A New Parareal Algorithm for Problems with Discontinuous Sources

TL;DR

This paper introduces a modified Parareal algorithm tailored for differential equations with discontinuous sources, improving convergence analysis and demonstrating effectiveness through numerical experiments and an application to induction machine simulation.

Contribution

It presents a novel Parareal algorithm that handles discontinuous right-hand sides by using a smooth input for the coarse problem, with theoretical error estimates and practical validation.

Findings

Error estimates show input reduction impacts convergence

Numerical experiments confirm theoretical predictions

Algorithm successfully applied to induction machine simulation

Abstract

The Parareal algorithm allows to solve evolution problems exploiting parallelization in time. Its convergence and stability have been proved under the assumption of regular (smooth) inputs. We present and analyze here a new Parareal algorithm for ordinary differential equations which involve discontinuous right-hand sides. Such situations occur in various applications, e.g., when an electric device is supplied with a pulse-width-modulated signal. Our new Parareal algorithm uses a smooth input for the coarse problem with reduced dynamics. We derive error estimates that show how the input reduction influences the overall convergence rate of the algorithm. We support our theoretical results by numerical experiments, and also test our new Parareal algorithm in an eddy current simulation of an induction machine.