We present an optimal rate convergence analysis for a second order accurate in time, fully discrete finite difference scheme for the Cahn-Hilliard-Navier-Stokes (CHNS) system, combined with logarithmic Flory-Huggins energy potential. The numerical scheme has been recently proposed, and the positivity-preserving property of the logarithmic arguments, as well as the total energy stability, have been theoretically justified. In this paper, we rigorously prove second order convergence of the proposed numerical scheme, in both time and space. Since the CHNS is a coupled system, the standard $\ell^\infty (0, T; \ell^2) \cap \ell^2 (0, T; H_h^2)$ error estimate could not be easily derived, due to the lack of regularity to control the numerical error associated with the coupled terms. Instead, the $\ell^\infty (0, T; H_h^1) \cap \ell^2 (0, T; H_h^3)$ error analysis for the phase variable and the $\ell^\infty (0, T; \ell^2)$ analysis for the velocity vector, which shares the same regularity as the energy estimate, is more suitable to pass through the nonlinear analysis for the error terms associated with the coupled physical process. Furthermore, the highly nonlinear and singular nature of the logarithmic error terms makes the convergence analysis even more challenging, since a uniform distance between the numerical solution and the singular limit values of is needed for the associated error estimate. Many highly non-standard estimates, such as a higher order asymptotic expansion of the numerical solution (up to the third order accuracy in time and fourth order in space), combined with a rough error estimate (to establish the maximum norm bound for the phase variable), as well as a refined error estimate, have to be carried out to conclude the desired convergence result.

翻译：本文针对耦合对数Flory-Huggins能量势的Cahn-Hilliard-Navier-Stokes (CHNS)系统，提出了一种时间二阶精度的全离散有限差分格式的最优收敛率分析。该数值格式为近期提出，其对数参数的正性保持性质及总能量稳定性已得到理论证明。本文严格证明了所提数值格式在时间和空间上的二阶收敛性。由于CHNS为耦合系统，标准$\ell^\infty (0, T; \ell^2) \cap \ell^2 (0, T; H_h^2)$误差估计因缺乏控制耦合项数值误差的正则性而难以直接推导。为此，采用与能量估计具有相同正则性的相变量$\ell^\infty (0, T; H_h^1) \cap \ell^2 (0, T; H_h^3)$误差分析及速度向量$\ell^\infty (0, T; \ell^2)$分析，更适用于处理耦合物理过程相关误差项的非线性分析。此外，对数误差项的高度非线性和奇异性使得收敛分析更具挑战性，因为相关误差估计需要确保数值解与奇异极限值之间的均匀距离。为得到预期的收敛结果，必须实施多项高度非标准估计，包括结合数值解的高阶渐近展开（时间三阶精度、空间四阶精度）与粗糙误差估计（建立相变量的最大模界）及精细化误差估计。