Stabilized linear semi-implicit schemes for the nonlocal Cahn–Hilliard equation

Qiang Du, Lili Ju, Xiao Li, Zhonghua Qiao

Research output: Journal article publicationJournal articleAcademic researchpeer-review

72 Citations (Scopus)


Comparing with the well-known classic Cahn–Hilliard equation, the nonlocal Cahn–Hilliard equation is equipped with a nonlocal diffusion operator and can describe more practical phenomena for modeling phase transitions of microstructures in materials. On the other hand, it evidently brings more computational costs in numerical simulations, thus efficient and accurate time integration schemes are highly desired. In this paper, we propose two energy-stable linear semi-implicit methods with first and second order temporal accuracies respectively for solving the nonlocal Cahn–Hilliard equation. The temporal discretization is done by using the stabilization technique with the nonlocal diffusion term treated implicitly, while the spatial discretization is carried out by the Fourier collocation method with FFT-based fast implementations. The energy stabilities are rigorously established for both methods in the fully discrete sense. Numerical experiments are conducted for a typical case involving Gaussian kernels. We test the temporal convergence rates of the proposed schemes and make a comparison of the nonlocal phase transition process with the corresponding local one. In addition, long-time simulations of the coarsening dynamics are also performed to predict the power law of the energy decay.
Original languageEnglish
Pages (from-to)39-54
Number of pages16
JournalJournal of Computational Physics
Publication statusPublished - 15 Jun 2018


  • Energy stability
  • Fast Fourier transform
  • Gaussian kernel
  • Nonlocal Cahn–Hilliard equation
  • Nonlocal diffusion operator
  • Stabilized linear scheme

ASJC Scopus subject areas

  • Physics and Astronomy (miscellaneous)
  • Computer Science Applications


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