| Abstract: | 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 in time derivation
combining with Douglas-Dupont regularization term. In addition, we present a pointwise 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 $?^∞$(0,$T$;$H_h^{-1}$)∩$?^2$(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. |
