## A fast algorithm for the energy space boson Boltzmann collision operator

HTML articles powered by AMS MathViewer

- 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.

## Additional Information

**Jingwei Hu**- Affiliation: Institute for Computational Engineering and Sciences (ICES), The University of Texas at Austin, 1 University Station, C0200, Austin, Texas 78712
- Email: hu@ices.utexas.edu
**Lexing Ying**- Affiliation: Department of Mathematics and Institute for Computational and Mathematical Engineering (ICME), Stanford University, 450 Serra Mall, Bldg 380, Stanford, California 94305
- Email: lexing@math.stanford.edu
- Received by editor(s): June 2, 2012
- Received by editor(s) in revised form: December 12, 2012, and December 27, 2012
- Published electronically: March 21, 2014
- Additional Notes: The first author was supported by an ICES Postdoctoral Fellowship

The second author was partially supported by NSF under CAREER award DMS-0846501 - © Copyright 2014 American Mathematical Society
- Journal: Math. Comp.
**84**(2015), 271-288 - MSC (2010): Primary 35Q20, 82C10, 65D32, 44A35, 65T50
- DOI: https://doi.org/10.1090/S0025-5718-2014-02824-X
- MathSciNet review: 3266960