Numerical Approximations for the Cahn-Hilliard phase field model of the binary fluid-surfactant system
This work provides practical numerical methods for simulating phase field models of fluid-surfactant systems, addressing the challenge of energy stability in coupled nonlinear equations.
The authors developed linear, decoupled time-stepping schemes for the binary fluid-surfactant Cahn-Hilliard model, achieving unconditional energy stability with first- and second-order accuracy. Numerical simulations confirmed stability and accuracy.
In this paper, we consider the numerical approximations for the commonly used binary fluid-surfactant phase field model that consists two nonlinearly coupled Cahn-Hilliard equations. The main challenge in solving the system numerically is how to develop easy-to-implement time stepping schemes while preserving the unconditional energy stability. We solve this issue by developing two linear and decoupled, first order and a second order time-stepping schemes using the so-called "Invariant Energy Quadratization" approach for the double well potentials and a subtle explicit-implicit technique for the nonlinear coupling potential. Moreover, the resulting linear system is well-posed and the linear operator is symmetric positive definite. We rigorously prove the first order scheme is unconditionally energy stable. Various numerical simulations are presented to demonstrate the stability and the accuracy thereafter.