A unified framework on the original energy laws of three effective classes of Runge-Kutta methods for phase field crystal type models

TL;DR

This paper develops a unified theoretical framework to prove energy dissipation laws for three classes of Runge-Kutta methods applied to phase field crystal models, overcoming previous limitations related to nonlinear boundedness assumptions.

Contribution

It introduces a general approach to establish energy stability of Runge-Kutta methods for phase-field equations without requiring global Lipschitz conditions.

Findings

Proves energy dissipation preservation when differentiation matrices are positive definite.

Revisits existing Runge-Kutta methods to evaluate stability conditions.

Provides a new perspective on nonlinear stability for dissipative parabolic problems.

Abstract

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.