Comparative study of the lattice Boltzmann models for Allen-Cahn and Cahn-Hilliard equations

H L Wang; Z H Chai; B C Shi; H Liang

doi:10.1103/PhysRevE.94.033304

Comparative study of the lattice Boltzmann models for Allen-Cahn and Cahn-Hilliard equations

Phys Rev E. 2016 Sep;94(3-1):033304. doi: 10.1103/PhysRevE.94.033304. Epub 2016 Sep 23.

Authors

H L Wang¹, Z H Chai^{1

2}, B C Shi^{1

2}, H Liang³

Affiliations

¹ School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China.
² State Key Laboratory of Coal Combustion, Huazhong University of Science and Technology, Wuhan 430074, China.
³ Department of Physics, Hangzhou Dianzi University, Hangzhou 310018, China.

PMID: 27739765
DOI: 10.1103/PhysRevE.94.033304

Abstract

In this paper, a comparative study of the lattice Boltzmann (LB) models for the Allen-Cahn (A-C) and Cahn-Hilliard (C-H) equations is conducted. To this end, a new LB model for the A-C equation is first proposed, where the equilibrium distribution function and the source term distribution function are delicately designed to recover the A-C equation correctly. The gradient term in this model can be computed by the nonequilibrium part of the distribution function such that the collision process can be implemented locally. Then a detailed numerical study on several classical problems is performed to give a comparison between the present model for the A-C equation and the recently developed LB model [H. Liang et al., Phys. Rev. E 89, 053320 (2014)PLEEE81539-375510.1103/PhysRevE.89.053320] for the C-H equation in terms of tracking the interface of two-phase flow. The results show that the present LB model for the A-C equation is more accurate and more stable, and also has a second-order convergence rate in space, while the convergence rate of the previous LB model for the C-H equation is only about 1.5.