IMEX Runge-Kutta Parareal for Non-Diffusive Equations

TL;DR

This paper investigates the stability and convergence of IMEX Runge-Kutta Parareal methods for non-diffusive equations, identifying configurations that enable effective parallel speedup through combined analytical and numerical analysis.

Contribution

It provides a novel analysis of Parareal methods on non-diffusive equations, highlighting stable configurations with low iteration counts and large block sizes for parallel speedup.

Findings

Certain Parareal configurations achieve parallel speedup on non-diffusive equations.

Stable configurations have low iteration counts, large block sizes, and many processors.

Numerical examples with nonlinear Schrödinger equation support the analytical results.

Abstract

Parareal is a widely studied parallel-in-time method that can achieve meaningful speedup on certain problems. However, it is well known that the method typically performs poorly on non-diffusive equations. This paper analyzes linear stability and convergence for IMEX Runge-Kutta Parareal methods on non-diffusive equations. By combining standard linear stability analysis with a simple convergence analysis, we find that certain Parareal configurations can achieve parallel speedup on non-diffusive equations. These stable configurations all posses low iteration counts, large block sizes, and a large number of processors. Numerical examples using the nonlinear Schrodinger equation demonstrate the analytical conclusions.