Integrating Factor Methods as Exponential Integrators

Recently a lot of effort has been placed in the construction and implementation of a class of methods called exponential integrators. These methods are preferable when one has to deal with stiff and highly oscillatory semilinear problems, which often arise after spatial discretization of Partial Differential Equations (PDEs). The main idea behind the methods is to use the exponential and some closely related functions inside the numerical scheme. In this note we show that the integrating factor methods, introduced by Lawson in 1967, are also examples of exponential integrators with very special structure for the related exponential functions. In order to prove this relation, we use the approach based on bi-coloured rooted trees and B-series. We also show under what conditions every bi-coloured rooted tree can be express as a linear combination of standard non-coloured rooted trees.