A Structure-Preserving, Upwind-SAV Scheme for the Degenerate Cahn-Hilliard Equation with Applications to Simulating Surface Diffusion

This paper establishes a structure-preserving numerical scheme for the Cahn-Hilliard equation with degenerate mobility. First, by applying a finite volume method with upwind numerical fluxes to the degenerate Cahn-Hilliard equation rewritten by the scalar auxiliary variable (SAV) approach, we creatively obtain an unconditionally bound-preserving, energy-stable and fully-discrete scheme, which, for the first time, addresses the boundedness of the classical SAV approach under H-1-gradient flow. Then, a dimensional-splitting technique is introduced in high-dimensional cases, which greatly reduces the computational complexity while preserves original structural properties. Numerical experiments are presented to verify the bound-preserving and energy-stable properties of the proposed scheme. Finally, by applying the proposed structure-preserving scheme, we numerically demonstrate that surface diffusion is approximated by the Cahn-Hilliard equation with degenerate mobility and Flory-Huggins potential, when the absolute temperature is sufficiently low, which agrees well with the theoretical result by using formal asymptotic analysis.

Automated summary

This paper establishes a structure-preserving numerical scheme for the Cahn-Hilliard equation with degenerate mobility. The proposed scheme is unconditionally bound-preserving, energy-stable, and fully-discrete, and it addresses the boundedness issue of the classical scalar auxiliary variable (SAV) approach. A dimensional-splitting technique is introduced to reduce computational complexity while preserving original structural properties. Numerical experiments verify the effectiveness of the proposed scheme, and numerical simulations demonstrate the approximation of surface diffusion by the Cahn-Hilliard equation with degenerate mobility and Flory-Huggins potential under low temperature conditions.