The main theoretical obstacle to establish the original energy dissipation laws of Runge-Kutta methods for phase-field equations is to verify the maximum norm boundedness of the stage solutions without assuming global Lipschitz continuity of the nonlinear bulk. We present a unified theoretical framework for the energy stability of three effective classes of Runge-Kutta methods, including the additive implicit-explicit Runge-Kutta, explicit exponential Runge-Kutta and corrected integrating factor Runge-Kutta methods, for the Swift-Hohenberg and phase field crystal models. By the standard discrete energy argument, it is proven that the three classes of Runge-Kutta methods preserve the original energy dissipation laws if the associated differentiation matrices are positive definite. Our main tools include the differential form with the associated differentiation matrix, the discrete orthogonal convolution kernels and the principle of mathematical induction. Many existing Runge-Kutta methods in the literature are revisited by evaluating the lower bound on the minimum eigenvalues of the associated differentiation matrices. Our theoretical approach paves a new way for the internal nonlinear stability of Runge-Kutta methods for dissipative semilinear parabolic problems.

翻译：建立相场方程龙格-库塔方法原始能量耗散律的主要理论障碍在于，需在不假设非线性体项全局Lipschitz连续的前提下，验证阶段解的最大范数有界性。本文针对Swift-Hohenberg模型和相场晶体模型，提出了三类有效龙格-库塔方法能量稳定性的统一理论框架，包括加性隐式-显式龙格-库塔方法、显式指数龙格-库塔方法与修正积分因子龙格-库塔方法。通过标准离散能量论证，证明当关联微分矩阵正定时，这三类龙格-库塔方法均保持原始能量耗散律。我们的主要工具包括带关联微分矩阵的微分形式、离散正交卷积核以及数学归纳法。通过评估关联微分矩阵最小特征值的下界，重新审视了文献中许多现有龙格-库塔方法。该理论途径为耗散半线性抛物问题龙格-库塔方法的内部非线性稳定性研究开辟了新路径。