BAMPHI: Matrix-free and transpose-free action of linear combinations of φ-functions from exponential integrators

The time integration of stiff systems of Ordinary Differential Equations (ODEs), usually arising from the spatial discretization of Partial Differential Equations (PDEs), constitutes a hot topic in numerical analysis. In particular, exponential-type integrators have attracted much attention for their capacity to effectively handle the stiffness, allowing integration with large time steps. The efficiency of exponential-type integrators strongly relies on the fast computation of the action of the exponential of matrices which often change only slightly during the integration. The authors exploited this characteristic of exponential integrators to develop a backward stable algorithm, bamphi, which is designed to reuse the information gathered through the exponential integration steps, reaching unmatched levels of speed and accuracy on a variety of numerical experiments.