@Article{CSIAM-AM-6-1,
author = {Wang, Haifeng and Sun, Jingwei and Zhang, Hong and Xu, Qian},
title = {A Novel up to Fourth-Order Equilibria-Preserving and Energy-Stable Exponential Runge-Kutta Framework for Gradient Flows},
journal = {CSIAM Transactions on Applied Mathematics},
year = {2025},
volume = {6},
number = {1},
pages = {106--147},
abstract = {<p style="text-align: justify;">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.</p>},
issn = {2708-0579},
doi = {https://doi.org/10.4208/csiam-am.SO-2024-0032},
url = {https://global-sci.com/article/91670/a-novel-up-to-fourth-order-equilibria-preserving-and-energy-stable-exponential-runge-kutta-framework-for-gradient-flows}
}