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Mathematics of Computation

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A fast algorithm for the energy space boson Boltzmann collision operator

Authors: Jingwei Hu and Lexing Ying
Journal: Math. Comp. 84 (2015), 271-288
MSC (2010): Primary 35Q20, 82C10, 65D32, 44A35, 65T50
Published electronically: March 21, 2014
MathSciNet review: 3266960
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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.

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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

Lexing Ying
Affiliation: Department of Mathematics and Institute for Computational and Mathematical Engineering (ICME), Stanford University, 450 Serra Mall, Bldg 380, Stanford, California 94305

Keywords: Quantum Boltzmann equation, energy space boson Boltzmann equation, recursive domain decomposition, double convolution, fast Fourier transform
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
Article copyright: © Copyright 2014 American Mathematical Society