  An iterative scheme can be used to find a steady-state solution to the
Boltzmann equation, however, it is very slow to converge in the near-continuum
flow regime. In this paper, a synthetic iterative scheme is developed to speed
up the solution of the linearized Boltzmann equation. The velocity distribution
function is first solved by the conventional iterative scheme, then it is
corrected such that the macroscopic flow velocity is governed by a diffusion
equation which is asymptotic-preserving in the Navier-Stokes limit. The
efficiency of the new scheme is verified by calculating the eigenvalue of the
iteration, as well as solving for Poiseuille and thermal transpiration flows.
The synthetic iterative scheme is significantly faster than the conventional
iterative scheme in both the transition and the near-continuum flow regimes.
Moreover, due to the asymptotic-preserving properties, the SIS needs less
spatial resolution in the near-continuum flow regimes, which makes it even
faster than the conventional iterative scheme. Using this synthetic iterative
scheme, and the fast spectral approximation of the linearized Boltzmann
collision operator, Poiseuille and thermal transpiration flows between two
parallel plates, through channels of circular/rectangular cross sections, and
various porous media are calculated over the whole range of gas rarefaction.
Finally, the flow of a Ne-Ar gas mixture is solved based on the linearized
Boltzmann equation with the Lennard-Jones potential for the first time, and the
difference between these results and those using hard-sphere intermolecular
potential is discussed.
