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 nume

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