NORGES TEKNISK-NATURVITENSKAPELIGE UNIVERSITET A review of exponential integrators for first order semi-linear problems

Recently, there has been a great deal of interest in the construction of exponential integrators. These integrators, as their name suggests, use the exponential function (and related functions) of the Jacobian or an approximation to it, inside the numerical method. However, unlike some of the recent literature suggests, integrators based on this philosophy have been known since at least 1960. The aim of this paper is to review exponential integrators, designed for first order problems, however, we will also briefly discuss some recent research into the construction of exponential integrators, for special problems. Our hope is, that with this article, by reviewing as much of the history of exponential integrators as reasonable possible, we can point interested readers to appropriate references and hopefully reduce the reinvention of known results.