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 that of state-of-the-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 proportion of the computational savings that they provide and illustrate how the structure of these methods ensures such efficiency. In addition, we demonstrate the computational advantages of the newly introduced Tokman and Loffeld (2010) [17] exponential propagation Runge-Kutta (EpiRK) fifth-order methods. The detailed analysis of the performance of the methods that is presented provides guidelines for the construction and implementation of efficient exponential methods and the quantitative comparisons inform the selection of appropriate schemes for other problems. (C) 2012 Elsevier By. All rights reserved.