Efficient computation of phi-functions in exponential integrators

A crucial point for the computational efficiency of the exponential-type integrators for differential equations is the effective evaluation of the integrals involving matrix exponential called phi-functions. Focusing on exponential integrators for differential equations of small dimensions, for which the Krylov-based approximations for the phi-functions are not viable, in this article refined algorithms are introduced for speeding up both, the evaluation of several phi-functions simultaneously and the computation of a linear combination of phi-functions times matrices. The algorithms are derived from a meticulous approximation to the exponential of certain partitioned matrix by adapting the conventional Padé method to the special structure of this matrix. Numerical simulations are provided to evaluate the performance of the proposed algorithms and compare them with others in the literature.