In this paper, we develop a general framework for constructing higher-order, unconditionally energy-stable exponential time differencing Runge-Kutta methods applicable to a range of gradient flows. Specifically, we identify conditions sufficient for ETDRK schemes to maintain the original energy dissipation. Our analysis reveals that commonly used third-order and fourth-order ETDRK schemes fail to meet these conditions. To address this, we introduce new third-order ETDRK schemes, designed with appropriate stabilization, which satisfy these conditions and thus guarantee the unconditional energy decaying property. We conduct extensive numerical experiments with these new schemes to verify their accuracy, stability, behavior under large time steps, long-term evolution, and adaptive time stepping strategy across various gradient flows. This study is the first to examine the unconditional energy stability of high-order ETDRK methods, and we are optimistic that our framework will enable the development of ETDRK schemes beyond the third order that are unconditionally energy stable.

翻译：本文构建了一个通用框架，用于开发适用于多种梯度流的高阶无条件能量稳定指数时间差分龙格-库塔方法。具体而言，我们确定了ETDRK格式维持原始能量耗散所需的充分条件。分析表明，常用的三阶和四阶ETDRK格式不满足这些条件。为此，我们引入新的三阶ETDRK格式，通过适当的稳定化设计使其满足条件，从而保证无条件降能特性。我们通过广泛的数值实验验证了新格式在多种梯度流中的精度、稳定性、大时间步长行为、长期演化及自适应时间步进策略。本研究首次探讨了高阶ETDRK方法的无条件能量稳定性，我们有信心该框架将能够开发出超越三阶且无条件能量稳定的ETDRK格式。