet, C., Cordier, S. (1998): Conservative and entropy decaying numerical scheme for the isotropic Fokker-Planck-Landau equation. J. Comput. Phys. 145, 228–245MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    Buet, C; Cordier, S., Degond, P., Lemou, M. (1997): Fast algorithms for numerical, conservative, and entropy approximations of the Fokker-Planck-Landau equation. J. Comput. Phys. 133, 310–322MathSciNetMATHCrossRefGoogle Scholar
  12. [12]
    Buet, C., Cordier, S., Filbet, E (1999): Comparison of numerical schemes for Fokker-Planck-Landau equation. In: Coquel, E, Cordier, S. (eds.): CEMRACS. (ESAIM Proceedings, vol. 10). Société de Mathemátiques Appliquées et Industrielles, Paris, pp. 161–181. http://www.emath.fr/Maths/Proc/Vol.10Google Scholar
  13. [13]
    Buet, C., Cordier, S., Lucquin-Desreux, B. (2001): The grazing collision limit for the Boltzmann-Lorentz model. Asymptot. Anal. 25, 93–107MathSciNetMATHGoogle Scholar
  14. [14]
    Buet, C., Cordier, S., Lucquin-Desreux, B., Mancini, S. (2002): Diffusion limit of the Lorentz model: asymptotic preserving schemes. M2AN Math. Model. Numer. Anal. 36, 631–655MathSciNetMATHCrossRefGoogle Scholar
  15. [15]
    Caflisch, R.E., Pareschi, L. (1999): An implicit Monte Carlo method for rarefied gas dynamics. I. The space homogeneous case. J. Comput. Phys. 154, 90–116MathSciNetMATHCrossRefGoogle Scholar
  16. [16]
    Cercignani, C. (1988): The Boltzmann equation and its applications. Springer, BerlinMATHCrossRefGoogle Scholar
  17. [17]
    Cercignani, C., Illner, R., Pulvirenti, M. (1994): The mathematical theory of dilute gases. Springer, New YorkMATHGoogle Scholar
  18. [18]
    Chen, G.Q., Levermore, C.D., Liu, T.-P. (1994): Hyperbolic conservation laws with stiff relaxation terms and entropy. Comm. Pure Appl. Math. 47, 787–830MathSciNetMATHCrossRefGoogle Scholar
  19. [19]
    Cordier, S., Lucquin-Desreux, B., Mancini, S. (2002): Focalization: a numerical test for smoothing effects of collision operators. J. Sci. Comput., submittedGoogle Scholar
  20. [20]
    Cordier, S., Lucquin-Desreux, B., Sabry, A. (1999): Numerical method for VlasovLorentz models. In: Coquel, E, Cordier, S. (eds.): CEMRACS. (ESAIM Proceedings, vol. 10). Société de Mathématiques Appliquées et Industrielles, Paris, pp. 201–210. http://www.emath.fr/Maths/Proc/Vol.10Google Scholar
  21. [21]
    Coron, E, Perthame, B. (1991): Numerical passage from kinetic to fluid equations. SIAM J. Numer. Anal. 28, 26–42MathSciNetMATHCrossRefGoogle Scholar
  22. [22]
    Degond, P., Lucquin-Desreux, B. (1992): The Fokker-Planck asymptotics of the Boltzmann collision operator in the Coulomb case. Math. Models Methods Appl. Sci. 2, 167–182MathSciNetMATHCrossRefGoogle Scholar
  23. [23]
    Degond, P., Lucquin-Desreux, B. (1994): An entropy scheme for the Fokker-Planck collision operator of plasma kinetic theory. Numer. Math. 68, 239–262MathSciNetMATHCrossRefGoogle Scholar
  24. [24]
    Degond, P., Lucquin-Desreux, B. (1996): The asymptotics of collision operators for two species of particles of disparate masses. Math. Models Methods Appl. Sci. 6, 405–436MathSciNetMATHCrossRefGoogle Scholar
  25. [25]
    Desvillettes, L. (1992): On asymptotics of the Boltzmann equation when the collisions become grazing. Transport Theory Statist. Phys. 21, 259–276MathSciNetMATHCrossRefGoogle Scholar
