Micro-macro decomposition based asymptotic-preserving numerical schemes and numerical moments conservation for collisional nonlinear kinetic equations
This work provides a numerical framework for efficiently solving multiscale collisional kinetic equations with exact moment conservation, benefiting computational physics and engineering applications involving rarefied gas dynamics and plasma physics.
The authors extend the micro-macro decomposition method to general collisional kinetic equations (Boltzmann and Fokker-Planck-Landau), enabling efficient computation across all regimes including fluid regimes without resolving small Knudsen numbers. The schemes preserve moments exactly and can be implemented explicitly despite containing implicit terms.
In this paper, we first extend the micro-macro decomposition method for multiscale kinetic equations from the BGK model to general collisional kinetic equations, including the Boltzmann and the Fokker-Planck Landau equations. The main idea is to use a relation between the (numerically stiff) linearized collision operator with the nonlinear quadratic ones, the later's stiffness can be overcome using the BGK penalization method of Filbet and Jin for the Boltzmann, or the linear Fokker-Planck penalization method of Jin and Yan for the Fokker-Planck Landau equations. Such a scheme allows the computation of multiscale collisional kinetic equations efficiently in all regimes, including the fluid regime in which the fluid dynamic behavior can be correctly computed even without resolving the small Knudsen number. A distinguished feature of these schemes is that although they contain implicit terms, they can be implemented explicitly. These schemes preserve the moments (mass, momentum and energy) exactly thanks to the use of the macroscopic system which is naturally in a conservative form. We further utilize this conservation property for more general kinetic systems, using the Vlasov-Ampére and Vlasov-Ampére-Boltzmann systems as examples. The main idea is to evolve both the kinetic equation for the probability density distribution and the moment system, the later naturally induces a scheme that conserves exactly the moments numerically if they are physically conserved.