The present study is an extension of the work done in Parareal convergence for oscillatory pdes with finite time-scale separation (2019), A. G. Peddle, T. Haut, and B. Wingate, [16], and An asymptotic parallel-in-time method for highly oscillatory pdes (2014), T. Haut and B. Wingate, [10], where a two-level Parareal method with averaging is examined. The method proposed in this paper is a multi-level Parareal method with arbitrarily many levels, which is not restricted to the two-level case. We give an asymptotic error estimate which reduces to the two-level estimate for the case when only two levels are considered. Introducing more than two levels has important consequences for the averaging procedure, as we choose separate averaging windows for each of the different levels, which is an additional new feature of the present study. The different averaging windows make the proposed method especially appropriate for multi-scale problems, because we can introduce a level for each intrinsic scale of the problem and adapt the averaging procedure such that we reproduce the behavior of the model on the particular scale resolved by the level. The computational complexity of the new method is investigated and the efficiency is studied on several examples.

翻译：本研究是对《有限时间尺度分离下振荡偏微分方程的帕拉雷尔收敛性》(2019, A.G. Peddle, T. Haut, B. Wingate, [16]) 和《高振荡偏微分方程的渐近并行时间方法》(2014, T. Haut, B. Wingate, [10]) 的扩展延伸。前述工作研究了带有平均算子的双层帕拉雷尔方法，而本文提出的方法是一种可支持任意层级的多层帕拉雷尔方法，不再局限于双层情形。我们给出了渐近误差估计式，该估计在仅考虑双层情形时可退化为双层估计。引入两层以上层级对平均过程具有重要影响，我们为不同层级分别选择独立的平均窗口——这是本研究的另一新特性。不同平均窗口使所提方法特别适用于多尺度问题：可针对问题的每个固有尺度引入对应层级，并通过调整平均过程，使模型在层级所解析的特定尺度上复现其动态行为。本文研究了新方法的计算复杂度，并通过多个算例验证其计算效率。