  The structural flexibility of the exponential propagation iterative methods
of Runge-Kutta type (EPIRK) enables construction of particularly efficient
exponential time integrators. While the EPIRK methods have been shown to
perform well on stiff problems, all of the schemes proposed up to now have been
derived using classical order conditions. In this paper we extend the stiff
order conditions and the convergence theory developed for the exponential
Rosenbrock methods to the EPIRK integrators. We derive stiff order conditions
for the EPIRK methods and develop algorithms to solve them to obtain specific
schemes. Moreover, we propose a new approach to constructing particularly
efficient EPIRK integrators that are optimized to work with an adaptive Krylov
algorithm. We use a set of numerical examples to illustrate the computational
advantages that the newly constructed EPIRK methods offer compared to
previously proposed exponential integrators.
