A New Parareal Algorithm for Time-Periodic Problems with Discontinuous Inputs

TL;DR

This paper introduces a novel Parareal algorithm tailored for efficiently solving time-periodic problems with rapidly oscillating discontinuous inputs, such as those in power engineering, by leveraging low-frequency approximations.

Contribution

It develops and analyzes a new Parareal method that uses a smooth low-frequency input for the coarse problem to handle highly-oscillatory discontinuous sources.

Findings

Effective in simulating induction machines

Handles highly-oscillatory discontinuous inputs

Improves computational efficiency for time-periodic problems

Abstract

The Parareal algorithm, which is related to multiple shooting, was introduced for solving evolution problems in a time-parallel manner. The algorithm was then extended to solve time-periodic problems. We are interested here in time-periodic systems which are forced with quickly-switching discontinuous inputs. Such situations occur, e.g., in power engineering when electric devices are excited with a pulse-width-modulated signal. In order to solve those problems numerically we consider a recently introduced modified Parareal method with reduced coarse dynamics. Its main idea is to use a low-frequency smooth input for the coarse problem, which can be obtained, e.g., from Fourier analysis. Based on this approach, we present and analyze a new Parareal algorithm for time-periodic problems with highly-oscillatory discontinuous sources. We illustrate the performance of the method via its application to the simulation of an induction machine.