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We present and analyze a new second-order finite difference scheme for theMacromolecular Microsphere Composite hydrogel, Time-Dependent Ginzburg-Landau(MMC-TDGL) equation, a Cahn-Hilliard equation with Flory-Huggins-deGennesenergy potential. This numerical scheme with unconditional energy stability isbased on the Backward Differentiation Formula (BDF) method time derivationcombining with Douglas-Dupont regularization term. In addition, we present apoint-wise bound of the numerical solution for the proposed scheme in thetheoretical level. For the convergent analysis, we treat three nonlinearlogarithmic terms as a whole and deal with all logarithmic terms directly byusing the property that the nonlinear error inner product is alwaysnon-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 usethe local Newton approximation and multigrid method to solve the nonlinearnumerical scheme, and various numerical results are presented, including thenumerical convergence test, positivity-preserving property test, spinodaldecomposition, energy dissipation and mass conservation properties.

A positivity-preserving second-order BDF scheme for the Cahn-Hilliard equation with variable interfacial parameters