A Unified Framework on the Original Energy Laws of Three Effective Classes of Runge–Kutta Methods for Phase Field Crystal Type Models

Abstract

SIAM Journal on Numerical Analysis, Volume 63, Issue 4, Page 1808-1832, August 2025.
Abstract. The main theoretical obstacle to establishing the original energy dissipation laws of Runge–Kutta methods for phase field equations is verifying 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 law if the associated differentiation matrices are positive definite. Our main tools include the differential form with the associated differentiation matrix, the discrete orthogonal convolution kernel, 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 toward the internal nonlinear stability of Runge–Kutta methods for dissipative semilinear parabolic problems.