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

A new class of uniformly accurate numerical schemes for highly oscillatory evolution equations