On the maximum principle preserving schemes for the generalized Allen-Cahn equation

Jie Shen, Tao Tang, Jiang Yang

Research output: Contribution to journalJournal articlepeer-review

107 Citations (Scopus)


This paper is concerned with the generalized Allen-Cahn equation with a nonlinear mobility that may be degenerate, which also includes an advection term appearing in many phasefield models for multi-component fluid flows. A class of maximum principle preserving schemes will be studied for the generalized Allen-Cahn equation, with either the commonly used polynomial free energy or the logarithmic free energy, and with a nonlinear degenerate mobility. For time discretization, the standard semi-implicit scheme as well as the stabilized semi-implicit scheme will be adopted, while for space discretization, the central finite difference is used for approximating the diffusion term and the upwind scheme is employed for the advection term. We establish the maximum principle for both semi-discrete (in time) and fully discretized schemes. We also provide an error estimate by using the established maximum principle which plays a key role in the analysis. Several numerical experiments are carried out to verify our theoretical results.

Original languageEnglish
Pages (from-to)1517-1534
Number of pages18
JournalCommunications in Mathematical Sciences
Issue number6
Publication statusPublished - 12 Aug 2016

Scopus Subject Areas

  • Mathematics(all)
  • Applied Mathematics

User-Defined Keywords

  • Allen-Cahn equation
  • Error estimate
  • Finite difference
  • Maximum principle
  • Stability


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