A Convergence Theorem for the Parareal Algorithm Revisited
Ernest Scheiber
TL;DR
This paper revisits the Gander–Hairer convergence framework for the parareal algorithm, focusing on the practical pairing of a cheap Euler coarse propagator $C_{I_n}$ and a high-accuracy four-stage Runge–Kutta fine propagator $F_{I_n}$. Under a Lipschitz condition on $f$, the authors verify the theorem’s core bounds, establishing that $\|C_{I_n}(u_1)-C_{I_n}(u_2)\| \le (1+hL)\|u_1-u_2\|$, $\|F_{I_n}(u)-C_{I_n}(u)\| \le h^{2} \Lambda$ (with $\alpha=1$), and a Lipschitz bound on the discrepancy term with a suitable $c_3$, which together yield convergence of the parareal iterates as the time-step $h \to 0$. The analysis further extends to cases where several RK4 steps are taken per coarse interval and discusses backward-Euler as an alternative coarse solver, showing that the convergence conditions remain verifiable and robust. Overall, the work provides concrete, verifiable criteria for ensuring parareal convergence in practical simulations using Euler as the coarse solver and RK4 as the fine solver, strengthening the method’s reliability for parallel-in-time computations.
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.