Parareal methods for highly oscillatory dynamical systems

TL;DR

This paper presents a novel parareal multiscale method for highly oscillatory systems, enabling efficient parallel computation of slow variables and phase alignment without system splitting, with proven convergence for almost-periodic solutions.

Contribution

It introduces a new coupling strategy combining parareal and multiscale integrators, including an alignment algorithm for phase variables, applicable to systems with hidden slow dynamics and resonances.

Findings

Convergence proved for almost-periodic solutions.

Method effectively handles hidden slow variables.

Numerical examples demonstrate applicability.

Abstract

We introduce a new strategy for coupling the parallel in time (parareal) iterative methodology with multiscale integrators. Following the parareal framework, the algorithm computes a low-cost approximation of all slow variables in the system using an appropriate multiscale integrator, which is refined using parallel fine scale integrations. Convergence is obtained using an alignment algorithm for fast phase-like variables. The method may be used either to enhance the accuracy and range of applicability of the multiscale method in approximating only the slow variables, or to resolve all the state variables. The numerical scheme does not require that the system is split into slow and fast coordinates. Moreover, the dynamics may involve hidden slow variables, for example, due to resonances. We propose an alignment algorithm for almost-periodic solution, in which case convergence of the parareal iterations is proved. The applicability of the method is demonstrated in numerical examples.