Direction splitting of ϕ-functions in exponential integrators for d-dimensional problems in Kronecker form

In this manuscript, we propose an efficient, practical and easy-to-implement way to ap-proximate actions ofφ-functions for matrices withd-dimensional Kronecker sum structurein the context of exponential integrators up to second order. The method is based on a di-rection splitting of the involved matrix functions, which lets us exploit the highly efficientlevel 3 BLAS for the actual computation of the required actions in aμ-mode fashion. Theapproach has been successfully tested on two- and three-dimensional problems with var-ious exponential integrators, resulting in a consistent speedup with respect to a techniquedesigned to approximate actions ofφ-functions for Kronecker sums.