Communications in Mathematical Sciences

Volume 19 (2021)

Number 2

Efficient numerical scheme for the anisotropic modified phase-field crystal model with a strong nonlinear vacancy potential

Pages: 355 – 381



Qi Li (School of Science, Chang’an University, Xi’an, Shaanxi, China)

Xiaofeng Yang (Department of Mathematics, University of South Carolina, Columbia, S.C., U.S.A.)

Liquan Mei (School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, Shaanxi, China)


In this paper, we consider numerical approximations for the anisotropic modified phasefield crystal model with a strong nonlinear vacancy potential, which describes microscopic phenomena involving atomic hopping and vacancy diffusion. The model is a nonlinear damped wave equation that includes an anisotropic Laplacian and a strong nonlinear vacancy term. To develop an easy to implement time marching scheme with unconditional energy stability, we combine the multiple scalar auxiliary variable (MSAV) approach with stabilization technique for achieving an efficient and linear numerical scheme, in which two new scalar auxiliary variables are introduced to reformulate the model and a linear stabilization term is added to enhance the stability and keep the required accuracy while using the large time steps. The scheme leads to decoupled linear equations with constant coefficients at each time step, and its unique solvability and unconditional energy stability are proved. Various numerical experiments are performed to show the accuracy, stability, and efficiency of the proposed scheme.


modified phase-field crystal model, anisotropic, vacancy, unconditionally energy stable, MSAV approach

2010 Mathematics Subject Classification

65M06, 65M70

The work of Q. Li is supported by the China Scholarship Council (CSC No. 201806280137). The work of X. Yang is supported by the NSF Grants DMS-1720212, DMS-1818783, and DMS-2012490. The work of L. Mei is supported by the Science Challenge Project (No. TZ2016002).

Received 20 November 2019

Accepted 7 September 2020

Published 12 April 2021