A Positivity-Preserving Second-Order BDF Scheme for the Cahn-Hilliard Equation with Variable Interfacial Parameters (Communications in Computational Physics, 2020)
We present and analyze a new second-order finite difference scheme forthe 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 derivationcombining 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 anddeal with all logarithmic terms directly by using the property that the nonlinear errorinner product is always non-negative. Moreover, we present the detailed convergentanalysis in $ℓ^∞$(0,$T$;$H_h^{-1}$)∩$ℓ^2$(0,$T$;$H_h^1$)  norm. At last, we use the local Newton approximation and multigrid method to solve the nonlinear numerical scheme, and variousnumerical results are presented, including the numerical convergence test, positivity-preserving property test, spinodal decomposition, energy dissipation and mass conservation properties.
Keywords: Cahn-Hilliard equation; Flory-Huggins energy; deGennes diffusive coefficient; energy stability; positivity preserving; convergence analysis.