Analysis of the parareal algorithm for linear parametric differential equations

Abstract

This paper presents a parareal algorithm with parameterized propagators for linear parametric differential equations over a wide range of parameters. Through transforming the initial value problem into nonparametric ODEs based on Taylor series, we construct the general parameterized fine and coarse propagators for the parareal algorithm in each time subinterval. Furthermore, to accelerate the convergence of the algorithm, the coarse propagator based on the waveform relaxation method is proposed. By analysing the computational complexity of the WR propagator and the general coarse propagator, we find these two propagators are appropriate for the different situations. Finally, the convergence analysis of the parareal algorithm with these propagators is presented and our analysis is illustrated with two numerical experiments.

Keywords:

• Parareal algorithm
• waveform relaxation method
• linear parametric differential equations
• convergence analysis
• numerical simulation

2000 AMS SUBJECT CLASSIFICATIONS:

• 65L05
• 68W10
• 65B05

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the International Science and Technology Cooperation Program of Shaanxi Key Research & Development Plan under grant 2019KWZ-08, the Natural Science Foundation of China (NSFC) under grant 12271426, and the Natural Science Basic Research Plan in Shaanxi Province of China under grant 2019JQ-617.