Efficient computation of phifunctions in exponential integrators Abstract   A crucial point for the computational efficiency of the exponentialtype integrators for differential equations is the effective evaluation of the integrals involving matrix exponential called phifunctions. Focusing on exponential integrators for differential equations of small dimensions, for which the Krylovbased approximations for the phifunctions are not viable, in this article refined algorithms are introduced for speeding up both, the evaluation of several phifunctions simultaneously and the computation of a linear combination of phifunctions times matrices. The algorithms are derived from a meticulous approximation to the exponential of certain partitioned matrix by adapting the conventional Pade 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.

Adaptive Exponential Integrators for MCTDHF We compare exponentialtype integrators for the numerical timepropagation of the equations of motion arising in the multiconfiguration timedependent HartreeFock method for the approximation of the highdimensional multiparticle Schrodinger equation. We find that among the most widely used integrators like RungeKutta, exponential splitting, exponential RungeKutta, exponential multistep and Lawson methods, exponential Lawson multistep methods with one predictorcorrector step provide optimal stability and accuracy at the least computational cost, taking into account that the evaluation of the nonlocal potential terms is by far the computationally most expensive part of such a calculation. Moreover, the predictor step provides an estimator for the timestepping error at no additional cost, which enables adaptive timestepping to reliably control the accuracy of a computation.