  We propose a unified theoretical framework to examine the energy dissipation
properties at all stages of explicit exponential Runge-Kutta (EERK) methods for
gradient flow problems. The main part of the novel framework is to construct
the differential form of EERK method by using the difference coefficients of
method and the so-called discrete orthogonal convolution kernels. As the main
result, we prove that an EERK method can preserve the original energy
dissipation law unconditionally if the associated differentiation matrix is
positive semi-definite. A simple indicator, namely average dissipation rate, is
also introduced for these multi-stage methods to evaluate the overall energy
dissipation rate of an EERK method such that one can choose proper parameters
in some parameterized EERK methods or compare different kinds of EERK methods.
Some existing EERK methods in the literature are evaluated from the perspective
of preserving the original energy dissipation law and the energy dissipation
rate. Some numerical examples are also included to support our theory.
