Uniformly stable numerical schemes for the Boltzmann equation preserving the compressible Navier–Stokes asymptotics

Mounir Bennoune (1) , Mohammed Lemou (2) , Luc Mieussens (3)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 IRMAR - Institut de Recherche Mathématique de Rennes
3 IMB - Institut de Mathématiques de Bordeaux

In this paper, we develop a numerical method to solve Boltzmann like equations of kinetic theory which is able to capture the compressible Navier–Stokes dynamics at small Knudsen numbers. Our approach is based on the micro/macro decomposition technique, which applies to general collision operators. This decomposition is performed in all the phase space and leads to an equivalent formulation of the Boltzmann (or BGK) equation that couples a kinetic equation with macroscopic ones. This new formulation is then discretized with a semi-implicit time scheme combined with a staggered grid space discretization. Finally, several numerical tests are presented in order to illustrate the efficiency of our approach. Incidentally, we also introduce in this paper a modification of a standard splitting method that allows to preserve the compressible Navier–Stokes asymptotics in the case of the simplified BGK model. Up to our knowledge, this property is not known for general collision operators.

Mots clés: Boltzmann equation; BGK equation; Compressible Navier–Stokes equations; Chapman–Enskog expansion; Asymptotic preserving schemes; Micro–macro decomposition

Domaines: Analyse numérique [math.NA]

Mounir Bennoune, Mohammed Lemou, Luc Mieussens. Uniformly stable numerical schemes for the Boltzmann equation preserving the compressible Navier–Stokes asymptotics. Journal of Computational Physics, 2008, 227 (8), pp.3781-3803. ⟨10.1016/j.jcp.2007.11.032⟩. ⟨hal-00348598⟩