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We propose and analyze a first-order finite difference scheme for thefunctionalized Cahn-Hilliard (FCH) equation with a logarithmic Flory-Hugginspotential. The semi-implicit numerical scheme is designed based on a suitableconvex-concave decomposition of the FCH free energy. We prove uniquesolvability of the numerical algorithm and verify its unconditional energystability without any restriction on the time step size. Thanks to the singularnature of the logarithmic part in the Flory-Huggins potential near the purestates $\pm 1$, we establish the so-called positivity-preserving property forthe phase function at a theoretic level. As a consequence, the numericalsolutions will never reach the singular values $\pm 1$ in the point-wise senseand the fully discrete scheme is well defined at each time step. Next, wepresent a detailed optimal rate convergence analysis and derive error estimatesin $l^{\infty}(0,T;L_h^2)\cap l^2(0,T;H^3_h)$ under a linear refinementrequirement $\Delta t\leq C_1 h$. To achieve the goal, a higher orderasymptotic expansion (up to the second order temporal and spatial accuracy)based on the Fourier projection is utilized to control the discrete maximumnorm of solutions to the numerical scheme. We show that if the exact solutionto 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 positivedistance that is uniform with respect to the size of the time step and thegrid. Finally, a few numerical experiments are presented. Convergence test isperformed to demonstrate the accuracy and robustness of the proposed numericalscheme. Pearling bifurcation, meandering instability and spinodal decompositionare observed in the numerical simulations.

journal of scientific computing

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