  In the previous studies, the high-order gas-kinetic schemes (HGKS) have
achieved successes for unsteady flows on three-dimensional unstructured meshes.
In this paper, to accelerate the rate of convergence for steady flows, the
implicit non-compact and compact HGKSs are developed. For non-compact scheme,
the simple weighted essentially non-oscillatory (WENO) reconstruction is used
to achieve the spatial accuracy, where the stencils for reconstruction contain
two levels of neighboring cells. Incorporate with the nonlinear generalized
minimal residual (GMRES) method, the implicit non-compact HGKS is developed. In
order to improve the resolution and parallelism of non-compact HGKS, the
implicit compact HGKS is developed with Hermite WENO (HWENO) reconstruction, in
which the reconstruction stencils only contain one level of neighboring cells.
The cell averaged conservative variable is also updated with GMRES method.
Simultaneously, a simple strategy is used to update the cell averaged gradient
by the time evolution of spatial-temporal coupled gas distribution function. To
accelerate the computation, the implicit non-compact and compact HGKSs are
implemented with the graphics processing unit (GPU) using compute unified
device architecture (CUDA). A variety of numerical examples, from the subsonic
to supersonic flows, are presented to validate the accuracy, robustness and
efficiency of both inviscid and viscous flows.
