  In this paper, a class of compact higher-order gas-kinetic schemes (GKS) with
spectral resolution will be presented. Based on the high-order gas evolution
model in GKS, both the interface flux function and conservative flow variables
can be evaluated explicitly from the time-accurate gas distribution function.
As a result, inside each control volume both the cell-averaged flow variables
and their cell-averaged gradients can be updated within each time step. The
flow variable update and slope update are coming from the same physical
solution at the cell interface. Different from many other approaches, such as
HWENO, there are no additional governing equations in GKS for the slopes or
equivalent degrees of freedom independently inside each cell. For nonlinear gas
dynamic evolution, the above compact linear reconstruction from the symmetric
stencil can be divided into sub-stencils and apply a biased nonlinear WENO-Z
reconstruction. In GKS, the time evolution solution of the gas distribution
function at a cell interface covers a physical process from an initial
non-equilibrium state to a final equilibrium one. The GKS evolution models
unifies the nonlinear and linear reconstructions in gas evolution process for
the determination of a time-dependent gas distribution function. This
dynamically adaptive model helps to solve a long lasting problem in the
development of high-order schemes about the choices of the linear and nonlinear
reconstructions. Compared with discontinuous Galerkin (DG) scheme, the current
compact GKS uses the same local and compact stencil, achieves the 6th-order and
8th-order accuracy, uses a much larger time step with CFL number $\geq 0.3$,
and gets accurate solutions in both shock and smooth regions without
introducing trouble cell and additional limiting process. At the same time, the
current scheme solves the Navier-Stokes equations.
