On the Stability of Exponential Integrators for Non-Diffusive Equations

TL;DR

This paper analyzes the stability issues of exponential integrators for non-diffusive equations and introduces a repartitioning method that enhances stability and efficiency without high-order derivatives.

Contribution

It proposes a simple repartitioning approach to stabilize exponential integrators for non-diffusive equations, improving their efficiency and convergence at large timesteps.

Findings

Repartitioning stabilizes exponential integrators on non-diffusive problems.

Repartitioning restores convergence at large timesteps.

Unlike hyperviscosity, repartitioning does not require high-order derivatives.

Abstract

Exponential integrators are a well-known class of time integration methods that have been the subject of many studies and developments in the past two decades. Surprisingly, there have been limited efforts to analyze their stability and efficiency on non-diffusive equations to date. In this paper we apply linear stability analysis to showcase the poor stability properties of exponential integrators on non-diffusive problems. We then propose a simple repartitioning approach that stabilizes the integrators and enables the efficient solution of stiff, non-diffusive equations. To validate the effectiveness of our approach, we perform several numerical experiments that compare partitioned exponential integrators to unmodified ones. We also compare repartitioning to the well-known approach of adding hyperviscosity to the equation right-hand-side. Overall, we find that the repartitioning restores convergence at large timesteps and, unlike hyperviscosity, it does not require the use of high-order spatial derivatives.