  Traditional step size controllers make the tacit assumption that the cost of
a time step is independent of the step size. This is reasonable for explicit
and implicit integrators that use direct solvers. In the context of exponential
integrators, however, an iterative approach, such as Krylov methods or
polynomial interpolation, to compute the action of the required matrix
functions is usually employed. In this case, the assumption of constant cost is
not valid. This is, in particular, a problem for higher-order exponential
integrators, which are able to take relatively large time steps based on
accuracy considerations. In this paper, we consider an adaptive step size
controller for exponential Rosenbrock methods that determines the step size
based on the premise of minimizing computational cost. The largest allowed step
size, given by accuracy considerations, merely acts as a constraint. We test
this approach on a range of nonlinear partial differential equations. Our
results show significant improvements (up to a factor of 4 reduction in the
computational cost) over the traditional step size controller for a wide range
of tolerances.
