  Using uniform global Carleman estimates for discrete elliptic and
semi-discrete hyperbolic equations, we study Lipschitz and logarithmic
stability for the inverse problem of recovering a potential in a semi-discrete
wave equation, discretized by finite differences in a 2-d uniform mesh, from
boundary or internal measurements. The discrete stability results, when
compared with their continuous counterparts, include new terms depending on the
discretization parameter h. From these stability results, we design a numerical
method to compute convergent approximations of the continuous potential.
