Explicit Exponential Runge--Kutta Methods for Semilinear Parabolic Problems

The aim of this paper is to analyze explicit exponential Runge--Kutta methods for the time integration of semilinear parabolic problems. The analysis is performed in an abstract Banach space framework of sectorial operators and locally Lipschitz continuous nonlinearities. We commence by giving a new and short derivation of the classical (nonstiff) order conditions for exponential Runge--Kutta methods, but the main interest of our paper lies in the stiff case. By expanding the errors of the numerical method in terms of the solution, we derive new order conditions that form the basis of our error bounds for parabolic problems. We show convergence for methods up to order four, and we analyze methods that were recently presented in the literature. These methods have classical order four, but they do not satisfy some of the new conditions. Therefore, an order reduction is expected. We present numerical experiments which show that this order reduction in fact arises in practical examples. Based on our new conditions, we finally construct methods that do not suffer from order reduction.