Communications in Mathematical Sciences

Volume 14 (2016)

Number 6

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

Pages: 1517 – 1534



Jie Shen (Department of Mathematics, Purdue University, West Lafayette, Indiana, U.S.A.)

Tao Tang (Dept. of Mathematics, South University of Science and Technology, Shenzhen, Guangdong, China; Dept. of Mathematics and Institute for Computational and Theoretical Studies, Hong Kong Baptist University, Kowloon Tong, Hong Kong)

Jiang Yang (Department of Applied Physics and Applied Mathematics, Columbia University, New York, N.Y., U.S.A.)


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.


Allen–Cahn equation, stability, error estimate, maximum principle, finite difference

2010 Mathematics Subject Classification

65G20, 65M06, 65M12

Published 12 August 2016