A positivity-preserving, second-order energy stable and convergent numerical scheme for a ternary system of macromolecular microsphere composite hydrogels

TL;DR

This paper introduces a second-order, energy-stable numerical scheme for a ternary MMC hydrogel system that preserves positivity and converges optimally, supported by rigorous theoretical analysis and numerical validation.

Contribution

It is the first to combine positivity-preserving, energy stability, and optimal convergence in a second-order scheme for the ternary MMC system.

Findings

The scheme is positivity-preserving for all singular terms.

It achieves energy stability and unique solvability.

Numerical results confirm theoretical properties.

Abstract

A second order accurate numerical scheme is proposed and analyzed for the periodic three-component Macromolecular Microsphere Composite(MMC) hydrogels system, a ternary Cahn-Hilliard system with a Flory-Huggins-deGennes free energy potential. This numerical scheme with energy stability is based on the Backward Differentiation Formula(BDF) method in time derivation combining with Douglas-Dupont regularization term, combined the finite difference method in space. We provide a theoretical justification of positivity-preserving property for all the singular terms, i.e., not only the two phase variables are always between $0$ and $1$, but also the sum of the two phase variables is between $0$ and $1$, at a point-wise level. In addition, an optimal rate convergence analysis is provided in this paper, in which a higher order asymptotic expansion of the numerical solution, the rough error estimate and refined error estimate techniques have to be included to accomplish such an analysis. This paper will be the first to combine the following theoretical properties for a second order accurate numerical scheme for the ternary MMC system: (i) unique solvability and positivity-preserving property; (ii) energy stability; (iii) and optimal rate convergence. A few numerical results are also presented.