Global-in-Time Energy Stability for a General Class of Stabilization Single-Step Schemes Applied to the Swift-Hohenberg Equation

Abstract

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 ${\mathcal{l}}^{\mathrm{\infty }}$ 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 singlestep schemes employing spatial Fourier pseudo-spectral discretization. First, assuming that all stage solutions are bounded in the ${\mathcal{l}}^{\mathrm{\infty }}$ 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 ${\mathcal{l}}^{\mathrm{\infty }}$ bound assumption of solutions, we use the thirdorder exponential-time-differencing Runge-Kutta scheme as a case study to establish a uniform-in-time discrete H² bound for stage solutions under an $\mathcal{O}$(1) time step constraint. This leads to a uniform ${\mathcal{l}}^{\mathrm{\infty }}$ 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.

Keywords

Swift-Hohenberg equation / single-step scheme / energy-stability-preserving coefficient / global-in-time energy stability