Semi-explicit Parareal method based on convergence acceleration technique

TL;DR

This paper introduces a semi-explicit Parareal method that employs convergence acceleration techniques to enhance the accuracy of parallel-in-time solutions for differential equations, maintaining simplicity with explicit solvers.

Contribution

It presents a novel Parareal algorithm variant that integrates convergence acceleration via series extrapolation, improving solution precision while using simple explicit methods.

Findings

Improved convergence speed demonstrated with explicit Euler scheme

Enhanced solution accuracy through series-based refinement

Numerical examples validate the method's effectiveness

Abstract

The Parareal algorithm is used to solve time-dependent problems considering multiple solvers that may work in parallel. The key feature is a initial rough approximation of the solution that is iteratively refined by the parallel solvers. We report a derivation of the Parareal method that uses a convergence acceleration technique to improve the accuracy of the solution. Our approach uses firstly an explicit ODE solver to perform the parallel computations with different time-steps and then, a decomposition of the solution into specific convergent series, based on an extrapolation method, allows to refine the precision of the solution. Our proposed method exploits basic explicit integration methods, such as for example the explicit Euler scheme, in order to preserve the simplicity of the global parallel algorithm. The first part of the paper outlines the proposed method applied to the simple explicit Euler scheme and then the derivation of the classical Parareal algorithm is discussed and illustrated with numerical examples.