A sturcture-preserving, upwind-SAV scheme for the degenerate Cahn--Hilliard equation with applications to simulating surface diffusion

TL;DR

This paper develops a novel, structure-preserving numerical scheme for the degenerate Cahn--Hilliard equation that ensures stability, boundedness, and efficiency, and demonstrates its application to surface diffusion simulations.

Contribution

It introduces an unconditionally bound-preserving, energy-stable scheme using a finite volume method with upwind fluxes and a scalar auxiliary variable approach, including a dimensional-splitting technique for high dimensions.

Findings

The scheme is unconditionally energy-stable and bound-preserving.

Numerical experiments confirm the scheme's stability and accuracy.

Application to surface diffusion aligns with theoretical asymptotic analysis.

Abstract

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.