\theta-parareal schemes

TL;DR

This paper introduces a weighted parareal scheme for parallel-in-time computation, analyzing its stability and optimizing weights for multiscale coupling, demonstrated through numerical examples including nonlinear Hamiltonian systems.

Contribution

It proposes a weighted parareal method with stability analysis and weight optimization, enhancing multiscale coupling capabilities in parallel-in-time algorithms.

Findings

Favorable stability properties with marginal accuracy loss

Effective weight optimization using past iteration data

Successful application to nonlinear Hamiltonian systems

Abstract

A weighted version of the parareal method for parallel-in-time computation of time dependent problems is presented. Linear stability analysis for a scalar weighing strategy shows that the new scheme may enjoy favorable stability properties with marginal reduction in accuracy at worse. More complicated matrix-valued weights are applied in numerical examples. The weights are optimized using information from past iterations, providing a systematic framework for using the parareal iterations as an approach to multiscale coupling. The advantage of the method is demonstrated using numerical examples, including some well-studied nonlinear Hamiltonian systems.