We construct a novel second-order time marching scheme with the full decoupling structure to solve a highly coupled nonlinear two-phase fluid flow system consisting of the nonlocal mass-conserved Allen–Cahn equation where two types of flow regimes are considered (Navier–Stokes and Darcy). We achieve the full decoupled structure by introducing a nonlocal variable and designing an additional ordinary differential equation for it which plays the key role to maintain the unconditional energy stability. The whole scheme is built upon the pressure correction/ quadratization approach for the fluid equation and nonlinear double-well potential, respectively. At each time step, one only needs to solve several independent elliptic equations with constant coefficients illustrating the high practical efficiency. We strictly prove that the scheme satisfies the unconditional energy stability, and carry out various numerical simulations to prove its stability and accuracy numerically, such as spinodal decomposition and fingering instability due to the continuous injection flow, etc.

我们构造了具有完全去耦结构的新型二阶时间行进方案，以解决由非局部质量守恒的Allen-Cahn方程组成的高度耦合的非线性两相流体流动系统，其中考虑了两种流动形式（Navier-Stokes和达西）。我们通过引入一个非局部变量并为其设计一个附加的常微分方程来实现完全解耦的结构，该方程对于保持无条件能量稳定性起关键作用。整个方案建立在分别针对流体方程和非线性双井势的压力校正/平方方法上。在每一时间步上，只需要求解几个独立的椭圆方程具有恒定的系数说明了较高的实际效率。我们严格证明该方案满足无条件的能量稳定性，并进行了各种数值模拟，以数值证明其稳定性和准确性，例如旋节线分解和连续注入流造成的指状不稳定性等。