A positivity-preserving scheme for the nonlocal Cahn-Hilliard equation with logarithmic potential

In this paper, we present and analyze a finite difference numerical scheme for the nonlocal Cahn-Hilliard equation with logarithmic Flory-Huggins energy potential. In particular, variable mobility is considered in the gradient flow system for all numerical analyses and tests. To ensure the unique solvability and energy stability, the convex splitting method is applied to the chemical potential, which leads to the implicit treatment for the nonlinear logarithmic terms, the surface diffusion term and the nonlocal term, while the expansive concave term and the mobility are treated explicitly. The positivity preserving property is always preserved at a point-wise level thanks to the singular nature of the logarithmic terms around the values of -1 and 1. In addition, an optimal rate convergence analysis is provided for the proposed numerical scheme. Due to the existence of variable mobility, a higher order asymptotic expansion and the rough error estimate are performed to ensure the uniform bound for numerical solution. Separation property in hand, we finally accomplish the convergence analysis with refined error estimate. Several numerical experiments, especially those designed for nonlocal term, are also presented in this paper, which demonstrate the robustness of the proposed numerical scheme and validate our previous theoretical analysis.