A Uniquely Solvable, Positivity-Preserving and Unconditionally Energy Stable Numerical Scheme for the Functionalized Cahn-Hilliard Equation with Logarithmic Potential

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