A number of highly accurate analytic approximations to solitary waves on lattices are detailed, and the results of numerical tests presented. Two types of approximation are considered. The first set of approximations are generalizations of the continuum approximation, based around various Pade expansions of the operator. These are convergent to the exact solution in the limit of wavespeed c to 1, or a dense lattice. The author also considers ways of constructing approximations to fast solitary waves from a continuum-type theory. The two types are closely related. In both cases significant improvements are made on existing approximations. A symplectic Hamiltonian integration scheme is used to perform computer simulations. A new method of measuring the accuracy of a predicted waveform is also used on new and old approximations alike, to compare the various methods considered.