Wave propagation characteristics of Parareal

TL;DR

This paper analyzes the wave propagation and stability issues of the Parareal parallel-in-time method for hyperbolic equations, identifying phase errors as a key factor and suggesting tailored coarse methods as a solution.

Contribution

It provides a detailed dispersion relation analysis of Parareal, revealing the cause of stability problems and proposing potential remedies through improved coarse propagators.

Findings

Instability arises from amplification factor convergence from above at high wave numbers.

Phase errors in the coarse propagator are identified as the main cause of instability.

Tailored coarse level methods could mitigate stability issues.

Abstract

The paper derives and analyses the (semi-)discrete dispersion relation of the Parareal parallel-in-time integration method. It investigates Parareal's wave propagation characteristics with the aim to better understand what causes the well documented stability problems for hyperbolic equations. The analysis shows that the instability is caused by convergence of the amplification factor to the exact value from above for medium to high wave numbers. Phase errors in the coarse propagator are identified as the culprit, which suggests that specifically tailored coarse level methods could provide a remedy.