  In this paper, two finite difference numerical schemes are proposed and
analyzed for the droplet liquid film model, with a singular Leonard-Jones
energy potential involved. Both first and second order accurate temporal
algorithms are considered. In the first order scheme, the convex potential and
the surface diffusion terms are implicitly, while the concave potential term is
updated explicitly. Furthermore, we provide a theoretical justification that
this numerical algorithm has a unique solution, such that the positivity is
always preserved for the phase variable at a point-wise level, so that a
singularity is avoided in the scheme. In fact, the singular nature of the
Leonard-Jones potential term around the value of 0 prevents the numerical
solution reaching such singular value, so that the positivity structure is
always preserved. Moreover, an unconditional energy stability of the numerical
scheme is derived, without any restriction for the time step size. In the
second order numerical scheme, the BDF temporal stencil is applied, and an
alternate convex-concave decomposition is derived, so that the concave part
corresponds to a quadratic energy. In turn, the combined Leonard-Jones
potential term is treated implicitly, and the concave part the is approximated
by a second order Adams-Bashforth explicit extrapolation, and an artificial
Douglas-Dupont regularization term is added to ensure the energy stability. The
unique solvability and the positivity-preserving property for the second order
scheme could be similarly established. In addition, optimal rate convergence
analysis is provided for both the first and second order accurate schemes. A
few numerical simulation results are also presented, which demonstrate the
robustness of the numerical schemes.
