Exponential Runge-Kutta Parareal for Non-Diffusive Equations

TL;DR

This paper introduces exponential Parareal integrators to improve parallel-in-time solutions for non-diffusive equations, demonstrating enhanced convergence and reduced computational time in complex wave and dispersive systems.

Contribution

It proposes and analyzes exponential Parareal methods specifically designed for non-diffusive equations, addressing convergence issues of traditional methods.

Findings

Exponential Parareal improves convergence for hyperbolic and dispersive equations.

Numerical experiments show faster solutions compared to serial exponential integrators.

Analysis confirms stability and convergence properties of the proposed methods.

Abstract

Parareal is a well-known parallel-in-time algorithm that combines a coarse and fine propagator within a parallel iteration. It allows for large-scale parallelism that leads to significantly reduced computational time compared to serial time-stepping methods. However, like many parallel-in-time methods it can fail to converge when applied to non-diffusive equations such as hyperbolic systems or dispersive nonlinear wave equations. This paper explores the use of exponential integrators within the Parareal iteration. Exponential integrators are particularly interesting candidates for Parareal because of their ability to resolve fast-moving waves, even at the large stepsizes used by coarse propagators. This work begins with an introduction to exponential Parareal integrators followed by several motivating numerical experiments involving the nonlinear Schr\"odinger equation. These experiments are then analyzed using linear analysis that approximates the stability and convergence properties of the exponential Parareal iteration on nonlinear problems. The paper concludes with two additional numerical experiments involving the dispersive Kadomtsev-Petviashvili equation and the hyperbolic Vlasov-Poisson equation. These experiments demonstrate that exponential Parareal methods offer improved time-to-solution compared to serial exponential integrators when solving certain non-diffusive equations.