Abstract
The non-cutoff Boltzmann equation can be simulated using the DSMC method, by a truncation of the collision term. However, even for computing stationary solutions this may be very time consuming, in particular in situations far from equilibrium. By adding an appropriate diffusion, to the DSMC-method, the rate of convergence when the truncation is removed, may be greatly improved. We illustrate the technique on a toy model, the Kac equation, as well as on the full Boltzmann equation in a special case.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Asmussen, S., Rosiński, J.: Approximations of small jumps of Lévy processes with a view towards simulation. J. Appl. Probab. 38, 482–493 (2001)
Babovsky, H.: On a simulation scheme for the Boltzmann equation. Math. Methods Appl. Sci. 8, 223–233 (1986)
Babovsky, H., Illner, R.: A convergence proof for Nanbu’s simulation method for the full Boltzmann equation. SIAM J. Numer. Anal. 26(1), 45–65 (1989)
Bird, G.A.: Molecular Gas Dynamics. Oxford University Press, London (1976)
Bagland, V., Wennberg, B., Wondmagegne, Y.: Stationary states for the non-cutoff Kac equation with a Gaussian thermostat. Nonlinearity 20(3), 583–604 (2007)
Bobylev, A.V., Gamba, I.M., Panferov, V.A.: Moment inequalities and high-energy tails for Boltzmann equations with inelastic interactions. J. Stat. Phys. 116(5, 6), 1651–1682 (2004)
Bobylev, A.V., Mossberg, E., Potapenko, I.F.: A DSMC method for the Landau-Fokker-Planck equation. In: Proc. 25th International Symposium on Rarefied Gas Dynamics. St. Petersburg, Russia, July (2006)
Brilliantov, N.V., Pöschel, T.: Kinetic Theory of Granular Gases. Oxford University Press, Oxford (2004)
Desvillettes, L.: About the regularizing properties of the non-cut-off Kac equation. Commun. Math. Phys. 168(2), 417–440 (1995)
Desvillettes, L., Graham, C., Méléard, S.: Probabilistic interpretation and numerical approximation of a Kac equation without cutoff. Stoch. Process. Appl. 84(1), 115–135 (1999)
Filbet, F., Pareschi, L.: Numerical solution of the Fokker-Planck-Landau equation by spectral methods. Commun. Math. Sci. 1(1), 206–207 (2003)
Filbet, F., Mouhot, C., Pareschi, L.: Solving the Boltzmann equation in Nlog N. SIAM J. Sci. Comput. 28(3), 1029–1053 (2006)
Fournier, N., Méléard, S.: Monte-Carlo approximations and fluctuations for 2D Boltzmann equations without cutoff. Markov Process. Relat. Fields 7, 159–191 (2001)
Fournier, N., Méléard, S.: A stochastic particle numerical method for 3D Boltzmann equations without cutoff. Math. Comput. 71, 583–604 (2002)
Goldhirsch, I.: Inelastic kinetic theory: the granular gas. In: Topics in Kinetic Theory, pp. 289–312. AMS, Providence (2005)
GNU scientific library. See http://directory.fsf.org/GNU/gsl.html
Kac, M.: Foundations of kinetic theory. In: Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954–1955, vol. III, pp. 171–197. University of California Press, Berkeley (1956)
Kiessling, M., Lancellotti, C.: On the Master-Equation approach to kinetic theory: linear and nonlinear Fokker-Planck equations. Transp. Theory Stat. Phys. 33, 379–401 (2004)
LAPACK: Linear Algebra PACKage. Available at http://www.netlib.org/lapack/
Marsaglia, G., Tsang, W.W.: The ziggurat method for generating random variables. J. Stat. Softw. 5(8), 1–7 (2000)
McKean, H.P.: Speed of approach to equilibrium for Kac’s caricature of a Maxwellian gas. Arch. Ration. Mech. Anal. 21, 343–367 (1966)
Mouhot, C., Pareschi, L.: Fast algorithms for computing the Boltzmann collision operator. Math. Comput. 75(256), 1833–1852 (2006)
Nanbu, K.: Direct simulation scheme derived from the Boltzmann equation. J. Phys. Soc. Jpn. 49, 2042–2049 (1980)
Øksendal, B.: Stochastic Differential Equations, 6th edn. Springer, Berlin (2003)
Pareschi, L., Russo, G.: Time relaxed Monet Carlo methods for the Boltzmann equation. SIAM J. Sci. Comput. 23, 1253–1273 (2001)
Pareschi, L., Trazzi, S.: Numerical solution of the Boltzmann equation by time relaxed Monte Carlo (TRMC) methods. Int. J. Numer. Methods Fluids 48, 947–983 (2005)
Pareschi, L., Wennberg, B.: A recursive Monte Carlo method for the Boltzmann equation in the Maxwellian case. Monte Carlo Methods Appl. 7, 349–357 (2001)
Pulvirenti, M., Wagner, W., Zavelani Rossi, M.B.: Convergence of particle schemes for the Boltzmann equation. Eur. J. Mech. B Fluids 13(3), 339–351 (1994)
Rjasanow, S., Wagner, W.: Stochastic Numerics for the Boltzmann Equation. Springer Series in Computational Mathematics, vol. 37. Springer, Berlin (2005)
Rosiński, J., Cohen, S.: Gaussian approximation of multivariate Lévy processes with applications to simulation of tempered and operator stable processes. Preprint
Stroock, D.: On the growth of stochastic integrals. Z. Wahrscheinlichkeitstheorie Verw. Gebiete 18, 340–344 (1971)
Sundén, M., Wennberg, B.: The Kac master equation with unbounded collision rate. In preparation
Villani, C.: A review of mathematical topics in collisional kinetic theory. In: Friedlander, S., Serre, D. (eds.) Handbook of Mathematical Fluid Mechanics. North-Holland, Amsterdam (2002)
Wennberg, B., Wondmagegne, Y.: The Kac equation with a thermostatted force field. J. Stat. Phys. 124(2–4), 859–880 (2006)
Wennberg, B., Wondmagegne, Y.: Stationary states for the Kac equation with a Gaussian thermostat. Nonlinearity 17, 633–648 (2004)
Wondmagegne, Y.: Kinetic equations with a Gaussian thermostat. Doctoral thesis, Department of Mathematical Sciences, Chalmers University of Technology and Göteborg University, Göteborg (2005)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Sundén, M., Wennberg, B. Brownian Approximation and Monte Carlo Simulation of the Non-Cutoff Kac Equation. J Stat Phys 130, 295–312 (2008). https://doi.org/10.1007/s10955-007-9424-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-007-9424-8