  We investigate the nonisospectral effects of a semi-discrete nonlinear
Schr\"{o}dinger equation, which is a direct integrable discretisation of its
continuous counterpart. Bilinear form and double casoratian solution of the
equation are presented. Dynamics of solutions are analyzed. Both solitons and
multiple pole solutions admit space-time localized rogue wave behavior. And
more interestingly, the solutions allow blow-up at finite time $t$.
