The article Convolutive decomposition and fast summation methods for discrete-velocity approximations of the Boltzmann equation, written in collaboration with C. Mouhot and L. Pareschi, has been accepted for publication in ESAIM: Mathematical Modeling and Numerical Analysis. Yay!
Here is a short abstract of this work:
This paper deals with a class of algorithms for computing the Boltzmann collision operator for hard spheres, and therefore computing the solutions to the Boltzmann equation. This class of algorithms is known as discrete-velocity methods, and is highly stable, due to the nice positivity and conservation properties of the schemes. The direct numerical evaluation of such methods involve a prohibitive cost, typically O(N2d+1) where d is the dimension of the velocity space. In this paper, we derive fast summation techniques for the evaluation of discrete-velocity schemes which permits to reduce the computational cost from O(N2d+1) to O(Ndlog N), with almost no loss of accuracy. It involves discrete (and fast) Fourier transforms and Farey series in number theory. We show some detailed numerical simulations supporting the interest of these numerical methods.