  Body force modelling in lattice Boltzmann method (LBM) has been extensively
studied in the incompressible limit but rarely discussed for thermal
compressible flows. Here we present a systematic approach of incorporating body
force in LBM which is valid for thermal compressible and non-equilibrium flows.
In particular, a LBM forcing scheme accurate for the energy equation with
second-order time accuracy is given. New and essential in this scheme is the
third-moment contribution of the force term. It is shown via Chapman-Enskog
analysis that the absence of this contribution causes an erroneous heat flux
quadratic in Mach number and linear in temperature variation. The theoretical
findings are verified and the necessity of the third-moment contribution
demonstrated by numerical simulations.
