Higher-order energy-decreasing exponential time differencing Runge-Kutta methods for gradient flows

In this paper, we develop a general framework for constructing higher-order, unconditionally energy-decreasing exponential time differencing Runge-Kutta (ETDRK) 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 the widely-employed third- 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 decay 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 offers the first framework 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.