In this article, we consider a one-dimensional Timoshenko system subject to different types of dissipation (linear and nonlinear damping). 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.
Finite difference Galerkin approximation Finite element Damped Timoshenko system Hyperbolic equations and systems
Mathematics Subject Classification
65M60 65M06 35B35 58J45
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A part of this work was performed while the first author was visiting LAMFA CNRS UMR 7352 CNRS UPJV. The first author would like to thank all the LAMFA members for their hospitality and their help with warm thanks to Professor Olivier Goubet for many fruitful discussions.
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