In this work, it is shown that the similarities between discretized and continuum models of water are heavily dependent upon the exact value of the discretization parameter, ζ, defined as the ratio of the Lennard-Jones size parameter σ to the lattice spacing. For example, some discretized models layered into a hexagonal ice-like structure at temperatures up to 550K. A discretization parameter of ζ = 13 was chosen such that the radial distribution function of the liquid at 298K for the lattice model closely matched the continuum. This discretized model is capable of reproducing the low-temperature phase diagram with the advantage of significant CPU time savings.