Solving the Boltzmann equation in N log N

Francis Filbet (1) , Clément Mouhot (2) , Lorenzo Pareschi (3)

1 MIP - Mathématiques pour l'Industrie et la Physique
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
3 Department of Mathematics [Ferrara]

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].

Mots clés: Boltzmann equation; Spectral methods; Fast algorithms; Entropy

Domaines: Equations aux dérivées partielles [math.AP]; Analyse numérique [math.NA]

Francis Filbet, Clément Mouhot, Lorenzo Pareschi. Solving the Boltzmann equation in N log N. SIAM Journal on Scientific Computing, 2006, 28, pp.1029-1053. ⟨10.1137/050625175⟩. ⟨hal-00087321⟩