This paper introduces a novel paradigm for constructing linearly implicit and high-order unconditionally energy-stable schemes for general gradient flows, utilizing the scalar auxiliary variable (SAV) approach and the additive Runge-Kutta (ARK) methods. We provide a rigorous proof of energy stability, unique solvability, and convergence. The proposed schemes generalizes some recently developed high-order, energy-stable schemes and address their shortcomings. On the one other hand, the proposed schemes can incorporate existing SAV-RK type methods after judiciously selecting the Butcher tables of ARK methods \cite{sav_li,sav_nlsw}. The order of a SAV-RKPC method can thus be confirmed theoretically by the order conditions of the corresponding ARK method. Several new schemes are constructed based on our framework, which perform to be more stable than existing SAV-RK type methods. On the other hand, the proposed schemes do not limit to a specific form of the nonlinear part of the free energy and can achieve high order with fewer intermediate stages compared to the convex splitting ARK methods \cite{csrk}. Numerical experiments demonstrate stability and efficiency of proposed schemes.

翻译：本文提出了一种基于标量辅助变量（SAV）方法与加性龙格-库塔（ARK）方法构建线性隐式、高阶无条件能量稳定格式的新范式，适用于一般梯度流问题。我们给出了能量稳定性、唯一可解性及收敛性的严格证明。所提出的格式推广了近年来发展的若干高阶能量稳定格式，并弥补了其不足之处。另一方面，通过合理选取ARK方法的布彻表（Butcher tables）\cite{sav_li,sav_nlsw}，本文格式可纳入现有SAV-RK类方法。因此，SAV-RKPC方法的阶数可通过对应ARK方法阶数条件从理论上予以确认。基于本文框架，我们构建了多个新格式，其表现比现有SAV-RK类方法更为稳定。此外，所提格式不局限于自由能非线性部分的特定形式，且相较于凸分裂ARK方法\cite{csrk}能以更少的中间阶段实现高阶精度。数值实验验证了所提格式的稳定性与计算效率。