Communications in Mathematical Sciences

Volume 17 (2019)

Number 3

Energy-stable second-order linear schemes for the Allen–Cahn phase-field equation

Pages: 609 – 635



Lin Wang (Beijing Computational Science Research Center, Beijing, China)

Haijun Yu (Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Beijing, China; and School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, China)


Phase-field model is a powerful mathematical tool to study the dynamics of interface and morphology changes in fluid mechanics and material sciences. However, numerically solving a phase field model for a real problem is a challenging task due to the non-convexity of the bulk energy and the small interface thickness parameter in the equation. In this paper, we propose two stabilized second-order semi-implicit linear schemes for the Allen–Cahn phase-field equation based on backward differentiation formula and Crank–Nicolson method, respectively. In both schemes, the nonlinear bulk force is treated explicitly with two second-order stabilization terms, which make the schemes unconditionally energy-stable and numerically efficient. By using a known result of the spectrum estimate of the linearized Allen–Cahn operator and some regularity estimates of the exact solution, we obtain an optimal second-order convergence in time with a prefactor depending on the inverse of the characteristic interface thickness only in some lower polynomial order. Both $2$-dimensional and $3$-dimensional numerical results are presented to verify the accuracy and efficiency of proposed schemes.


Allen–Cahn equation, energy-stable, stabilized semi-implicit scheme, second-order scheme, error estimate

2010 Mathematics Subject Classification

65M12, 65M15, 65P40

Copyright © 2019 Lin Wang and Haijun Yu

This work is partially supported by NNSFC Grants 11771439, U1530401, 91852116 and China National Program on Key Basic Research Project 2015CB856003.

Received 14 February 2018

Accepted 17 October 2018

Published 30 August 2019