Multi-level Parareal algorithm with Averaging for Oscillatory Problems

TL;DR

This paper introduces a multi-level Parareal algorithm with adaptive averaging windows tailored for multi-scale oscillatory PDEs, extending previous two-level methods and demonstrating improved efficiency and scalability.

Contribution

It develops a multi-level Parareal method with variable averaging windows for each level, enhancing the handling of multi-scale oscillatory problems beyond existing two-level approaches.

Findings

Asymptotic error estimate reduces to two-level case

Method effectively captures multiple intrinsic scales

Demonstrated improved efficiency on example problems

Abstract

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.