TY - JOUR
T1 - Global-in-Time Energy Stability for a General Class of Stabilization Single-Step Schemes Applied to the Swift-Hohenberg Equation
AU - Zhang , Hong
AU - Wang , Haifeng
AU - Wang , Yan
AU - Qian , Xu
JO - CSIAM Transactions on Applied Mathematics
VL - 3
SP - 555
EP - 592
PY - 2025
DA - 2025/09
SN - 6
DO - http://doi.org/10.4208/csiam-am.SO-2024-0069
UR - https://global-sci.org/intro/article_detail/csiam-am/24375.html
KW - Swift-Hohenberg equation, single-step scheme, energy-stability-preserving coefficient, global-in-time energy stability.
AB - <p style="text-align: justify;">The linear stabilization approach is well-known for facilitating the use of
large time steps in solving gradient flows while maintaining stability. However, the
up-to-date analysis of energy stability relies on either a global Lipschitz nonlinearity
or an $ℓ^∞$ bound assumption of numerical solutions. Considering the Swift-Hohenberg
equation that lacks a global Lipschitz nonlinearity, we develop a unified framework to
analyze the energy stability and characterize the stabilization size for a class of single-step schemes employing spatial Fourier pseudo-spectral discretization. First, assuming that all stage solutions are bounded in the $ℓ^∞$ norm, we illustrate that the energy obtained from a single-step scheme with non-negative energy-stability-preserving coefficient is unconditionally dissipative, as long as a sufficiently large stabilization parameter is employed. To justify the $ℓ^∞$ bound assumption of solutions, we use the third-order exponential-time-differencing Runge-Kutta scheme as a case study to establish
a uniform-in-time discrete $H^2$ bound for stage solutions under an $\mathcal{O}(1)$ time step constraint. This leads to a uniform $ℓ^∞$ bound of stage solutions through discrete Sobolev
embedding. Consequently, we achieve a stabilization parameter of $\mathcal{O}(1),$ which is independent of the time step, thereby ensuring the energy stability. The global-in-time
energy stability analysis and characterization of the stabilization parameter represent
significant advancements for general single-step schemes applied to a gradient flow
without the global Lipschitz continuity.</p>