Exponential Runge-Kutta methods for parabolic problems

The aim of this paper is to construct exponential Runge-Kutta methods of collocation type and to analyze their convergence properties for linear and semilinear parabolic problems. For the analysis, an abstract Banach space framework of sectorial operators and locally Lipschitz continuous nonlinearities is chosen. This framework includes interesting examples like reaction-diffusion equations. It is shown that the methods converge at least with their stage order, and that convergence of higher order (up to the classical order) occurs, if the problem has sufficient temporal and spatial smoothness. The latter, however, might require the source function to fulfil unnatural boundary conditions. Therefore, the classical order is not always obtained and an order reduction must be expected, in general.