  In this paper an even higher-order compact GKS up to sixth order of accuracy
will be constructed for the shock and acoustic wave computation on unstructured
mesh. The compactness is defined by the physical domain of dependence for an
unstructured triangular cell, which may involve the closest neighbors of
neighboring cells. The compactness and high-order accuracy of the scheme are
coming from the consistency between the high-order initial reconstruction and
the high-order gas evolution model under GKS framework. The high-order
evolution solution at a cell interface provides not only a time-accurate flux
function, but also the time-evolving flow variables. Therefore, the
cell-averaged flow variables and their gradients can be explicitly updated at
the next time level from the moments of the same time-dependent gas
distribution function. Based on the cell averages and cell-averaged
derivatives, both linear and nonlinear high-order reconstruction can be
obtained for macroscopic flow variables in the evaluation of local equilibrium
and non-equilibrium states. The current nonlinear reconstruction is a
combination of WENO and ENO methodology. The initial piecewise discontinuous
reconstruction is used for the determination non-equilibrium state and an
evolved smooth reconstruction for the equilibrium state. The evolution model in
gas-kinetic scheme is based on a relaxation process from non-equilibrium to
equilibrium state. The accuracy, efficiency, and robustness of the scheme have
been validated. The main conclusion of the paper is that beyond the first-order
Riemann solver, the use of high-order gas evolution model seems necessary in
the development of high-order schemes.
