We present and analyze a new second-order finite difference scheme for the Macromolecular Microsphere Composite hydrogel, Time-Dependent Ginzburg-Landau (MMC-TDGL) equation, a Cahn-Hilliard equation with Flory-Huggins-deGennes energy potential. This numerical scheme with unconditional energy stability is based on the Backward Differentiation Formula (BDF) method time derivation combining with Douglas-Dupont regularization term. In addition, we present a point-wise bound of the numerical solution for the proposed scheme in the theoretical level. For the convergent analysis, we treat three nonlinear logarithmic terms as a whole and deal with all logarithmic terms directly by using the property that the nonlinear error inner product is always non-negative. Moreover, we present the detailed convergent analysis in $\ell^\infty (0,T; H_h^{-1}) \cap \ell^2 (0,T; H_h^1)$ norm. At last, we use the local Newton approximation and multigrid method to solve the nonlinear numerical scheme, and various numerical results are presented, including the numerical convergence test, positivity-preserving property test, spinodal decomposition, energy dissipation and mass conservation properties.

中文翻译：

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具有可变界面参数的 Cahn-Hilliard 方程的保正二阶 BDF 格式

我们提出并分析了大分子微球复合水凝胶、瞬态 Ginzburg-Landau (MMC-TDGL) 方程、具有 Flory-Huggins-deGennes 能量势的 Cahn-Hilliard 方程的新二阶有限差分格式。这种具有无条件能量稳定性的数值方案是基于反向微分公式（BDF）方法时间推导结合道格拉斯-杜邦正则项。此外，我们在理论水平上提出了所提出方案的数值解的逐点界限。对于收敛分析，我们将三个非线性对数项作为一个整体，利用非线性误差内积始终为非负的性质，直接处理所有对数项。此外，我们在 $\ell^\infty (0,T; H_h^{-1}) \cap \ell^2 (0,T; H_h^1)$ 范数。最后，我们使用局部牛顿近似和多重网格方法对非线性数值方案进行求解，并给出了各种数值结果，包括数值收敛性测试、正性保持性测试、旋节分解、能量耗散和质量守恒性质。