Communications in Mathematical Sciences

Volume 17 (2019)

Number 4

A positivity-preserving, energy stable and convergent numerical scheme for the Cahn–Hilliard equation with a Flory–Huggins–Degennes energy

Pages: 921 – 939

DOI: https://dx.doi.org/10.4310/CMS.2019.v17.n4.a3

Authors

Lixiu Dong (School of Mathematical Sciences, Beijing Normal University, Beijing, China)

Cheng Wang (Department of Mathematics, University of Massachusetts, North Dartmouth, Mass., U.S.A.)

Hui Zhang (Laboratory of Mathematics and Complex Systems, Ministry of Education and School of Mathematical Sciences, Beijing Normal University, Beijing, China)

Zhengru Zhang (Laboratory of Mathematics and Complex Systems, Ministry of Education and School of Mathematical Sciences, Beijing Normal University, Beijing, China)

Abstract

This article is focused on the bound estimate and convergence analysis of an unconditionally energy-stable scheme for the MMC-TDGL equation, a Cahn–Hilliard equation with a Flory–Huggins–deGennes energy. The numerical scheme, a finite difference algorithm based on a convex splitting technique of the energy functional, was proposed in [Sci. China Math., 59:1815, 2016]. We provide a theoretical justification of the unique solvability for the proposed numerical scheme, in which a well-known difficulty associated with the singular nature of the logarithmic energy potential has to be handled. Meanwhile, a careful analysis reveals that, such a singular nature prevents the numerical solution of the phase variable reaching the limit singular values, so that the positivitypreserving property could be proved at a theoretical level. In particular, the natural structure of the deGennes diffusive coefficient also ensures the desired positivity-preserving property. In turn, the unconditional energy stability becomes an outcome of the unique solvability and the convex-concave decomposition for the energy functional. Moreover, an optimal rate convergence analysis is presented in the $\ell^\infty (0, T; H^{-1}_h) \cap \ell^2 (0, T; H^1_h)$ norm, in which the the convexity of nonlinear energy potential has played an essential role. In addition, a rewritten form of the surface diffusion term has facilitated the convergence analysis, in which we have made use of the special structure of concentration-dependent deGennes type coefficient. Some numerical results are presented as well.

Keywords

Cahn–Hilliard equation, Flory–Huggins energy, deGennes diffusive coefficient, energy stability, positivity-preserving, convergence analysis

2010 Mathematics Subject Classification

60F10, 60J75, 62P10, 92C37

Full Text (PDF format)

This work is supported in part by the grants NSF DMS-1418689 (C. Wang), NSFC-11471046 and 11571045 (H. Zhang), NSFC-11571045, the Science Challenge Project TZ2018002 and the Fundamental Research Funds for the Central Universities (Z. Zhang).

Received 10 June 2018

Accepted 15 February 2019

Published 25 October 2019