Abstract
Exponential integrators have emerged as an efficient alternative to commonly used time integrators. Recently a new class of exponential propagation iterative methods of Runge--Kutta type (EPIRK) has been introduced [M. Tokman, J. Comput. Phys., 230 (2011), pp. 8762--8778]. These schemes possess a structure that makes them computationally advantageous compared to other exponential methods. In addition, the general EPIRK formulation offers flexibility that allows derivation of new efficient techniques. In this paper, we use this feature to derive new EPIRK methods which are particularly designed to take advantage of the adaptive Krylov algorithm [J. Niesen and W. Wright, http://arxiv.org/abs/0907.4631 (2010)]. The adaptive Krylov method significantly reduces the computational complexity of evaluating products of matrix $\varphi$-functions and vectors necessary for implementing an exponential integrator. We present the derivation of the new adaptive EPIRK methods, construct new schemes, and illustrate the computational savings they offer using numerical examples.
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