A new approach to constructing efficient stiffly accurate EPIRK methods

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.