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

TL;DR

This paper rigorously proves the long-time stability and energy dissipation of a second-order explicit exponential Runge-Kutta scheme for the Cahn-Hilliard equation, ensuring reliable simulations over extended periods.

Contribution

It establishes unconditional energy stability and uniform boundedness of the scheme without previous boundedness assumptions, and provides optimal error estimates.

Findings

Proves uniform-in-time boundedness in discrete norms.

Demonstrates unconditional energy dissipation.

Derives optimal-order error estimates.

Abstract

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.