Exponential integrators are a numerical scheme for computing the temporal evolution of the solution of differential equations. They are most effective for stiff problems arising from a wide variety of fields such as physics, chemistry, biology and many other disciplines. Exponential integrators were long considered as not implementable as they require a reliable and efficient implementation of the action of the matrix exponential and related functions.
While matrix functions have been studied for a long time, the computation of their action came into focus more recently. For large scale matrices, when the actual matrix function is too expensive to compute, the computation or appropriate approximation of the action is often still feasible. Nowadays, various methods exist to perform this task.
In this talk we consider polynomial interpolation based on Leja points. We give an introduction to exponential integrators and an overview on recent developments of the Leja method with a closer look at a newly developed backwards error analysis for the action of the matrix exponential.