  In this article, we consider a one-dimensional Timoshenko system subject to
different types of dissipation (linear and nonlinear dampings). Based on a
combination between the finite element and the finite difference methods, we
design a discretization scheme for the different Timoshenko systems under
consideration. We first come up with a numerical scheme to the free-undamped
Timoshenko system. Then, we adapt this numerical scheme to the corresponding
linear and nonlinear damped systems. Interestingly, this scheme reaches to
reproduce the most important properties of the discrete energy. Namely, we show
for the discrete energy the positivity, the energy conservation property and
the different decay rate profiles. We numerically reproduce the known
analytical results established on the decay rate of the energy associated with
each type of dissipation.
