  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.
