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 can be 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.wn theoretically by formal matched asymptotics.

翻译：本文建立了一种针对具有简并迁移率的Cahn-Hilliard方程的结构保持数值格式。首先，通过将有限体积法与上风数值通量应用于经标量辅助变量（SAV）方法重新表述的简并Cahn-Hilliard方程，我们创造性地得到了一种无条件保界、能量稳定的全离散格式，该格式首次解决了经典SAV方法在$H^{-1}$梯度流下的有界性问题。随后，在高维情形下引入了一种维数分裂技术，该技术在大幅降低计算复杂度的同时保留了原有的结构性质。通过数值实验验证了所提格式的保界和能量稳定特性。最后，应用所提出的结构保持格式，我们数值证明了当绝对温度足够低时，表面扩散可由具有简并迁移率和Flory-Huggins势的Cahn-Hilliard方程近似，这与通过形式渐近分析得到的理论结果高度吻合。