  In the study of rarefied gas dynamics, the discrete velocity method (DVM) has
been widely employed to solve the gas kinetic equations. Although various
versions of DVM have been developed, their performance, in terms of accuracy
and computational efficiency, is yet to be compreheively studied in the whole
flow regime. Here, the traditional third-order time-implicit Godunov DVM (GDVM)
and the recently developed discrete unified gas-kinetic scheme (DUGKS) are
analysed in finding steady-state solutions of the force-driven Poiseuille and
lid-driven cavity flows. With the molecular collision and free streaming being
treated simultaneously, the DUGKS preserves the second-order accuracy in the
spatial and temporal discretizations in all flow regimes. Towards the
hydrodynamic flow regime, the DUGKS is not only faster than the GDVM when using
the same spatial mesh, but also requires less spatial resolution than that of
the GDVM to achieve the same numerical accuracy. From the slip to free
molecular flow regimes, however, the DUGKS is slower than the GDVM, due to the
complicated flux evaluation and the time step is less than the maximum
effective time step of the GDVM. Therefore, the DUGKS is preferable for
problems involving different flow regimes, particularly when the hydrodynamic
flow regime is dominant. For highly rarefied gas flows, if the steady-state
solution is concerned, the implicit DVM, which can boost the convergence
significantly, is a better choice.
