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 Parareal convergence for oscillatory pdes with finite time-scale separation (2019), A. G. Peddle, T. Haut, and B. Wingate, [16] 以及 An asymptotic parallel-in-time method for highly oscillatory pdes (2014), T. Haut and B. Wingate, [10] 研究的扩展。在这些研究中，研究者们考察了带有平均化的两层 Parareal 方法。本文所提出的方法是带有任意层数量的多层 Parareal 方法，不受限于两层的情况。我们给出了一个渐近误差估计，该估计在只考虑两层的情况下可以简化。引入多于两层对于平均化过程具有重要的影响，因为我们为每个不同层级选择各自的平均化窗口，这是本研究的一个额外新特点。不同的平均化窗口使得我们提出的方法特别适合于多尺度问题，因为我们可以为每个固有尺度引入一个层级，并根据该层级所分辨的特定尺度来调整平均化过程以复现模型的行为。本文还研究了新方法的计算复杂度和在几个示例上的效率。
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