We propose and analyze a first-order finite difference scheme for the functionalized Cahn-Hilliard (FCH) equation with a logarithmic Flory-Huggins potential. The semi-implicit numerical scheme is designed based on a suitable convex-concave decomposition of the FCH free energy. We prove unique solvability of the numerical algorithm and verify its unconditional energy stability without any restriction on the time step size. Thanks to the singular nature of the logarithmic part in the Flory-Huggins potential near the pure states $\pm 1$, we establish the so-called positivity-preserving property for the phase function at a theoretic level. As a consequence, the numerical solutions will never reach the singular values $\pm 1$ in the point-wise sense and the fully discrete scheme is well defined at each time step. Next, we present a detailed optimal rate convergence analysis and derive error estimates in $l^{\infty}(0,T;L_h^2)\cap l^2(0,T;H^3_h)$ under a mild CFL condition $C_1 h\leq\Delta t\leq C_2 h$. To achieve the goal, a higher order asymptotic expansion (up to the second order temporal and spatial accuracy) based on the Fourier projection is utilized to control the discrete maximum norm of solutions to the numerical scheme. We show that if the exact solution to the continuous problem is strictly separated from the pure states $\pm 1$, then the numerical solutions can be kept away from $\pm 1$ by a positive distance that is uniform with respect to the size of the time step and the grid. Finally, a few numerical experiments are presented to demonstrate the accuracy and robustness of the proposed numerical scheme.

翻译：我们为功能化 Cahn- Hilliard (FCH) 方程式提出并分析一阶定值差异方案。 半隐含数字方案的设计基于FCH 自由能量的合适的 convex- concave 分解。 我们证明数字算法的独特可溶性, 并核查其无条件的能源稳定性, 不受时间步数的限制 。 由于在纯状态附近的 Floriy- huggins (FCH) 方程式的对数部分的奇特性质, 我们为阶段函数设定了所谓的正态- pm 1 $ 。 我们为级函数设定了所谓的正态- pm 1 软化- hugggins 方程式保存属性 。 因此, 数字解决方案将永远无法达到点感奇值$\ pm 1, 完全离差值的公式将显示为 $@ infty introfty fty} (0, T;; L_h/2\\\\\\\\\\\\\\\ cap creal deal deal deal deal deal deal deal deal deal deal deal deal deal deal) as the a mal deal deal deal a mal a mal demod the a.