A finite difference numerical scheme is proposed and analyzed for the Cahn-Hilliard-Stokes system with Flory-Huggins energy functional. A convex splitting is applied to the chemical potential, which in turns leads to the implicit treatment for the singular logarithmic terms and the surface diffusion term, and an explicit update for the expansive concave term. The convective term for the phase variable, as well as the coupled term in the Stokes equation, are approximated in a semi-implicit manner. In the spatial discretization, the marker and cell (MAC) difference method is applied, which evaluates the velocity components, the pressure and the phase variable at different cell locations. Such an approach ensures the divergence-free feature of the discrete velocity, and this property plays an important role in the analysis. The positivity-preserving property and the unique solvability of the proposed numerical scheme are theoretically justified, utilizing the singular nature of the logarithmic term as the phase variable approaches the singular limit values. An unconditional energy stability analysis is standard, as an outcome of the convex-concave decomposition technique. A convergence analysis with accompanying error estimate is provided for the proposed numerical scheme. In particular, a higher order consistency analysis, accomplished by supplementary functions, is performed to ensure the separation properties of numerical solution. In turn, using the approach of rough and refined error (RRE) estimates, we are able to derive an optimal rate convergence. To conclude, several numerical experiments are presented to validate the theoretical analysis.

翻译：针对具有Flory-Huggins能量泛函的Cahn-Hilliard-Stokes系统，提出并分析了一种有限差分数值格式。对化学势采用凸分裂方法，从而对奇异对数项和表面扩散项进行隐式处理，而对膨胀凹项进行显式更新。相变量的对流项以及Stokes方程中的耦合项采用半隐式近似。在空间离散中，应用了标记点与网格（MAC）差分方法，该方法在网格不同位置处计算速度分量、压力和相变量。该途径确保了离散速度的无散性质，这一性质在分析中起着重要作用。利用相变量趋近奇异极限值时对数项的奇异性质，从理论上证明了所提数值格式的保正性和唯一可解性。作为凸-凹分解技术的结果，无条件能量稳定性分析是标准的。给出了所提数值格式的收敛性分析及相应的误差估计。特别地，通过辅助函数完成了一致性高阶分析，以确保数值解的分离性质。进而，采用粗糙与精细误差（RRE）估计方法，推导出最优收敛速率。最后，通过多个数值实验验证了理论分析结果。