There are a couple of books that describe how asymptotic methods developed for differential equations can be extended to difference equations
Bender Orszag Advanced Math methods points out that even the simplest nonlinear difference eqns are rarely exactly solvable because they and also do not exhibit movable singularities (unlike PDE). No general procedure exists to determine local behavior of nonlinear difference equation. There are very few techniques for closed form solutionssolns. Some of the widely applicable techniques are substitution, use of known nonlinear functional relations and use of generating functions.
Holmes in Perturbation Methods Sec 4.8, 3.9 and 2.7 discusses how WKB, multiple scales and matched asymptotic expansion can be applied to solve difference equations.
See also http://www.math.niu.edu/~rusin/known-math/99/difference_eq for two books which lucubrate on the analogy between difference and differential equations
