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We introduce a new numerical strategy to solve a class of oscillatorytransport PDE models which is able to captureaccurately the solutions withoutnumerically resolving the high frequency oscillations {\em in both space andtime}.Such PDE models arise in semiclassical modeling of quantum dynamics withband-crossings, and otherhighly oscillatory waves. Our first main idea is touse the nonlinear geometric optics ansatz, which builds theoscillatory phaseinto an independent variable. We then choose suitable initial data, based onthe Chapman-Enskog expansion, for the new model. For a scalar model, we provethat so constructed model will have certain smoothness, and consequently, for afirst order approximation scheme we prove uniform error estimates independentof the (possibly small) wave length. The method is extended to systems arisingfrom a semiclassical model for surface hopping, a non-adiabatic quantum dynamicphenomenon. Numerous numerical examples demonstrate that the method has thedesired properties.

Nonlinear Geometric Optics method based multi-scale numerical schemes for highly-oscillatory transport equations