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 linear refinement requirement $\Delta t\leq C_1 h$. 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. Convergence test is performed to demonstrate the accuracy and robustness of the proposed numerical scheme. Pearling bifurcation, meandering instability and spinodal decomposition are observed in the numerical simulations.

翻译：本文针对带有对数Flory-Huggins势的泛函Cahn-Hilliard（FCH）方程，提出并分析了一种一阶有限差分格式。基于FCH自由能的适当凸凹分解，我们设计了该半隐式数值格式。我们证明了该数值算法的唯一可解性，并验证了其在时间步长无限制条件下的无条件能量稳定性。由于Flory-Huggins势在对数部分于纯态$\pm 1$附近具有奇异性，我们从理论层面建立了相函数的保正性性质。因此，数值解在逐点意义上永远不会达到奇异值$\pm 1$，且全离散格式在每个时间步均明确定义。进一步地，我们给出了详细的最优收敛阶分析，并在线性细化条件$\Delta t\leq C_1 h$下，推导了$l^{\infty}(0,T;L_h^2)\cap l^2(0,T;H^3_h)$空间中的误差估计。为实现这一目标，我们利用基于傅里叶投影的高阶渐近展开（至二阶时间与空间精度）来控制数值解离散最大范数。我们证明：若连续问题的精确解严格远离纯态$\pm 1$，则数值解可与$\pm 1$保持一个正距离，该距离关于时间步长和网格尺寸一致。最后，我们给出了若干数值实验。通过收敛性测试验证了所提数值格式的精度与鲁棒性。数值模拟中观测到串珠分叉、蜿蜒不稳定性及旋节线分解等现象。