  We introduce a higher order phase averaging method for nonlinear oscillatory
systems. Phase averaging is a technique to filter fast motions from the
dynamics whilst still accounting for their effect on the slow dynamics. Phase
averaging is useful for deriving reduced models that can be solved numerically
with more efficiency, since larger timesteps can be taken. Recently, Haut and
Wingate (2014) introduced the idea of computing finite window numerical phase
averages in parallel as the basis for a coarse propagator for a
parallel-in-time algorithm. In this contribution, we provide a framework for
higher order phase averages that aims to better approximate the unaveraged
system whilst still filtering fast motions. Whilst the basic phase average
assumes that the solution is independent of changes of phase, the higher order
method expands the phase dependency in a basis which the equations are
projected onto. We illustrate the properties of this method on an ODE that
describes the dynamics of a swinging spring due to Lynch (2002). Although
idealized, this model shows an interesting analogy to geophysical flows as it
exhibits a slow dynamics that arises through the resonance between fast
oscillations. On this example, we show convergence to the non-averaged (exact)
solution with increasing approximation order also for finite averaging windows.
At zeroth order, our method coincides with a standard phase average, but at
higher order it is more accurate in the sense that solutions of the phase
averaged model track the solutions of the unaveraged equations more accurately.
