  The kinetic theory provides a physical basis for developing multiscal methods
for gas flows covering a wide range of flow regimes. A particular challenge for
kinetic schemes is whether they can capture the correct hydrodynamic behaviors
of the system in the continuum regime (i.e., as the Knudsen number $\epsilon\ll
1$ ) without enforcing kinetic scale resolution. At the current stage, {the
main approach to analyze such a property is the asymptotic preserving (AP)
concept, which aims to show whether a kinetic scheme reduces to a solver for
the hydrodynamic equations as $\epsilon \to 0$, such as the shock capturing
scheme for the Euler equations. However, the detailed asymptotic properties of
the kinetic scheme are indistinguishable when $\epsilon$ is small but finite
under the AP framework}. In order to distinguish different characteristics of
kinetic schemes, in this paper we introduce the concept of unified preserving
(UP) aiming at assessing asmyptotic orders of a kinetic scheme by employing the
modified equation approach and Chapman-Enskon analysis. It is shown that the UP
properties of a kinetic scheme generally depend on the spatial/temporal
accuracy and closely on the inter-connections among the three scales (kinetic
scale, numerical scale, and hydrodynamic scale) and their corresponding coupled
dynamics. Specifically, the numerical resolution and specific discretization of
particle transport and collision determine the flow physics of the scheme in
different regimes, especially in the near continuum limit. As two examples, the
UP methodology is applied to analyze the discrete unified gas-kinetic scheme
and a second-order implicit-explicit Runge-Kutta scheme in their asymptotic
behaviors in the continuum limit.
