Upwind-SAV approach for constructing bound-preserving and energy-stable schemes of the Cahn-Hilliard equation with degenerate mobility

This paper establishes an unconditionally bound-preserving and energy-stable scheme for the Cahn-Hilliard equation with degenerate mobility. More specifically, by applying a finite volume method (FVM) with up-wind numerical fluxes to the degenerate Cahn-Hilliard equation rewritten by the scalar auxiliary variable (SAV) approach, we 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. Furthermore, the dimensional-splitting technique is introduced in high-dimensional spaces, which greatly reduces the computational complexity while preserving original structural properties. Several numerical experiments are presented to verify the bound-preserving and energy-stable properties of the proposed scheme. Moreover, by applying the scheme to the moving interface problem, we have numerically demonstrated that surface diffusion can be modeled by the Cahn-Hilliard equation with degenerate mobility and Flory-Huggins potential at low temperature, which was only shown theoretically by formal matched asymptotics.