# Discrete Energy behavior of a damped Timoshenko system

## TL;DR

This paper develops a numerical scheme combining finite element and finite difference methods to analyze the energy decay behavior of a damped Timoshenko system, accurately reproducing theoretical decay rates.

## Contribution

It introduces a novel discretization scheme that preserves key energy properties and adapts to various damping types in Timoshenko systems.

## Key findings

- The scheme preserves positivity and energy conservation.
- It accurately reproduces decay rate profiles.
- Numerical results match analytical decay rates.

## Abstract

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.