A fast algorithm for the energy space boson Boltzmann collision operator

Hu, Jingwei; Ying, Lexing

doi:10.1090/S0025-5718-2014-02824-X

A fast algorithm for the energy space boson Boltzmann collision operator
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by Jingwei Hu and Lexing Ying PDF

Math. Comp. 84 (2015), 271-288 Request permission

Abstract:

This paper introduces a fast algorithm for the energy space boson Boltzmann collision operator. Compared to the direct $O(N^3)$ calculation and the previous $O(N^2\log N)$ method [Markowich and Pareschi, 2005], the new algorithm runs in complexity $O(N\log ^2N)$, which is optimal up to a logarithmic factor ($N$ is the number of grid points in energy space). The basic idea is to partition the 3-D summation domain recursively into elementary shapes so that the summation within each shape becomes a special double convolution that can be computed efficiently by the fast Fourier transform. Numerical examples are presented to illustrate the efficiency and accuracy of the proposed algorithm.

References

Leif Arkeryd and Anne Nouri, Bose condensates in interaction with excitations: a kinetic model, Comm. Math. Phys. 310 (2012), no. 3, 765–788. MR 2891873, DOI 10.1007/s00220-012-1415-1
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, DOI 10.1016/S0997-7546(99)00121-1
C. Connaughton and Y. Pomeau, Kinetic theory and Bose-Einstein condensation. C. R. Pysique, 5:91–106, 2004.
M. Escobedo, S. Mischler, and M. A. Valle, Homogeneous Boltzmann equation in quantum relativistic kinetic theory. J. Differ. Equ., Monograph 4, 2003.
Miguel Escobedo, Federica Pezzotti, and Manuel Valle, Analytical approach to relaxation dynamics of condensed Bose gases, Ann. Physics 326 (2011), no. 4, 808–827. MR 2771726, DOI 10.1016/j.aop.2010.11.001
Irene M. Gamba and Sri Harsha Tharkabhushanam, Spectral-Lagrangian methods for collisional models of non-equilibrium statistical states, J. Comput. Phys. 228 (2009), no. 6, 2012–2036. MR 2500671, DOI 10.1016/j.jcp.2008.09.033
A. L. Garcia and W. Wagner, Direct simulation Monte Carlo method for the Uehling-Uhlenbeck-Boltzmann equation. Phys. Rev. E, 68:056703, 2003.
Jingwei Hu and Lexing Ying, A fast spectral algorithm for the quantum Boltzmann collision operator, Commun. Math. Sci. 10 (2012), no. 3, 989–999. MR 2911206, DOI 10.4310/CMS.2012.v10.n3.a13
I. Ibragimov and S. Rjasanow, Numerical solution of the Boltzmann equation on the uniform grid, Computing 69 (2002), no. 2, 163–186. MR 1954793, DOI 10.1007/s00607-002-1458-9
Boris N. Khoromskij, Structured data-sparse approximation to high order tensors arising from the deterministic Boltzmann equation, Math. Comp. 76 (2007), no. 259, 1291–1315. MR 2299775, DOI 10.1090/S0025-5718-07-01901-1
Robert Lacaze, Pierre Lallemand, Yves Pomeau, and Sergio Rica, Dynamical formation of a Bose-Einstein condensate, Phys. D 152/153 (2001), 779–786. Advances in nonlinear mathematics and science. MR 1837939, DOI 10.1016/S0167-2789(01)00211-1
E. M. Lifshitz and L. P. Pitaevskiĭ, Course of theoretical physics [“Landau-Lifshits”]. Vol. 9, Pergamon Press, Oxford-Elmsford, N.Y., 1980. Statistical physics. Part 2. Theory of the condensed state; Translated from the Russian by J. B. Sykes and M. J. Kearsley. MR 586944
Xuguang Lu and Xiangdong Zhang, On the Boltzmann equation for 2D Bose-Einstein particles, J. Stat. Phys. 143 (2011), no. 5, 990–1019. MR 2811470, DOI 10.1007/s10955-011-0221-z
Peter A. Markowich and Lorenzo Pareschi, Fast conservative and entropic numerical methods for the boson Boltzmann equation, Numer. Math. 99 (2005), no. 3, 509–532. MR 2117737, DOI 10.1007/s00211-004-0570-5
Clément Mouhot and Lorenzo Pareschi, Fast algorithms for computing the Boltzmann collision operator, Math. Comp. 75 (2006), no. 256, 1833–1852. MR 2240637, DOI 10.1090/S0025-5718-06-01874-6
L. W. Nordheim, On the kinetic method in the new statistics and its application in the electron theory of conductivity. Proc. R. Soc. London, Ser. A, 119:689–698, 1928.
Lorenzo Pareschi and Benoit Perthame, A Fourier spectral method for homogeneous Boltzmann equations, Proceedings of the Second International Workshop on Nonlinear Kinetic Theories and Mathematical Aspects of Hyperbolic Systems (Sanremo, 1994), 1996, pp. 369–382. MR 1407541, DOI 10.1080/00411459608220707
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, DOI 10.1137/S0036142998343300
R. K. Pathria, Statistical Mechanics. Butterworth-Heinemann, second edition, 1996.
D. V. Semikoz and I. I. Tkachev, Kinetics of Bose condensation. Phys. Rev. Lett., 74:3093–3097, 1995.
D. V. Semikoz and I. I. Tkachev, Condensation of bosons in the kinetic regime. Phys. Rev. D, 55:489–502, 1997.
Herbert Spohn, Kinetics of the Bose-Einstein condensation, Phys. D 239 (2010), no. 10, 627–634. MR 2601928, DOI 10.1016/j.physd.2010.01.018
E. A. Uehling and G. E. Uhlenbeck, Transport phenomena in Einstein-Bose and Fermi-Dirac gases. I. Phys. Rev., 43:552–561, 1933.