Explicit exponential Runge–Kutta methods of high order for parabolic problems

Abstract

Exponential Runge–Kutta methods constitute efficient integrators for semilinear stiff problems. So far, however, explicit exponential Runge–Kutta methods are available in the literature up to order 4 only. The aim of this paper is to construct a fifth-order method. For this purpose, we make use of a novel approach to derive the stiff order conditions for high-order exponential methods. This allows us to obtain the conditions for a method of order 5 in an elegant way. After stating the conditions, we first show that there does not exist an explicit exponential Runge–Kutta method of order 5 with less than or equal to 6 stages. Then, we construct a fifth-order method with 8 stages and prove its convergence for semilinear parabolic problems. Finally, a numerical example is given that illustrates our convergence bound.