  In the frame of the Boltzmann equation, wall-bounded flows of rarefied gases
require the implementation of boundary conditions at the kinetic level. Such
boundary conditions induce a discontinuity in the distribution function with
respect to the component of the momentum which is normal to the boundary.
Expanding the distribution function with respect to half-range polynomials
allows this discontinuity to be captured. The implementation of this concept
has been reported in the literature only for force-free flows. In the case of
general forces which can have non-zero components in the direction
perpendicular to the walls, the implementation of the force term requires
taking the momentum space gradient of a discontinuous function. Our proposed
method deals with this difficulty by employing the theory of distributions. We
validate our procedure by considering the simple one-dimensional flow between
diffuse-reflective walls of equal or different temperatures driven by the
constant gravitational force. For this flow, a comparison between the results
obtained with the full-range and the half-range Gauss-Hermite LB models is also
presented.
