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.

翻译：在本文中,我们构建并分析一个独特的可溶性、真实性保护和无条件能源稳定的有限差异计划,用于三成分子微粒复合体(MMC)水凝系统,即具有Flory-Huggins-deGennes自由能源潜力的永久Cahn-Hilliard系统。拟议计划的基础是用两个变量对给定的能源功能进行共振分解,并在空间中采用核心差异法。我们提供了一个理论依据,说明这个数字方案有一套独特的解决方案,即所有单词的假设性总是保存在0美元到1美元之间,而且两个阶段变量的总和在点水平上在0美元到1美元之间。此外,我们使用本地的Newton近似和多格方法来解决这个非线性数字法,并提供了各种数字结果,包括数字趋同测试、假设性保留地产测试、能量分解和质量保护。