A class of explicit exponential general linear methods

In this paper, we consider a class of explicit exponential integrators that includes as special cases the explicit exponential Runge-Kutta and exponential Adams-Bashforth methods. The additional freedom in the choice of the numerical schemes allows, in an easy manner, the construction of methods of arbitrarily high order with good stability properties. We provide a convergence analysis for abstract evolution equations in Banach spaces including semilinear parabolic initial-boundary value problems and spatial discretizations thereof. From this analysis, we deduce order conditions which in turn form the basis for the construction of new schemes. Our convergence results are illustrated by numerical examples.