We construct a novel fully-decoupled and second-order accurate time marching numerical scheme with unconditional energy stability for the Cahn–Hilliard–Darcy phase-field model of the two-phase Hele–Shaw flow, in which, the key idea to realize the full decoupling structure is to use the so-called “zero-energy-contribution” function and design a special ordinary differential equation to deal with the nonlinear coupling terms between the flow field and the phase-field variable. Compared with the existing decoupling type schemes, the scheme developed here is more effective, efficient and easy to implement. At each time step, one only needs to solve a few fully-decoupled linear equations only with constant coefficients. We also strictly prove the solvability and unconditional energy stability of the scheme, and implement various numerical simulations in 2D and 3D to show the efficiency and stability of the proposed scheme numerically.

我们为两相Hele-Shaw流的Cahn-Hilliard-Darcy相场模型构造了一个新的具有无条件能量稳定性的完全解耦和二阶精确时间行进数值方案，其中，实现该过程的关键思想是完全的去耦结构是使用所谓的“零能量贡献”并设计一个特殊的常微分方程来处理流场和相场变量之间的非线性耦合项。与现有的去耦类型方案相比，此处开发的方案更加有效，高效且易于实施。在每一时间步上，仅需求解几个常数系数完全解耦的线性方程即可。我们还严格证明了该方案的可解性和无条件能量稳定性，并在2D和3D中进行了各种数值模拟，以数字方式显示了该方案的效率和稳定性。