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.