  We consider the asymptotic behaviour of the second discrete Painlev\'{e}
equation in the limit as the independent variable becomes large. Using
asymptotic power series, we find solutions that are asymptotically pole-free
within some region of the complex plane. These asymptotic solutions exhibit
Stokes phenomena, which is typically invisible to classical power series
methods. We subsequently apply exponential asymptotic techniques to investigate
such phenomena, and obtain mathematical descriptions of the rapid switching
behaviour associated with Stokes curves. Through this analysis, we determine
the regions of the complex plane in which the asymptotic approximations are
valid, and find that the behaviour of these asymptotic solutions shares a
number of features with the tronqu\'{e}e and tri-tronqu\'{e}e solutions of the
second continuous Painlev\'{e} equation.
