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.

翻译：我们提出了一套统一的理论框架，用于研究求解梯度流问题的显式指数龙格-库塔（EERK）方法在各阶段保持能量耗散性质。该新框架的核心在于利用方法的差分系数与所谓的离散正交卷积核，构造出EERK方法的微分形式。作为主要结果，我们证明：若关联的微分矩阵为半正定，则EERK方法可无条件保持原始能量耗散律。此外，针对这类多阶段方法引入了一个简单指标——平均耗散率，用于评估EERK方法的整体能量耗散速率，从而可在参数化EERK方法中选择合适参数，或比较不同类型的EERK方法。我们从保持原始能量耗散律与能量耗散率的角度，对文献中若干现有EERK方法进行了评估。文中还包含部分数值算例以支撑理论分析。