  We introduce a new methodology to design uniformly accurate methods for
oscillatory evolution equations. The targeted models are envisaged in a wide
spectrum of regimes, from non-stiff to highly-oscillatory. Thanks to an
averaging transformation, the stiffness of the problem is softened, allowing
for standard schemes to retain their usual orders of convergence. Overall,
high-order numerical approximations are obtained with errors and at a cost
independent of the regime.
