Long-time stability analysis of an explicit exponential Runge-Kutta scheme for Cahn-Hilliard equations

In this paper, we present a comprehensive long-time stability analysis of a second-order explicit exponential Runge--Kutta (ERK2) method for the Cahn--Hilliard (CH) equation. By employing Fourier spectral collocation in space and a two-stage ERK2 scheme in time, we construct a fully discrete numerical method that preserves the original energy dissipation property. The uniform-in-time boundedness of the numerical solution is rigorously proven in the discrete $H^1$ and $H^2$ norms under a mild time-step condition, and an $\ell^\infty$ bound is derived via a discrete Sobolev embedding. These results remove the typical boundedness assumption required in previous energy-stability analyses, thereby establishing unconditional energy dissipation for the fully discrete scheme. Building on this uniform boundedness, we derive an optimal-order error estimate in the $\ell^2$ norm. The analytical framework developed herein is general and can be extended to higher-order exponential integrators for a broader class of phase-field models.