A Novel up to Fourth-Order Equilibria-Preserving and Energy-Stable Exponential Runge-Kutta Framework for Gradient Flows (CSIAM Transactions on Applied Mathematics, 2025)
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 globalLipschitz continuity. To address the exponential damping/growth behavior observedin Lawson’s integrating factor Runge-Kutta approach, we propose a novel strategy tomaintain 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 energystability 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 rateconvergence analysis and error estimate for a particular three-stage, third-order ERKscheme coupled with Fourier pseudo-spectral discretization. This is accomplishedthrough rigorous eigenvalue estimation and nonlinear analysis. Numerical experiments are presented to confirm the high-order accuracy and energy stability of theproposed schemes.