A high-order numerical method is developed for solving the Cahn-Hilliard-Navier-Stokes equations with the Flory-Huggins potential. The scheme is based on the $Q_k$ finite element with mass lumping on rectangular grids, the second-order convex splitting method, and the pressure correction method. The unique solvability, unconditional stability, and bound-preserving properties are rigorously established. The key to bound-preservation is the discrete $L^1$ estimate of the singular potential. Ample numerical experiments are performed to validate the desired properties of the proposed numerical scheme.

翻译：本文针对具有Flory-Huggins势的Cahn-Hilliard-Navier-Stokes方程，提出了一种高阶数值求解方法。该格式基于矩形网格上采用集中质量的$Q_k$有限元、二阶凸分裂方法以及压力修正方法。我们严格证明了格式的唯一可解性、无条件稳定性及保界特性。保界性质的关键在于奇异势的离散$L^1$估计。通过大量数值实验验证了所提数值格式的各项预期性质。