  In this paper, a compact high-order gas-kinetic scheme (GKS) with spectral
resolution will be presented and used in the simulation of acoustic and shock
waves. For accurate simulation, the numerical scheme is required to have
excellent dissipation-dispersion preserving property, while the wave modes,
propagation characteristics, and wave speed of the numerical solution should be
kept as close as possible to the exact solution of governing equations. For
compressible flow simulation with shocks, the numerical scheme has to be
equipped with proper numerical dissipation to make a crispy transition in the
shock layer. Based on the high-order gas evolution model, the GKS provides a
time accurate solution at a cell interface, from which both time accurate flux
function and the time evolving flow variables can be obtained. The GKS updates
explicitly both cell-averaged conservative flow variables and the cell-averaged
gradients by applying Gauss-theorem along the boundary of the control volume.
Based on the cell-averaged flow variables and cell-averaged gradients, a
reconstruction with compact stencil can be obtained. With the same stencil of a
second-order scheme, a reconstruction up to 8th-order spacial accuracy can be
constructed, which include the nonlinear and linear reconstruction for the
non-equilibrium and equilibrium states respectively. The GKS unifies the
nonlinear and linear reconstruction through a time evolution process at a cell
interface from the non-equilibrium state to an equilibrium one. In the region
between these two limits, the contribution from nonlinear and linear
reconstructions depends on the weighting functions of $\exp(-\Delta t/\tau)$
and $(1-\exp(-\Delta t /\tau))$, where $\Delta t$ is the time step and $\tau$
is the particle collision time, which is enhanced in the shock region. As a
result, both shock and acoustic wave can be captured accurately in GKS.
