This study focuses on the development and analysis of a group of high-order implicit-explicit (IMEX) Runge--Kutta (RK) methods that are suitable for discretizing gradient flows with nonlinearity that is Lipschitz continuous. We demonstrate that these IMEX-RK methods can preserve the original energy dissipation property without any restrictions on the time-step size, thanks to a stabilization technique. The stabilization constants are solely dependent on the minimal eigenvalues that result from the Butcher tables of the IMEX-RKs. Furthermore, we establish a simple framework that can determine whether an IMEX-RK scheme is capable of preserving the original energy dissipation property or not. We also present a heuristic convergence analysis based on the truncation errors. This is the first research to prove that a linear high-order single-step scheme can ensure the original energy stability unconditionally for general gradient flows. Additionally, we provide several high-order IMEX-RK schemes that satisfy the established framework. Notably, we discovered a new four-stage third-order IMEX-RK scheme that reduces energy. Finally, we provide numerical examples to demonstrate the stability and accuracy properties of the proposed methods.

翻译：本研究聚焦于一类适用于离散化具有Lipschitz连续非线性项的梯度流的高阶隐式-显式(IMEX)龙格-库塔(RK)方法的开发与分析。我们证明，通过稳定化技术，这些IMEX-RK方法能够在时间步长无任何限制的情况下保持原始能量耗散性质。稳定化常数仅取决于IMEX-RK的Butcher表所产生的最小特征值。此外，我们建立了一个简单框架，可用于判定IMEX-RK格式是否能够保持原始能量耗散性质。我们还提出了一种基于截断误差的启发式收敛性分析。这是首次证明线性高阶单步格式能够无条件保证一般梯度流的原始能量稳定性。同时，我们提供了若干满足该框架的高阶IMEX-RK格式。值得注意的是，我们发现了一种新的四阶段三阶IMEX-RK方案，该方案可实现能量递减。最后，我们通过数值算例展示了所提方法的稳定性和精度特性。