This paper deals with the numerical approximation of solutions to second-order evolution equations. In their approach, the authors introduce the numerical viscosity terms in the approximation scheme. They show that the exponential or polynomial decay of the discretized model occurs whenever the original continuous problem has this property and when the spectrum of the spatial operator associated with the undamped problem satisfies the generalized gap condition. The convergence of the discrete problem is shown by using the Trotter-Kato theorem. Various examples related to the wave equation are included.