  Computational fluid dynamics is a direct modeling of physical laws in a
discretized space. The basic physical laws include the mass, momentum and
energy conservations, physically consistent transport process, and similar
domain of dependence and influence between the physical reality and the
numerical representation. Therefore, a physically soundable numerical scheme
must be a compact one which involves the closest neighboring cells within the
domain of dependence for the solution update under a CFL number $(\sim 1 )$. In
the construction of explicit high-order compact scheme, subcell flow
distributions or the equivalent degree of freedoms beyond the cell averaged
flow variables must be evolved and updated, such as the gradients of the flow
variables inside each control volume. The direct modeling of flow evolution
under generalized initial condition will be developed in this paper. The direct
modeling will provide the updates of flow variables differently on both sides
of a cell interface and limit high-order time derivatives of the flux function
nonlinearly in case of discontinuity in time, such as a shock wave moving
across a cell interface within a time step. The direct modeling unifies the
nonlinear limiters in both space for the data reconstruction and time for the
time-dependent flux transport. Under the direct modeling framework, as an
example, the high-order compact gas-kinetic scheme (GKS) will be constructed.
The scheme shows significant improvement in terms of robustness, accuracy, and
efficiency in comparison with the previous high-order compact GKS.
