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

2025/12/05 by Jing Guo, Guo, Jing
Materials Science · Mathematics · Physics and Astronomy · #35K58 #65M12 #65M15 #65M70 #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods for differential equations #Quantum chaos and dynamical systems #Solidification and crystal growth phenomena

paper · pdf · doi:10.48550/arxiv.2512.05608

openalex publication_date 2025/12/05 · openalex created_date 2025/12/09 · openalex updated_date 2026/07/28

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 H¹ and H² norms under a mild time-step condition, and an ℓ^∞ 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 ℓ² 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.

Citations

• A uniquely solvable and positivity-preserving finite difference scheme for the Flory-Huggins-Cahn-Hilliard equation with dynamical boundary condition
• Convergence analysis of a positivity-preserving numerical scheme for the Cahn-Hilliard-Stokes system with Flory-Huggins energy potential
• The operator-splitting method for Cahn-Hilliard is stable
• Long time stability of a classical efficient scheme for two dimensional\n Navier-Stokes equations
• Explicit Exponential Runge--Kutta Methods for Semilinear Parabolic Problems

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