Analysis of spectral methods for the homogeneous Boltzmann equation

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

1 ICJ - Institut Camille Jordan
2 DAMTP - Department of Applied Mathematics and Theoretical Physics
3 DMA - Département de Mathématiques et Applications - ENS Paris

The development of accurate and fast algorithms for the Boltzmann collision integral and their analysis represent a challenging problem in scientific computing and numerical analysis. Recently, several works were devoted to the derivation of spectrally accurate schemes for the Boltzmann equation, but very few of them were concerned with the stability analysis of the method. In particular there was no result of stability except when the method was modified in order to enforce the positivity preservation, which destroys the spectral accuracy. In this paper we propose a new method to study the stability of homogeneous Boltzmann equations perturbed by smoothed balanced operators which do not preserve positivity of the distribution. This method takes advantage of the ''spreading'' property of the collision, together with estimates on regularity and entropy production. As an application we prove stability and convergence of spectral methods for the Boltzmann equation, when the discretization parameter is large enough (with explicit bound).

Mots clés: Boltzmann equation, spectral methods, numerical stability, asymptotic stability, Fourier-Galerkin method; Boltzmann equation, spectral methods, numerical stability, asymptotic stability, Fourier-Galerkin method

Domaines: Analyse numérique [math.NA]

Francis Filbet, Clément Mouhot. Analysis of spectral methods for the homogeneous Boltzmann equation. Transactions of the American Mathematical Society, 2011, 363 (4), pp.1947-1980. ⟨10.1090/S0002-9947-2010-05303-6⟩. ⟨hal-00659629⟩