Global-in-Time Energy Stability for a General Class of Stabilization Single-Step Schemes Applied to the Swift-Hohenberg Equation (CSIAM Transactions on Applied Mathematics, 2025)
The linear stabilization approach is well-known for facilitating the use oflarge time steps in solving gradient flows while maintaining stability. However, theup-to-date analysis of energy stability relies on either a global Lipschitz nonlinearityor an $ℓ^∞$ bound assumption of numerical solutions. Considering the Swift-Hohenbergequation that lacks a global Lipschitz nonlinearity, we develop a unified framework toanalyze 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 establisha 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 Sobolevembedding. 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-timeenergy stability analysis and characterization of the stabilization parameter representsignificant advancements for general single-step schemes applied to a gradient flowwithout the global Lipschitz continuity.
Keywords: Swift-Hohenberg equation; single-step scheme; energy-stability-preserving coefficient; global-in-time energy stability.