Efficient design of exponential-Krylov integrators for large scale computing

As a result of recent resurgence of interest in exponential integrators a number of such methods have been introduced in the literature. However, questions of what constitutes an efficient exponential method and how these techniques compare with commonly used schemes remain to be fully investigated. In this paper we consider exponential-Krylov integrators in the context of large scale applications and discuss what design principles need to be considered in construction of an efficient method of this type. Since the Krylov projections constitute the primary computational cost of an exponential integrator we demonstrate how an exponential-Krylov method can be structured to minimize the total number of Krylov projections per time step and the number of Krylov vectors each of the projections requires. We present numerical experiments that validate and illustrate these arguments. In addition, we compare exponential methods with commonly used implicit schemes to demonstrate their competitiveness.