In this paper, we develop a first order (in time) numerical scheme for the binary fluid surfactant phase field model. The free energy contains a double-well potential, a nonlinear coupling entropy and a Flory-Huggins potential. The resulting coupled system consists of two Cahn-Hilliard type equations. This system is solved numerically by finite difference spatial approximation, in combination with convex splitting temporal discretization. We prove the proposed scheme is unique solvable, positivity-preserving and unconditionally energy stable. In addition, an optimal rate convergence analysis is provided for the proposed numerical scheme, which will be the first such result for the binary fluid-surfactant system. Newton iteration is used to solve the discrete system. Some numerical experiments are performed to validate the accuracy and energy stability of the proposed scheme.

翻译：在本文中, 我们为二进制流体表面活性相位模型开发了第一个顺序( 及时) 数字方案。 免费能源包含双行潜力, 一个非线性联结的环球和 Flory- Huggins 潜力。 由此形成的组合系统由两个 Cahn- Hilliard 型方程式组成。 这个系统由有限的空间近似差和分解的分流时间离散以数字方式解决。 我们证明这个拟议方案是独一无二的可溶性、 相对性- 保护性和无条件的能量稳定。 此外, 为拟议的数字方案提供了最佳速率趋同率分析, 这将是二进制流- 流体活性系统的第一个结果。 牛顿迭代用于解决离散系统。 一些数字实验是为了验证拟议方案的准确性和能量稳定性。