Rosenau type approximations to the heat equation

My collaborator Giuseppe Toscani and myself submitted a new paper, entitled Large-time behavior of the solutions to Rosenau type approximations to the heat equation.
We study in this paper the validity of the approximation to the linear diffusion equation proposed by Rosenau as a regularized version of the Chapman-Enskog expansion of hydrodynamics. This approximation essentially is realized by substituting the heat equation with a linear kinetic equation of Boltzmann type, describing collisions of particles with a fixed background. This remark allows to consider the Rosenau approximation as a particular realization of a model Boltzmann equation, in which the background distribution is a general probability density with bounded variance. In addition to the Rosenau distribution, we considered also a point masses background, which furnishes the central difference scheme to solve numerically the linear diffusion equation.

A Rescaling Velocity Method for Dissipative Kinetic Equations

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.

Fast Algorithms for Discrete Velocity Models of Boltzmann Equation

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(N^2d+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(N^2d+1) to O(N^d log N), with almost no loss of accuracy.

Poster and teaching

I added a link to a poster I have presented last summer in the school Methods and Models on Kinetic Theory in Porto Ercole. I have also improved the content of the teaching page.

Opening of the website

This is the first version of my website. The content will improve soon. Feel free to give comment by email.