Abstract
We propose a simple fast spectral method for the Boltzmann collision operator with general collision kernels. In contrast to the direct spectral method [L. Pareschi and G. Russo, SIAM J. Numer. Anal., 37 (2000), pp. 1217{1245; I. M. Gamba and S. H. Tharkabhushanam, J. Comput. Phys., 228 (2009), pp. 2012{2036], which requires O(N6) memory to store precomputed weights and has O(N6) numerical complexity, the new method has complexity O(MN4 logN), where N is the number of discretization points in each of the three velocity dimensions and M is the total number of discretization points on the sphere and M ≪ N2. Furthermore, it requires no precomputation for the variable hard sphere model and only O(MN4) memory to store precomputed functions for more general collision kernels. Although a faster spectral method is available [C. Mouhot and L. Pareschi, Math. Comp., 75 (2006), pp. 1833{1852] (with complexity O(MN3 logN)), it works only for hard sphere molecules, thus limiting its use for practical problems. Our new method, on the other hand, can apply to arbitrary collision kernels. A series of numerical tests is performed to illustrate the efficiency and accuracy of the proposed method.
| Original language | English |
|---|---|
| Pages (from-to) | B658-B674 |
| Journal | SIAM Journal on Scientific Computing |
| Volume | 39 |
| Issue number | 4 |
| DOIs | |
| State | Published - 2017 |
Funding
∗Submitted to the journal’s Computational Methods in Science and Engineering section September 27, 2016; accepted for publication (in revised form) April 13, 2017; published electronically August 22, 2017. http://www.siam.org/journals/sisc/39-4/M109600.html Funding: This work was supported by Los Alamos Report LA-UR-16-26555 and by the U.S. Department of Energy at Los Alamos National Laboratory under contract DE-AC52-06NA25396. The first author’s research was partially supported by NSF grant DMS-1413064 and NSF RNMS (KI-Net) grant DMS-1107465. The second author’s research was partially supported by NSF grant DMS-1109625 and NSF RNMS (KI-Net) grant DMS-1107465. The third author’s research was supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Research. The fourth author’s research was supported by NSF grants DMS-1620250, DMS-1107291, RNMS KI-Net, and a startup grant from Purdue University. This manuscript has been authored, in part, by UT-Battelle, LLC, under contract DE-AC0500OR22725 with the U.S. Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a nonexclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for the United States Government purposes. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe-public-access-plan). This work also received support from the Institute of Computational Engineering and Sciences (ICES) at the University of Texas Austin.
Keywords
- Boltzmann collision integral
- Convolution
- Fast Fourier transform
- Lebedev quadrature
- Spectral method