  In rarefied gas flows, the spatial grid size could vary by several orders of
magnitude in a single flow configuration (e.g., inside the Knudsen layer it is
at the order of mean free path of gas molecules, while in the bulk region it is
at a much larger hydrodynamic scale). Therefore, efficient implicit numerical
method is urgently needed for time-dependent problems. However, the
integro-differential nature of gas kinetic equations poses a grand challenge,
as the gain part of the collision operator is non-invertible. Hence an
iterative solver is required in each time step, which usually takes a lot of
iterations in the (near) continuum flow regime where the Knudsen number is
small; worse still, the solution does not asymptotically preserve the fluid
dynamic limit when the spatial cell size is not refined enough. Inspired by our
general synthetic iteration scheme for steady-state solution of the Boltzmann
equation, we propose two numerical schemes to push the multiscale simulation of
unsteady rarefied gas flows to a new boundary, that is, the numerical solution
not only converges within dozens of iterations in each time step, but also
asymptotic preserves the Navier-Stokes-Fourier limit at coarse spatial grid,
even when the time step is large (e.g., in the extreme slow decay of
two-dimensional Taylor vortex, the time step is at the order of vortex decay
time). The properties of fast convergence and asymptotic preserving of the
proposed schemes are not only rigorously proven by the Fourier stability
analysis, but also demonstrated by solid numerical examples.