  26. [26]
    Desvillettes, L., Villani, C. (2000): On the spatially homogeneous Landau equation for hard potentials. I. Existence, uniqueness and smoothness. II. H-theorem and applications. Comm. Partial Differential Equations 25, 179–259, 261–298MathSciNetMATHCrossRefGoogle Scholar
  27. [27]
    Dubroca, B., Mieussens, L. (1999): Aconservative and entropic discrete-velocity model for rarefied polyatomic gases. In: Coquel, E, Cordier, S. (eds.): CEMRACS. (ESAIM Proceedings, vol. 10). Societe de Mathematiques Appliquees et Industrielles, Paris, pp. 127–139. http://www.emath.fr/Maths/Proc/Vol.10Google Scholar
  28. [28]
    Filbet, E, Russo, G.: High order numerical methods for the space homogeneous Boltzmann equation, in preparationGoogle Scholar
  29. [29]
    Filbet, E, Sonnendrücker, E. (2002): Numerical methods for the Vlasov equation. In: Brezzi, F. et al. (eds.): Numerical mathematics and advanced applications, Proceedings of ENUMATH 2001. Springer-Verlag Italia, Milano, pp. 459–468Google Scholar
  30. [30]
    Gabetta, E., Pareschi, L., Toscani, G. (1997): Relaxation schemes for nonlinear kinetic equations. SIAM J. Numer. Anal. 34, 2168–2194MathSciNetMATHCrossRefGoogle Scholar
  31. [31]
    Gosse, L., Toscani, G. (2002): An asymptotic preserving well-balanced scheme for the hyperbolic heat equations. C.R. Math. Acad. Sci. Paris 334, 337–342MathSciNetMATHCrossRefGoogle Scholar
  32. [32]
    Goudon, T. (1997): On Boltzmann equations and Fokker-Planck asymptotics: influence of grazing collisions. J. Statist. Phys. 89, 751–776MathSciNetMATHCrossRefGoogle Scholar
  33. [33]
    Jin, S. (1999): Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations. SIAM J. Sci. Comput. 21, 441–454MathSciNetMATHCrossRefGoogle Scholar
  34. [34]
    Jin, S., Pareschi, L. (2000): Discretization of the multiscale semiconductor Boltzmann equation by diffusive relaxation schemes. J. Comput. Phys. 161, 312–330MathSciNetMATHCrossRefGoogle Scholar
  35. [35]
    Klar, A. (1998): An asymptotic-induced scheme for nonstationary transport equations in the diffusive limit. SIAM J. Numer. Anal. 35, 1073–1094MathSciNetMATHCrossRefGoogle Scholar
  36. [36]
    Larsen, E.W. (1992): The asymptotic diffusion limit of discretized transport problems. Nuclear Sci. Engrg. 112, 336–346Google Scholar
  37. [37]
    Lemou, M. (1998): Multipole expansions for the Fokker-Planck-Landau operator. Numer. Math. 78, 597–618MathSciNetMATHCrossRefGoogle Scholar
  38. [38]
    Lucquin-Desreux, B. (2000): Diffusion of electrons by multicharged ions, Math. Models Methods Appl. Sci. 10, 409–440MathSciNetMATHGoogle Scholar
  39. [39]
    Lucquin-Desreux, B., Mancini, S. (2001): A finite element approximation of grazing collisions. Transport Theory Statist. Phys., submittedGoogle Scholar
  40. [40]
    Mallinger, F., Le Tallec, P. (1997): Coupling Boltzmann and Navier-Stokes equations by half fluxes. J. Comput. Phys. 136, 51–57MathSciNetMATHCrossRefGoogle Scholar
  41. [41]
    Markowich, P.A., Ringhofer, C.A., Schmeiser, C. (1990): Semiconductor equations, Springer, ViennaMATHCrossRefGoogle Scholar
  42. [42]
    Mieussens, L. (2000): Discrete-velocity models and numerical schemes for the Boltzmann-BGK equation in plane and axisymmetric geometries. J. Comput. Phys. 162, 429–466MathSciNetMATHCrossRefGoogle Scholar
