# Parareal Convergence for Oscillatory PDEs with Finite Time-scale Separation

## TL;DR

This paper extends the convergence analysis of the Parareal method for oscillatory PDEs to cases with finite time-scale separation, providing practical guidelines for parameter selection.

## Contribution

It proves convergence of the Parareal method for finite time-scale separation and introduces an optimization-based approach for choosing averaging parameters.

## Key findings

- Convergence is achievable with finite time-scale separation.
- An optimization problem determines the averaging window and time step.
- Guidelines for parameter selection improve practical implementation.

## Abstract

A variant of the Parareal method for highly oscillatory systems of PDEs was proposed by Haut and Wingate (2014). In that work they proved superlinear conver- gence of the method in the limit of infinite time scale separation. Their coarse solver features a coordinate transformation and a fast-wave averag- ing method inspired by analysis of multiple scales PDEs and is integrated using an HMM-type method. However, for many physical applications the timescale separation is finite, not infinite. In this paper we prove con- vergence for finite timescale separaration by extending the error bound on the coarse propagator to this case. We show that convergence requires the solution of an optimization problem that involves the averaging win- dow interval, the time step, and the parameters in the problem. We also propose a method for choosing the averaging window relative to the time step based as a function of the finite frequencies inherent in the problem.