Uniform exponential decay of the energy for a fully discrete wave equation with point-wise dissipation

In this paper, the uniform exponential decay of the energy for a fully discretization of the 1-D wave equation has been studied with a punctual interior damping at xi$$ \xi $$ (xi is an element of(0,1))$$ \left(\xi \in \left(0,1\right)\right) $$. It is known that the usual scheme obtained with a space discretization is not uniformly exponential decaying. In this direction, we have introduced an implicit finite difference scheme that differs from the usual centered one by additional terms of order h2$$ {h}<^>2 $$ and delta t2$$ \delta {t}<^>2 $$ (where h$$ h $$ and delta t$$ \delta t $$ represent the discretization parameters). In this approach, the fully discrete system was decomposed into two subsystems, namely conservative and nonconservative. According to the literature, it has been noted that under a numerical hypothesis on xi$$ \xi $$, a uniform observability inequality holds for a conservative system. The uniform exponential decay of the energy for the damped system was proved via the application of the observability inequality.