Why Improving the Accuracy of Exponential Integrators Can Decrease Their Computational Cost?

In previous papers, a technique has been suggested to avoid order reduction when integrating initial boundary value problems with several kinds of exponential methods. The technique implies in principle to calculate additional terms at each step from those already necessary without avoiding order reduction. The aim of the present paper is to explain the surprising result that, many times, in spite of having to calculate more terms at each step, the computational cost of doing it through Krylov methods decreases instead of increases. This is very interesting since, in that way, the methods improve not only in terms of accuracy, but also in terms of computational cost.