In this paper, we are interested to prove the uniform exponential decay of the energy for the finite difference fully-discretization of 1-D wave equation with an interior damping at \(\xi \). To this end, we decompose the fully discrete system into two subsystems: a conservative system, and a non conservative one. We show under a numerical hypothesis on \(\xi \) that a uniform observability inequality holds for a conservative system, when the mesh size tends to zero using Fourier series technique. Then, we use the observability inequality to prove the uniform exponential decay of the energy for the damped system. Finally, we describe some numerical experiments to illustrate the exponential decay of the energy.
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