Brownian approximation and Monte Carlo simulation of the Non-Cutoff Kac equation

Journal of Statistical Physics

Volumn 130, Issue 2, 2008, Pages 295-312

  Sundén, Mattias ^a,b   Wennberg, Bernt ^a,b  

^a CHALMERS UNIVERSITY OF TECHNOLOGY   (Sweden)

^b UNIVERSITY OF GOTHENBURG   (Sweden)

Author keywords

Diffusion approximation; Direct simulation Monte Carlo; Kac equation; Markov jump process; Non equilibrium stationary state; Thermostat

Indexed keywords

EID: 36649019726     PISSN: 00224715     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10955-007-9424-8     Document Type: Article

References (36)

1. Asmussen, S., Rosiński, J.: Approximations of small jumps of Lévy processes with a view towards simulation. J. Appl. Probab. 38, 482-493 (2001)
2. Babovsky, H.: On a simulation scheme for the Boltzmann equation. Math. Methods Appl. Sci. 8, 223-233 (1986)
3. 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)
4. Bird, G.A.: Molecular Gas Dynamics. Oxford University Press, London (1976)
5. Bagland, V., Wennberg, B., Wondmagegne, Y.: Stationary states for the non-cutoff Kac equation with a Gaussian thermostat. Nonlinearity 20(3), 583-604 (2007)
6. 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)
7. 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)
8. Brilliantov, N.V., Pöschel, T.: Kinetic Theory of Granular Gases. Oxford University Press, Oxford (2004)
9. Desvillettes, L.: About the regularizing properties of the non-cut-off Kac equation. Commun. Math. Phys. 168(2), 417-440 (1995)
10. 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)
11. Filbet, F., Pareschi, L.: Numerical solution of the Fokker-Planck-Landau equation by spectral methods. Commun. Math. Sci. 1(1), 206-207 (2003)
12. Filbet, F., Mouhot, C., Pareschi, L.: Solving the Boltzmann equation in Nlog∈N. SIAM J. Sci. Comput. 28(3), 1029-1053 (2006)
13. Fournier, N., Méléard, S.: Monte-Carlo approximations and fluctuations for 2D Boltzmann equations without cutoff. Markov Process. Relat. Fields 7, 159-191 (2001)
14. Fournier, N., Méléard, S.: A stochastic particle numerical method for 3D Boltzmann equations without cutoff. Math. Comput. 71, 583-604 (2002)
15. Goldhirsch, I.: Inelastic kinetic theory: the granular gas. In: Topics in Kinetic Theory, pp. 289-312. AMS, Providence (2005)
16. GNU scientific library. See http://directory.fsf.org/GNU/gsl.html
17. 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)
18. 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)
19. LAPACK: Linear Algebra PACKage. Available at http://www.netlib.org/ lapack/
20. Marsaglia, G., Tsang, W.W.: The ziggurat method for generating random variables. J. Stat. Softw. 5(8), 1-7 (2000)
21. McKean, H.P.: Speed of approach to equilibrium for Kac's caricature of a Maxwellian gas. Arch. Ration. Mech. Anal. 21, 343-367 (1966)
22. Mouhot, C., Pareschi, L.: Fast algorithms for computing the Boltzmann collision operator. Math. Comput. 75(256), 1833-1852 (2006)
23. Nanbu, K.: Direct simulation scheme derived from the Boltzmann equation. J. Phys. Soc. Jpn. 49, 2042-2049 (1980)
24. Øksendal, B.: Stochastic Differential Equations, 6th edn. Springer, Berlin (2003)
25. Pareschi, L., Russo, G.: Time relaxed Monet Carlo methods for the Boltzmann equation. SIAM J. Sci. Comput. 23, 1253-1273 (2001)
26. 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)
27. 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)
28. 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)
29. Rjasanow, S., Wagner, W.: Stochastic Numerics for the Boltzmann Equation. Springer Series in Computational Mathematics, vol. 37. Springer, Berlin (2005)
30. Rosiński, J., Cohen, S.: Gaussian approximation of multivariate Lévy processes with applications to simulation of tempered and operator stable processes. Preprint
31. Stroock, D.: On the growth of stochastic integrals. Z. Wahrscheinlichkeitstheorie Verw. Gebiete 18, 340-344 (1971)
32. Sundén, M., Wennberg, B.: The Kac master equation with unbounded collision rate. In preparation
33. 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)
34. Wennberg, B., Wondmagegne, Y.: The Kac equation with a thermostatted force field. J. Stat. Phys. 124(2-4), 859-880 (2006)
35. Wennberg, B., Wondmagegne, Y.: Stationary states for the Kac equation with a Gaussian thermostat. Nonlinearity 17, 633-648 (2004)
36. Wondmagegne, Y.: Kinetic equations with a Gaussian thermostat. Doctoral thesis, Department of Mathematical Sciences, Chalmers University of Technology and Göteborg University, Göteborg (2005)