My collaborators Francis Filbet, Lorenzo Pareschi and myself just wrote a note entitled On steady-state preserving spectral methods for homogeneous Boltzmann equations.

In this note, we present a general way to construct spectral methods for the collision operator of the Boltzmann equation which preserves exactly the Maxwellian steady-state of the system.

We show that the resulting method is able to approximate with spectral accuracy the solution uniformly in time.Our approach is based on a micro-macro decomposition of the solution to the kinetic equation considered, and is in particular not limited to the Boltzmann equation. Its numerical complexity is the same than the one of a classical spectral method.