  This work is devoted to the numerical simulation of nonlinear Schr\"odinger
and Klein-Gordon equations. We present a general strategy to construct
numerical schemes which are uniformly accurate with respect to the oscillation
frequency. This is a stronger feature than the usual so called "Asymptotic
preserving" property, the last being also satisfied by our scheme in the highly
oscillatory limit. Our strategy enables to simulate the oscillatory problem
without using any mesh or time step refinement, and the orders of our schemes
are preserved uniformly in all regimes. In other words, since our numerical
method is not based on the derivation and the simulation of asymptotic models,
it works in the regime where the solution does not oscillate rapidly, in the
highly oscillatory limit regime, and in the intermediate regime with the same
order of accuracy. In the same spirit as in [5], the method is based on two
main ingredients. First, we embed our problem in a suitable "two-scal"
reformulation with the introduction of an additional variable. Then a link is
made with classical strategies based on Chapman-Enskog expansions in kinetic
theory despite the dispersive context of the targeted equations, allowing to
separate the fast time scale from the slow one. Uniformly accurate (UA) schemes
are eventually derived from this new formulation and their properties and
performances are assessed both theoretically and numerically.
