Convolutive decomposition and fast summation methods for   discrete-velocity approximations of the Boltzmann equation
  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 [27,28], 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 $O(\bar{N}^d N^d\log_2 N)$, $\bar{N} << N$, with almost no loss of accuracy. 
Cl\'ement Mouhot, Lorenzo Pareschi (DPT OF MATH., UNIV. OF FERRARA),   Thomas Rey (ICJ)
math.NA math.AP