We consider the numerical approximation of the flow‐coupled phase‐field model of two‐phase incompressible flows using the mass‐conserved Allen‐Cahn equation. Due to the highly nonlinear nature of the coupling, how to develop an accurate and practically efficient scheme has always been a challenging problem. To solve this challenge, we construct a novel effective fully decoupled scheme that is linear, unconditional energy stable, and second‐order time accurate. The key idea of decoupling is to introduce a nonlocal variable and a related ordinary differential equation to deal with the nonlinear coupling terms that satisfy the so‐called “zero‐energy‐contribution” property. Thus, in actual calculations, this scheme only needs to solve several independent linear equations at each time step to obtain a numerical solution with the second‐order time accuracy. We strictly prove the solvability and unconditional energy stability and perform numerical simulations in 2D and 3D to verify the accuracy and stability of the scheme numerically.

中文翻译：

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Navier-Stokes方程的二阶时间精度和无条件能量稳定性的新型完全解耦方案，以及两相不可压缩流的质量守恒的Allen-Cahn相场模型

我们使用质量守恒的Allen-Cahn方程考虑了两相不可压缩流的流耦合相场模型的数值近似。由于联轴器的高度非线性特性，如何开发出准确而实用的方案一直是一个具有挑战性的问题。为了解决这一挑战，我们构建了一种新颖的有效的完全解耦方案，该方案具有线性，无条件能量稳定和二阶时间精确度。解耦的关键思想是引入一个非局部变量和一个相关的常微分方程来处理满足所谓“零能量贡献”性质的非线性耦合项。因此，在实际计算中 该方案仅需在每个时间步求解几个独立的线性方程，即可获得具有二阶时间精度的数值解。我们严格证明了可解性和无条件能量稳定性，并在2D和3D中进行了数值模拟，以数字方式验证该方案的准确性和稳定性。