Parallel in time algorithm with spectral-subdomain enhancement for Volterra integral equations

TL;DR

This paper introduces a novel parallel in time algorithm with spectral enhancement for Volterra integral equations, achieving high accuracy and reduced computational cost through iterative domain decomposition and spectral collocation methods.

Contribution

It presents the first time parareal method for Volterra equations combined with spectral accuracy enhancement and provides a rigorous convergence analysis.

Findings

Significant reduction in computational cost.

Achieves spectral rate of convergence.

First implementation of time parareal for Volterra equations.

Abstract

This paper proposes a parallel in time (called also time parareal) method to solve Volterra integral equations of the second kind. The parallel in time approach follows the same spirit as the domain decomposition that consists of breaking the domain of computation into subdomains and solving iteratively the sub-problems in a parallel way. To obtain high order of accuracy, a spectral collocation accuracy enhancement in subdomains will be employed. Our main contributions in this work are two folds: (i) a time parareal method is designed for the integral equations, which to our knowledge is the first of its kind. The new method is an iterative process combining a coarse prediction in the whole domain with fine corrections in subdomains by using spectral approximation, leading to an algorithm of very high accuracy; (ii) a rigorous convergence analysis of the overall method is provided. The numerical experiment confirms that the overall computational cost is considerably reduced while the desired spectral rate of convergence can be obtained.