This paper introduces a new algorithm to improve the accuracy of numerical phase-averaging in oscillatory, multiscale, differential equations. Phase-averaging is a technique that applies averaging to a mapped variable to remove highly oscillatory linear terms from the differential equation. This retains the main contribution of fast oscillations on the low frequencies without needing to resolve the rapid oscillations themselves. However, this comes at the cost of an averaging error, which we aim to offset with a modified mapping. The new mapping includes a mean correction which encodes an average measure of the nonlinear interactions. This mapping was introduced in Tao (2019) for weak nonlinearity and relied on classical time averaging. Our algorithm extends this work to the case where 1) the nonlinearity is not weak but the linear oscillations are fast and 2) finite averaging windows are applied via a smooth kernel, which has the advantage of retaining low frequencies whilst still eliminating the fastest oscillations. We show that the new algorithm reduces phase errors in the mapped variable for the swinging spring ODE. We also demonstrate accuracy improvements compared to standard phase-averaging in numerical experiments with the one-dimensional Rotating Shallow Water Equations, a useful test case for weather and climate applications. As the mean correction term can be computed in parallel, this new mapping has potential as a more accurate, yet still computationally cheap, coarse propagator for the oscillatory parareal method.

翻译：摘要：本文提出一种新算法，用于提升振荡多尺度微分方程数值相位平均的精度。相位平均是一种对映射变量施以平均处理，以消除微分方程中高度振荡线性项的技术。该方法保留了快速振荡对低频的主要贡献，而无需直接解析快速振荡本身。然而，这以引入平均误差为代价，本文旨在通过修正映射来抵消该误差。新映射包含一个均值校正项，该编码了非线性相互作用的平均度量。该映射最初由Tao(2019)针对弱非线性情况提出，并依赖于经典时域平均。我们的算法将工作扩展至以下情形：1)非线性非弱，但线性振荡为快速；2)通过光滑核函数施加有限平均窗口，其优势在于能在消除最快振荡的同时保留低频分量。我们证明，新算法可减小摆动弹簧常微分方程中映射变量的相位误差。在一维旋转浅水方程（天气与气候应用中具有代表性的测试案例）的数值实验中，我们也展示了相比标准相位平均的精度提升。由于均值校正项可并行计算，该新映射有潜力成为振荡并行算法中更精确且仍保持低计算成本的粗传播算子。