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.
翻译:我们引入了非线性悬浮系统的更高排序阶段平均平均法。 阶段平均法是一种从动态中过滤快速移动的技术, 同时也考虑到动态对慢动态的影响。 阶段平均法对于得出由于可以采取更大的时间步骤而能够以更高的时间步骤而以更高的效率以数字方式解决的简化模型很有用。 最近, 上和 永加( Wingate) (2014) 引入了计算定窗口数字阶段平均值的理念, 并同时作为平行时间算法的粗缩推进器的基础。 在这项贡献中, 我们为较高排序平均平均水平提供了一种框架, 目的是更好地接近非平均系统, 并同时过滤快速移动。 基本阶段平均平均平均平均平均水平假定解决方案的趋同性, 也就是在平均水平标准阶段的趋同性上, 我们展示了平均水平标准阶段的趋同性, 也就是平均水平标准阶段的趋同性, 也就是平均标准阶段的趋同性, 也就是平均标准阶段的趋同性, 也就是平均标准阶段的趋同性。