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 ℓ∞(0,T;Hh−1)∩ℓ2(0,T;Hh1) 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.
@article{arxiv.2004.03371,
title = {A positivity-preserving second-order BDF scheme for the Cahn-Hilliard equation with variable interfacial parameters},
author = {Lixiu Dong and Cheng Wang and Hui Zhang and Zhengru Zhang},
journal= {arXiv preprint arXiv:2004.03371},
year = {2020}
}
Comments
arXiv admin note: text overlap with arXiv:1712.03225