A Convergence Theorem for the Parareal Algorithm Revisited

TL;DR

This paper revisits the convergence conditions of the parareal algorithm, specifically analyzing the case with Euler as the coarse integrator and Runge-Kutta as the fine integrator, using Gander and Hairer's theorem.

Contribution

It provides a detailed verification of convergence conditions for the parareal algorithm with specific integrators, enhancing theoretical understanding.

Findings

Convergence conditions are verified for Euler and Runge-Kutta integrators.

The analysis confirms the applicability of Gander and Hairer's theorem to this setting.

Results improve theoretical foundations for parallel-in-time algorithms.

Abstract

The subject of the paper is to verify the convergence conditions for the parareal algorithm using Gander and Hairer's theorem . The analysis is conducted in the case where the coarse integrator is the Euler method and the high-accuracy integrator is an explicit Runge-Kutta type method.