B-series and Order Conditions for Exponential Integrators

Abstract

We introduce a general format of numerical ODE-solvers which include many of the recently proposed exponential integrators. We derive a general order theory for these schemes in terms of $B$-series and bicolored rooted trees. To ease the construction of specific schemes we generalize an idea of Zennaro [{\em {Math. Comp.,}} 46 (1986), pp. 119--133] and define natural continuous extensions in the context of exponential integrators. This leads to a relatively easy derivation of some of the most popular recently proposed schemes. The general format of schemes considered here makes use of coefficient functions which will usually be selected from some finite dimensional function spaces. We will derive lower bounds for the dimension of these spaces in terms of the order of the resulting schemes. Finally, we illustrate the presented ideas by giving examples of new exponential integrators of orders 4 and 5.