  In this paper, for the first time a compact third-order gas-kinetic scheme is
proposed on unstructured meshes for the compressible viscous flow computations.
The possibility to de sign such a third-order compact scheme is due to the
high-order gas evolution model, where a time-dependent gas distribution
function at a cell interface not only provides the fluxes across a cell
interface, but also the time evolution of the flow variables at the cell
interface as well. As a result, both cell averaged and cell interface flow
variables can be used for the initial data reconstruction at the beginning of
next time step. A weighted least-square reconstruction has been used for the
construction of a third-order initial condition. Therefore, a compact
third-order gas-kinetic scheme with the involvement of neighboring cells only
can be developed on unstructured meshes. In comparison with other conventional
high-order schemes, the current method avoids the use of Gaussian points for
the flux integration along a cell interface and the multi-stage Runge-Kutta
time stepping technique. The third-order compact scheme is numerically stable
under CFL condition above 0.5. Due to the multidimensional gas-kinetic
formulation and the coupling of inviscid and viscous terms, even with
unstructured meshes the boundary layer solution and the vortex structure can be
accurately captured in the current scheme. At the same time, the compact scheme
can capture strong shocks as well.
