In this article, we develop a new linear, decoupled, second-order accurate, and energy stable numerical method for a modified phase-field surfactant model (Xu et al., 2020). The proposed scheme is a simple and efficient variant of stabilized-scalar auxiliary variable (S-SAV) method. The proposed scheme not only retains all advantages of S-SAV method but also simplifies the solution algorithm. The phase-field, surfactant, and auxiliary variables are totally decoupled in time, thus we can solve the whole system in a step-by-step manner. The phase-field function $\varphi$ and surfactant $\psi$ can be separately updated by solving two linear semi-implicit systems with constant coefficients and then the auxiliary variable is directly updated in an explicit way. We analytically prove the energy stability and the unique solvability of the proposed method. The numerical experiments show the desired temporal accuracy and energy stability. We numerically investigate the proper stabilization coefficients for the present scheme with specific parameters. Furthermore, various two- and three-dimensional benchmark tests are performed to study the dynamics of surfactant-laden phase separation.

在本文中，我们为修正的相场表面活性剂模型开发了一种新的线性，解耦，二阶精确且能量稳定的数值方法（Xu等，2020）。提出的方案是稳定标量辅助变量（S-SAV）方法的一种简单有效的变体。所提出的方案不仅保留了S-SAV方法的所有优点，而且简化了求解算法。相场，表面活性剂和辅助变量在时间上完全解耦，因此我们可以逐步解决整个系统。相场功能$\varphi$ 和表面活性剂 $\psi$可以通过求解两个具有常数系数的线性半隐式系统分别进行更新，然后以显式方式直接更新辅助变量。我们通过分析证明了该方法的能量稳定性和独特的可溶性。数值实验表明所需的时间精度和能量稳定性。我们在数值上研究了具有特定参数的本方案的适当稳定系数。此外，进行了各种二维和三维基准测试，以研究负载表面活性剂的相分离的动力学。