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.

在本文中，我们构建并分析了周期性三组分大分子微球复合 (MMC) 水凝胶系统的唯一可解、正性保持和无条件能量稳定的有限差分格式，该系统是具有 Flory-Huggins-deGennes 的三元 Cahn-Hilliard 系统自由能潜力。所提出的方案基于具有两个变量的给定能量泛函的凸凹分解，并且在空间上采用中心差分法。我们提供了一个理论证明，即该数值方案具有一对唯一解，因此对于所有奇异项始终保持正性，即不仅两个相位变量始终在 0 和 1 之间，而且两个相位的总和变量在 0 和 1 之间，在逐点级别。此外，