Impact of spatial coarsening on Parareal convergence for the linear advection equation

TL;DR

This paper investigates how spatial coarsening affects the convergence of the Parareal method for the linear advection equation, revealing that traditional spectral radius measures are unreliable and proposing pseudo-spectral radius as a better predictor.

Contribution

It provides a theoretical analysis showing the limitations of the 2-norm for predicting convergence and introduces the pseudo-spectral radius as a more reliable indicator for hyperbolic problems with spatial coarsening.

Findings

2-norm of the iteration matrix does not predict convergence reliably.

Pseudo-spectral radius correlates with convergence behavior.

Numerical results support pseudo-spectral radius as a convergence predictor.

Abstract

The Parareal parallel-in-time integration method often performs poorly when applied to hyperbolic partial differential equations. This effect is even more pronounced when the coarse propagator uses a reduced spatial resolution. However, some combinations of spatial discretization and numerical time stepping nevertheless allow for Parareal to converge with monotonically decreasing errors. This raises the question how these configurations can be distinguished theoretically from those where the error initially increases, sometimes over many orders of magnitude. For linear problems, we prove a theorem that implies that the 2-norm of the Parareal iteration matrix is not a suitable tool to predict convergence for hyperbolic problems when spatial coarsening is used. We then show numerical results that suggest that the pseudo-spectral radius can reliably indicate if a given configuration of Parareal will show transient growth or monotonic convergence. For the studied examples, it also provides a good quantitative estimate of the convergence rate in the first few Parareal iterations.