  43. [43]
    Mischler, S. (1997): Convergence of discrete-velocity schemes for the Boltzmann equation Arch. Rational Mech. Anal. 140, 53–77MathSciNetMATHCrossRefGoogle Scholar
  44. [44]
    Naldi, G., Pareschi, L. (1998): Numerical schemes for kinetic equations in diffusive regimes. Appl. Math. Lett. 11, 29–35MathSciNetCrossRefGoogle Scholar
  45. [45]
    Nanbu, K. (1980): Direct simulation scheme derived from the Boltzmann equation. I. Monocomponent Gases. J. Phys. Soc. Japan 49, 2042–2049CrossRefGoogle Scholar
  46. [46]
    Neunzert, H., Struckmeier, J. (1995): Particle methods for the Boltzmann equation. In: Iserles, A. (ed.): Acta numerica. 1995. Cambridge University Press, Cambridge, pp. 417–457Google Scholar
  47. [47]
    Pareschi, L., Russo, G. (1999): An introduction to Monte Carlo methods for the Boltzmann equation. In: Coquel, F., Cordier, S. (eds.): CEMRACS. (ESAIM Proceedings, vol. 10). Société de Mathématiques Appliquées et Industrielles, Paris, pp. 35–76. http://www.emath.fr/Maths/Proc/Vol.l0Google Scholar
  48. [48]
    Pareschi, L., Russo, G. (2000): Asymptotic preserving Monte Carlo methods for the Boltzmann equation. Transport Theory Statist. Phys. 29, 415–430MathSciNetMATHCrossRefGoogle Scholar
  49. [49]
    Pareschi, L., Russo, G. (2000): Numerical solution of the Boltzmann equation, I. Spectrally accurate approximation of the collision operator. SIAM J. Numer. Anal. 37, 1217–1245MathSciNetMATHCrossRefGoogle Scholar
  50. [50]
    Pareschi, L., Russo, G., Toscani, G. (2000): Méthode spéctrale rapide pour l'équation de Fokker-Planck-Landau. C. R. Acad. Sci. Paris Ser. I Math. 330, 517–522MathSciNetMATHCrossRefGoogle Scholar
  51. [51]
    Pareschi, L., Wennberg, B. (2001): A recursive Monte Carlo method for the Boltzmann equation in the Maxwellian case. Monte Carlo Methods Appl. 7, 349–357MathSciNetMATHCrossRefGoogle Scholar
  52. [52]
    Pekker, M.S., Khudik, Y.N. (1984): Conservative difference schemes for the Fokker-Planck equation. (Russian). Zh. Vychisl. Mat. i Mat. Fiz. 24, 947–953; translated as U.S.S.R. Comput. Math. Math. Phys. 24, 206–210MathSciNetGoogle Scholar
  53. [53]
    Perthame, B. (1990): Boltzmann type schemes for gas dynamics and the entropy property. SIAM J. Numer. Anal. 27, 1405–1421MathSciNetMATHCrossRefGoogle Scholar
  54. [54]
    Perthame, B. (1998): An introduction to kinetic schemes for gas dynamics. In: Kröner, D. et al. (eds.): An introduction to recent developments in theory and numerics for conservation laws. (Lecture Notes in Computational Science and Engineering, vol. 5). Springer, Berlin, pp. 1–27CrossRefGoogle Scholar
  55. [55]
    Rogier, F., Schneider, J. (1994): A direct method for solving the Boltzmann equation. Transport Theory Statist. Phys. 23, 313–338MathSciNetMATHCrossRefGoogle Scholar
  56. [56]
    Villani, C. (2002): A review of mathematical topics in collisional kinetic theory. In: Friedlander, S., Serre, D. (eds.): Handbook of matematical fluid dynamics. Vol. I. Elsevier, Amsterdam, pp. 71–305Google Scholar

Copyright information

© Springer-Verlag Italia 2003

Authors and Affiliations

  • S. Cordier
    • 1
  • S. Mancini
    • 2
  1. 1.Laboratoire MAPMOUniversité d’OrléansOrléansFrance
  2. 2.Laboratoire J.-L. LionsUniversité Pierre et Marie CurieParisFrance

Personalised recommendations