The automatic selection of an appropriate time step size has been considered extensively in the literature. However, most of the strategies developed operate under the assumption that the computational cost (per time step) is independent of the step size. This assumption is reasonable for non-stiff ordinary differential equations and for partial differential equations where the linear systems of equations resulting from an implicit integrator are solved by direct methods. It is, however, usually not satisfied if iterative (for example, Krylov) methods are used. In this paper, we propose a step size selection strategy that adaptively reduces the computational cost (per unit time step) as the simulation progresses, constraint by the tolerance specified. We show that the proposed approach yields significant improvements in performance for a range of problems (diffusion-advection equation, Burgers equation with a reaction term, porous media equation, viscous Burgers equation, Allen--Cahn equation, and the two-dimensional Brusselator system). While traditional step size controllers have emphasized a smooth sequence of time step sizes, we emphasize the exploration of different step sizes which necessitates relatively rapid changes in the step size.
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