A Novel up to Fourth-Order Equilibria-Preserving and Energy-Stable Exponential Runge-Kutta Framework for Gradient Flows

Abstract

In this work, we develop and analyze a family of up to fourth-order, unconditionally energy-stable, single-step schemes for solving gradient flows with global Lipschitz continuity. To address the exponential damping/growth behavior observed in Lawson's integrating factor Runge-Kutta approach, we propose a novel strategy to maintain the original system's steady state, leading to the construction of an exponential Runge-Kutta (ERK) framework. By integrating the linear stabilization technique, we provide a unified framework for examining the energy stability of the ERK method. Moreover, we show that certain specific ERK schemes achieve unconditional energy stability when a sufficiently large stabilization parameter is utilized. As a case study, using the no-slope-selection thin film growth equation, we conduct an optimal rate convergence analysis and error estimate for a particular three-stage, third-order ERK scheme coupled with Fourier pseudo-spectral discretization. This is accomplished through rigorous eigenvalue estimation and nonlinear analysis. Numerical experiments are presented to confirm the high-order accuracy and energy stability of the proposed schemes.

Keywords

Gradient flows / exponential Runge-Kutta method / unconditional energy stability / error estimate

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