  For the simulations of unsteady flow, the global time step becomes really
small with a large variation of local cell size. In this paper, an implicit
high-order gas-kinetic scheme (HGKS) is developed to remove the restrictions on
the time step for unsteady simulations. In order to improve the efficiency and
keep the high-order accuracy, a two-stage third-order implicit time-accurate
discretization is proposed. In each stage, an artificial steady solution is
obtained for the implicit system with the pseudo-time iteration. In the
iteration, the classical implicit methods are adopted to solve the nonlinear
system, including the lower-upper symmetric Gauss-Seidel (LUSGS) and
generalized minimum residual (GMRES) methods. To achieve the spatial accuracy,
the HGKSs with both non-compact and compact reconstructions are constructed.
For the non-compact scheme, the weighted essentially non-oscillatory (WENO)
reconstruction is used. For the compact one, the Hermite WENO (HWENO)
reconstruction is adopted due to the updates of both cell-averaged flow
variables and their derivatives. The expected third-order temporal accuracy is
achieved with the two-stage temporal discretization. For the smooth flow, only
a single artificial iteration is needed. For uniform meshes, the efficiency of
the current implicit method improves significantly in comparison with the
explicit one. For the flow with discontinuities, compared with the well-known
Crank-Nicholson method, the spurious oscillations in the current schemes are
well suppressed. The increase of the artificial iteration steps introduces
extra reconstructions associating with a reduction of the computational
efficiency. Overall, the current implicit method leads to an improvement in
efficiency over the explicit one in the cases with a large variation of mesh
size.
