A positivity-preserving, energy stable scheme for a Ternary Cahn-Hilliard system with the singular interfacial parameters

TL;DR

This paper introduces a novel finite-difference scheme for a ternary Cahn-Hilliard system modeling MMC hydrogels, ensuring positivity, energy stability, and unique solvability, with comprehensive theoretical and numerical validation.

Contribution

It presents the first positivity-preserving, energy-stable, and uniquely solvable scheme for a singular ternary Cahn-Hilliard system with theoretical proof and efficient numerical methods.

Findings

The scheme guarantees positivity of all phase variables at all times.

Numerical results confirm energy dissipation and mass conservation.

The method achieves optimal convergence rates.

Abstract

In this paper, we construct and analyze a uniquely solvable, positivity preserving and unconditionally energy stable finite-difference scheme for the periodic three-component Macromolecular Microsphere Composite (MMC) hydrogels system, a ternary Cahn-Hilliard system with a Flory-Huggins-deGennes free energy potential. The proposed scheme is based on a convex-concave decomposition of the given energy functional with two variables, and the centered difference method is adopted in space. We provide a theoretical justification that this numerical scheme has a pair of unique solutions, such that the positivity is always preserved for all the singular terms, i.e., not only two phase variables are always between $0$ and $1$, but also the sum of two phase variables is between $0$ and $1$, at a point-wise level. In addition, we use the local Newton approximation and multigrid method to solve this nonlinear numerical scheme, and various numerical results are presented, including the numerical convergence test, positivity-preserving property test, energy dissipation and mass conservation properties.