This talk concerns some highly-oscillatory phenomena which present the numerical challenges of both stiffness and energy preservation. Even appropriate numerical methods often present deteriorated accuracy for some time-steps. The method of high-order averaging isolates non-stiff drift dynamics from stiff oscillations in a new differential equation that is easier to solve numerically. While this approach has successfully been extended to other contexts such as relaxation problems and multi-frequency forcings, its current reliance on heavy formal computations limits its implementations.

The goal here is to introduce a new framework for high-order averaging inspired by normal forms. After presenting the necessary notions of two-scale numerical accuracy and of geometric properties, the key assumptions and tools ensuring well-posedness and uniform accuracy will be detailed in the multi-frequency case. These results will be illustrated with preliminary numerical experiments, using automatic differentiation to avoid formal computations.