Projection-Based Parareal for Nonlinear Initial-Value Problems
Abstract
Parareal is a parallel-in-time method for initial-value problems that has found extensive applications and given rise to many variants. Krylov-enhanced parareal is a variant of parareal that uses projection techniques to accelerate convergence substantially for linear problems. In this paper, we study a projection-based parareal algorithm for nonlinear initial-value problems with a modified iteration scheme and an arbitrary orthogonal projection. For a general nonlinear problem, we prove a convergence estimate for each iteration step on a fixed time grid. We analyze the errors of the modified scheme and the scheme of Krylov-enhanced parareal relative to the old error as the coarse grid is refined. For a coarse solver of order p and step size h, we prove error bounds of O(hp+1) for the modified scheme and O(h) for that of Krylov-enhanced parareal at the first potentially inexact time point under mild assumptions. Numerical experiments on two nonlinear problems are consistent with the theoretical analysis.
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BibTeX
@misc{tan2026projectionbased,
title = {{Projection-Based Parareal for Nonlinear Initial-Value Problems}},
author = {Chenyi Tan and Yuncheng Xu and Yehao Zhang and Yangfeng Su},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21680},
eprint = {2610.21680},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21680},
}
RIS
TY - UNPB
TI - Projection-Based Parareal for Nonlinear Initial-Value Problems
AU - Chenyi Tan
AU - Yuncheng Xu
AU - Yehao Zhang
AU - Yangfeng Su
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.21680
UR - https://arxiv.org/abs/2610.21680
AB - Parareal is a parallel-in-time method for initial-value problems that has found extensive applications and given rise to many variants. Krylov-enhanced parareal is a variant of parareal that uses projection techniques to accelerate convergence substantially for linear problems. In this paper, we study a projection-based parareal algorithm for nonlinear initial-value problems with a modified iteration scheme and an arbitrary orthogonal projection. For a general nonlinear problem, we prove a convergence estimate for each iteration step on a fixed time grid. We analyze the errors of the modified scheme and the scheme of Krylov-enhanced parareal relative to the old error as the coarse grid is refined. For a coarse solver of order p and step size h, we prove error bounds of O(h^p+1) for the modified scheme and O(h) for that of Krylov-enhanced parareal at the first potentially inexact time point under mild assumptions. Numerical experiments on two nonlinear problems are consistent with the theoretical analysis.
ER -