SIAM Journal on Scientific Computing, Volume 43, Issue 2, Page B479-B507, January 2021.
The binary fluid surfactant phase-field model, coupled with two Cahn--Hilliard equations and Navier--Stokes equations, is a very complex nonlinear system, which poses many challenges to the design of numerical schemes. As far as the author knows, due to the highly nonlinear coupling nature, there is no fully decoupled scheme with second-order accuracy in time for numerical approximation. This paper proposes a novel decoupling approach by introducing a nonlocal auxiliary variable and its associated ODE to deal with the nonlinear coupling terms that satisfy the so-called zero-energy-contribution property. By combining it with other proven effective methods (the projection method of the Navier--Stokes equations and the SAV method of linearizing nonlinear potential), we arrive at a fully decoupled, linear, unconditionally energy stable scheme with second-order time accuracy. At each time step, only a few fully decoupled linear elliptic equations with constant coefficients are needed to be solved, which shows the advantages of ease of implementation and efficiency. We also prove the unconditional energy stability rigorously and provide various numerical simulations in two and three dimensions to demonstrate its stability and accuracy, numerically.

中文翻译：

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二元表面活性剂相场模型的新型完全解耦，二阶精确能量稳定数值格式

SIAM科学计算杂志，第43卷，第2期，第B479-B507页，2021年1月。
二元流体表面活性剂相场模型，再加上两个Cahn-Hilliard方程和Navier-Stokes方程，是一个非常复杂的非线性系统，给数值方案的设计带来了许多挑战。据作者所知，由于高度非线性的耦合特性，在数值上还没有时间上具有二阶精度的完全解耦方案。本文通过引入非局部辅助变量及其关联的ODE来提出一种新颖的解耦方法，以处理满足所谓零能量贡献性质的非线性耦合项。通过将其与其他经过验证的有效方法（Navier-Stokes方程的投影方法和线性化非线性势的SAV方法）相结合，我们得出了完全解耦的线性 具有二阶时间精度的无条件能量稳定方案。在每个时间步上，只需要求解几个具有常数系数的完全解耦的线性椭圆方程，这显示了易于实现和效率的优点。我们还严格证明了无条件的能量稳定性，并在二维和三维中提供了各种数值模拟，以数值方式证明其稳定性和准确性。