  Due to the limited cell resolution in the representation of flow variables, a
piecewise continuous initial reconstruction with discontinuous jump at a cell
interface is usually used in modern computational fluid dynamics methods.
Starting from the discontinuity, a Riemann problem in the Godunov method is
solved for the flux evaluation across the cell interface in a finite volume
scheme. With the increasing of Mach number in the CFD simulations, the
adaptation of the Riemann solver seems introduce intrinsically a mechanism to
develop instabilities in strong shock regions. Theoretically, the Riemann
solution of the Euler equations are based on the equilibrium assumption, which
may not be valid in the non-equilibrium shock layer. In order to clarify the
flow physics from a discontinuity, the unsteady flow behavior of
one-dimensional contact and shock wave is studied on a time scale of (0~10000)
times of the particle collision time. In the study of the non-equilibrium flow
behavior from a discontinuity, the collision-less Boltzmann equation is first
used for the time scale within one particle collision time, then the direct
simulation Monte Carlo (DSMC) method will be adapted to get the further
evolution solution. The transition from the free particle transport to the
dissipative Navier-Stokes (NS) solutions are obtained as an increasing of time.
The exact Riemann solution becomes a limiting solution with infinite number of
particle collisions. For the high Mach number flow simulations, the points in
the shock transition region, even though the region is enlarged numerically to
the mesh size, should be considered as the points inside a highly
non-equilibrium shock layer.
