Stability and Convergence Analysis of the Exponential Time Differencing Scheme for a Cahn–Hilliard Binary Fluid-Surfactant Model

In this paper, we focus on the Cahn–Hilliard type of binary fluid-surfactant model, which is derived as the H⁻¹ gradient flow system of a binary energy functional of the fluid density and the surfactant density. By introducing two stabilization terms appropriately, we give a linear convex splitting of the energy functional, and then establish the exponential time differencing scheme with first-order temporal accuracy in combination with the Fourier spectral approximation in space. To guarantee the energy stability, we treat the nonlinear term partially implicitly in the equation for the fluid and evaluate the nonlinear term in the equation for the surfactant completely explicitly. The developed scheme is linear and decoupled, and the unconditional energy stability, the mass conservation, and the convergence are proved rigorously in the fully discrete setting. Various numerical experiments illustrate the stability and convergence of proposed scheme, along with the effectiveness in the long-time simulations.