  In this paper we propose and analyze a finite difference numerical scheme for
the Flory-Huggins-Cahn-Hilliard equation with dynamical boundary condition. The
singular logarithmic potential is included in the Flory-Huggins energy
expansion. Meanwhile, a dynamical evolution equation for the boundary profile
corresponds to a lower-dimensional singular energy potential. In turn, a
theoretical analysis for the coupled system becomes very challenging, since it
contains nonlinear and singular energy potentials for both the interior region
and on the boundary. In the numerical design, a convex splitting approach is
applied to the chemical potential associated with the energy both at the
interior region and on the boundary: implicit treatments for the singular and
logarithmic terms, as well as the surface diffusion terms, combined with an
explicit treatment for the concave expansive term. In addition, the discrete
boundary condition for the phase variable is coupled with the evolutionary
equation of the boundary profile. The resulting numerical system turns out to
be highly nonlinear, singular and coupled. A careful finite difference
approximation and convexity analysis reveals that such a numerical system could
be represented as a minimization of a discrete numerical energy functional,
which contains both the interior and boundary integrals. More importantly, all
the singular terms correspond to a discrete convex functional. As a result, a
unique solvability and positivity-preserving analysis could be theoretically
justified, based on the subtle fact that the singular nature of the logarithmic
terms around the singular limit values prevent the numerical solutions reaching
these values. The total energy stability analysis could be established by a
careful estimate over the finite difference inner product. Some numerical
results are presented in this article.
