Convolutive decomposition and fast summation methods for discrete-velocity approximations of the Boltzmann equation

¹ DPMMS, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom.
C.Mouhot@dpmms.cam.ac.uk
² DMI, Università di Ferrara, Via Machiavelli 35, 44121 Ferrara, Italy.
lorenzo.pareschi@unife.it
³ CSCAMM, University of Maryland, CSIC Building, Paint Branch Drive, College Park, MD 20740, USA.
trey@cscamm.umd.edu

Received: 18 January 2012
Revised: 11 December 2012

Abstract

Discrete-velocity approximations represent a popular way for computing the Boltzmann collision operator. The direct numerical evaluation of such methods involve a prohibitive cost, typically O(N^2d + 1) where d is the dimension of the velocity space. In this paper, following the ideas introduced in [C. Mouhot and L. Pareschi, C. R. Acad. Sci. Paris Sér. I Math. 339 (2004) 71–76, C. Mouhot and L. Pareschi, Math. Comput. 75 (2006) 1833–1852], we derive fast summation techniques for the evaluation of discrete-velocity schemes which permits to reduce the computational cost from O(N^2d + 1) to O(N̅^dN^d log₂N), N̅ ≪ N, with almost no loss of accuracy.