Fast algorithms for computing the Boltzmann collision operator

The development of accurate and fast numerical schemes for the five fold Boltzmann collision integral represents a challenging problem in scientific computing. For a particular class of interactions, including the so-called hard spheres model in dimension three, we are able to derive spectral methods that can be evaluated through fast algorithms. These algorithms are based on a suitable representation and approximation of the collision operator. Explicit expressions for the errors in the schemes are given and spectral accuracy is proved. Parallelization properties and adaptivity of the algorithms are also discussed.Comment: 22 page

Solving the Boltzmann equation in N log N

In [C. Mouhot and L. Pareschi, "Fast algorithms for computing the Boltzmann collision operator," Math. Comp., to appear; C. Mouhot and L. Pareschi, C. R. Math. Acad. Sci. Paris, 339 (2004), pp. 71-76], fast deterministic algorithms based on spectral methods were derived for the Boltzmann collision operator for a class of interactions including the hard spheres model in dimension three. These algorithms are implemented for the solution of the Boltzmann equation in two and three dimension, first for homogeneous solutions, then for general non homogeneous solutions. The results are compared to explicit solutions, when available, and to Monte-Carlo methods. In particular, the computational cost and accuracy are compared to those of Monte-Carlo methods as well as to those of previous spectral methods. Finally, for inhomogeneous solutions, we take advantage of the great computational efficiency of the method to show an oscillation phenomenon of the entropy functional in the trend to equilibrium, which was suggested in the work [L. Desvillettes and C. Villani, Invent. Math., 159 (2005), pp. 245-316].Comment: 32 page

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.Comment: v2: 22 pages, improvement of the presentation and more details given in some proofs. arXiv admin note: text overlap with arXiv:1106.1020 by other author

On deterministic approximation of the Boltzmann equation in a bounded domain

In this paper we present a fully deterministic method for the numerical solution to the Boltzmann equation of rarefied gas dynamics in a bounded domain for multi-scale problems. Periodic, specular reflection and diffusive boundary conditions are discussed and investigated numerically. The collision operator is treated by a Fourier approximation of the collision integral, which guarantees spectral accuracy in velocity with a computational cost of $N\,\log(N)$, where $N$ is the number of degree of freedom in velocity space. This algorithm is coupled with a second order finite volume scheme in space and a time discretization allowing to deal for rarefied regimes as well as their hydrodynamic limit. Finally, several numerical tests illustrate the efficiency and accuracy of the method for unsteady flows (Poiseuille flows, ghost effects, trend to equilibrium)