  In this paper, the high-order compact gas-kinetic scheme (CGKS) on
three-dimensional hybrid unstructured mesh is further developed with the
p-multigrid technique for steady-state solution acceleration. The p-multigrid
strategy is a two-level algorithm. On the high-order level, the high-order CGKS
is used to evolve both cell-averaged conservative flow variables and their
gradients under high-order compact initial reconstruction at the beginning of
next time step. On the low-order level, starting from the high-order level
solution the cell-averaged conservative flow variables is evolved by a
first-order scheme, where implicit backward Euler smoother is adopted for
accelerating the convergence of steady-state solution. The final iterative
updating scheme becomes numerically simple and computationally efficient. The
effectiveness of the p-multigrid method is validated in both subsonic and
supersonic flow simulations in two- and three-dimensional space with hybrid
unstructured mesh. One order of magnitude speedup in the convergence rate has
be achieved by the approach in comparison with the explicit counterpart.
