Comparative performance of exponential, implicit, and explicit integrators for stiff systems of ODEs

Exponential integrators have enjoyed a resurgence of interest in recent years, but there is still limited understanding of how their performance compares with state-of-art integrators most notably the commonly used Newton-Krylov implicit methods. In this paper we present comparative performance analysis of Krylov-based exponential, implicit and explicit integrators on a suite of stiff test problems and demonstrate that exponential integrators have computational advantages compared to the other methods particularly as problems become larger and more stiff. We argue that the faster convergence of the Krylov iteration within exponential integrators accounts for the main portion of the computational savings they provide and illustrate how the structure of these methods ensures such efficiency. The presented detailed analysis of the methods’ performance provides guidelines for construction and implementation of efficient exponential methods and the quantitative comparisons instruct selection of appropriate schemes for other problems.