I finished a new paper with my collaborator (and advisor) Francis Filbet: A Rescaling Velocity Method for Dissipative Kinetic Equations – Applications to Granular Media.
We present in this paper a new numerical algorithm based on a relative energy scaling for collisional kinetic equations allowing to study numerically their long time behavior, without the usual problems related to the change of scales in velocity variables. Several applications are presented for Boltzmann like equations. This method is particularly efficient for numerical simulations of the granular gases equation with dissipative energy: it allows to study accurately the long time behavior of this equation and is very well suited for the study of clustering phenomena.
I added a link to a paper I wrote with C. Mouhot and L. Pareschi: Convolutive decomposition and fast summation methods for discrete-velocity approximations of the Boltzmann equation. It 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 (DVM), 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(Nd log N), with almost no loss of accuracy.