Communications in Mathematical Sciences

Volume 17 (2019)

Number 3

Numerical approximations for the hydrodynamics coupled binary surfactant phase field model: Second-order, linear, unconditionally energy-stable schemes

Pages: 835 – 858



Chen Xu (Institute of Intelligent Computing Science, Shenzhen University, Shenzhen, China)

Chuanjun Chen (School of Mathematics and Information Sciences, Yantai University, Yantai, China)

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

Xiaoming He (Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, Mo., U.S.A.; and School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, China)


In this paper, we consider numerical approximations of a binary fluid-surfactant phasefield model coupled with the fluid flow, in which the system consists of the incompressible Navier–Stokes equations and two Cahn–Hilliard type equations. We develop two linear and second order time marching schemes for solving this system by combining the “Invariant Energy Quadratization” approach for the nonlinear potentials, the projection method for the Navier–Stokes equation, and a subtle implicit-explicit treatment for the stress and convective terms. We prove the well-posedness of the linear system and its unconditional energy stability rigorously. Various 2D and 3D numerical experiments are performed to validate the accuracy and energy stability of the proposed schemes.


phase-field, fluid-surfactant, Navier–Stokes, Cahn–Hilliard, second order, energy stability

2010 Mathematics Subject Classification

60F10, 60J75, 62P10, 92C37

C. Xu is partially supported by NSFC-61872429.

C. Chen is partially supported by the NSFC-11771375 and NSFC-11571297, Shandong Province Natural Science Foundation ZR2018MAQ008.

X. Yang is partially supported by NSF DMS-1720212 and DMS-1818783.

X. He is partially supported by the National Science Foundation under grant number DMS-1722647.

Received 9 May 2018

Accepted 21 February 2019

Published 30 August 2019