Convergence analysis of a parareal algorithm with multistep fine propagator
For researchers in parallel-in-time integration, this provides a convergence-guaranteed method for multistep fine propagators, addressing a gap in existing theory.
The paper proposes a parareal algorithm for parabolic equations using a BDF2 fine propagator, achieving linear convergence with a novel correction for initialization steps. Numerical results show improvements over existing multistep strategies.
The parareal algorithm is a powerful parallel-in-time integration method that accelerates the numerical solution of evolution equations by iteratively combining a fine propagator and a coarse propagator. Although the convergence of the parareal algorithm has been extensively studied, most existing analyses assume that the fine propagator is either an exact solver or a single-step method. In this paper, we construct and analyze a parareal algorithm for solving parabolic equations, where the fine propagator is based on the two-step backward differentiation formula (BDF2), while the coarse propagator remains a single-step method. We propose a novel approach to design an effective correction for the initialization steps and establish linear convergence of the iteration. Numerical results fully support the theoretical findings, show clear improvements over existing multistep parareal strategies, and indicate that the proposed approach extends effectively to higher-order BDF methods and to nonlinear problems.