  The unified gas kinetic scheme (UGKS) is a direct modeling method based on
the gas dynamical model on the mesh size and time step scales. With the
implementation of particle transport and collision in a time-dependent flux
function, the UGKS can recover multiple flow physics from the kinetic particle
transport to the hydrodynamic wave propagation. In comparison with direct
simulation Monte Carlo (DSMC), the equations-based UGKS can use the implicit
techniques in the updates of macroscopic conservative variables and microscopic
distribution function. The implicit UGKS significantly increases the
convergence speed for steady flow computations, especially in the highly
rarefied and near continuum regime. In order to further improve the
computational efficiency, for the first time a geometric multigrid technique is
introduced into the implicit UGKS, where the prediction step for the
equilibrium state and the evolution step for the distribution function are both
treated with multigrid acceleration. The multigrid implicit UGKS (MIUGKS) is
used in the non-equilibrium flow study, which includes microflow, such as
lid-driven cavity flow and the flow passing through a finite-length flat plate,
and high speed one, such as supersonic flow over a square cylinder. The MIUGKS
shows 5 to 9 times efficiency increase over the previous implicit scheme. For
the low speed microflow, the efficiency of MIUGKS is several orders of
magnitude higher than the DSMC. Even for the hypersonic flow at Mach number 5
and Knudsen number 0.1, the MIUGKS is still more than 100 times faster than the
DSMC method for a convergent steady state solution.
