  We introduce a new numerical strategy to solve a class of oscillatory
transport PDE models which is able to captureaccurately the solutions without
numerically resolving the high frequency oscillations {\em in both space and
time}.Such PDE models arise in semiclassical modeling of quantum dynamics with
band-crossings, and otherhighly oscillatory waves. Our first main idea is to
use the nonlinear geometric optics ansatz, which builds theoscillatory phase
into an independent variable. We then choose suitable initial data, based on
the Chapman-Enskog expansion, for the new model. For a scalar model, we prove
that so constructed model will have certain smoothness, and consequently, for a
first order approximation scheme we prove uniform error estimates independent
of the (possibly small) wave length. The method is extended to systems arising
from a semiclassical model for surface hopping, a non-adiabatic quantum dynamic
phenomenon. Numerous numerical examples demonstrate that the method has the
desired properties.
