  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.
