TY - JOUR
T1 - A Novel up to Fourth-Order Equilibria-Preserving and Energy-Stable Exponential Runge-Kutta Framework for Gradient Flows
AU - Wang , Haifeng
AU - Sun , Jingwei
AU - Zhang , Hong
AU - Qian , Xu
JO - CSIAM Transactions on Applied Mathematics
VL - 1
SP - 106
EP - 147
PY - 2025
DA - 2025/02
SN - 6
DO - http://doi.org/10.4208/csiam-am.SO-2024-0032
UR - https://global-sci.org/intro/article_detail/csiam-am/23798.html
KW - Gradient flows, exponential Runge-Kutta method, unconditional energy stability, error estimate.
AB - <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>