A Parareal algorithm without Coarse Propagator?

TL;DR

This paper introduces a novel Parareal algorithm that operates without a coarse propagator, demonstrating its effectiveness for parabolic problems and providing new convergence proofs, supported by numerical experiments.

Contribution

It presents the first Parareal algorithm without a coarse propagator and offers a new convergence analysis for space-approximated coarse propagators.

Findings

Effective for parabolic problems

Convergence proven for space-based coarse propagators

Not suitable for hyperbolic problems

Abstract

The Parareal algorithm was invented in 2001 in order to parallelize the solution of evolution problems in the time direction. It is based on parallel fine time propagators called F and sequential coarse time propagators called G, which alternatingly solve the evolution problem and iteratively converge to the fine solution. The coarse propagator G is a very important component of Parareal, as one sees in the convergence analyses. We present here for the first time a Parareal algorithm without coarse propagator, and explain why this can work very well for parabolic problems. We give a new convergence proof for coarse propagators approximating in space, in contrast to the more classical coarse propagators which are approximations in time, and our proof also applies in the absence of the coarse propagator. We illustrate our theoretical results with numerical experiments, and also explain why this approach can not work for hyperbolic problems.