  In this work, a novel synthetic iteration scheme (SIS) is developed for the
LBE to find solutions to Kramer's problem accurately and efficiently: the
velocity distribution function is first solved by the conventional iterative
scheme, then it is modified such that in each iteration i) the flow velocity is
guided by an ordinary differential equation that is asymptotic-preserving at
the Navier-Stokes limit and ii) the shear stress is equal to the average shear
stress. Based on the Bhatnagar-Gross-Krook model, the SIS is assessed to be
efficient and accurate. Then we investigate the Kramer's problem for gases
interacting through the inverse power-law, shielded Coulomb, and Lennard-Jones
potentials, subject to diffuse-specular and Cercignani-Lampis gas-surface
boundary conditions. When the tangential momentum accommodation coefficient
(TMAC) is not larger than one, the Knudsen layer function is strongly affected
by the potential, where its value and width increase with the effective
viscosity index of gas molecules. Moreover, the Knudsen layer function exhibits
similarities among different values of TMAC when the intermolecular potential
is fixed. For Cercignani-Lampis boundary condition with TMAC larger than one,
both the viscous slip coefficient and Knudsen layer function are affected by
the intermolecular potential, especially when the "backward" scattering limit
is approached. With the asymptotic theory by Jiang and Luo for the singular
behavior of the velocity gradient in the vicinity of the solid surface, we find
that the whole Knudsen layer function can be well fitted by power series.
