Numerical solutions for a Timoshenko-type system with thermoelasticity with second sound

TL;DR

This paper investigates the numerical solutions of a nonlinear Timoshenko system with thermoelasticity and second sound, validating theoretical energy decay results through finite difference simulations.

Contribution

It introduces a fourth order finite difference scheme to numerically analyze energy decay in a nonlinear Timoshenko system with thermoelasticity and second sound, confirming theoretical predictions.

Findings

Numerical simulations confirm energy decay depending on stability parameters.

The finite difference scheme accurately captures the asymptotic behavior.

Results validate theoretical energy decay rates in various cases.

Abstract

In this work, we consider a nonlinear vibrating Timoshenko system with thermoelasticity with second sound. We recall first the results of well-posdness and regularity and the asymptotic behavior of the energy obtained in \cite{Ayadi}. Then, we use a fourth order finite difference scheme to compute the numerical solutions and thus we show the energy decay in several cases depending on the stability number. R\'esum\'e : Dans ce travail, on consid\`ere le syst\`eme de Timoshenko non-lin\'eaire avec Thermo-\'elasticit\'e et deuxi\`eme son. On rappelle d'abord les r\'esultats d'existence, de r\'egularit\'e et du comportement asymptotique de l'\'energie obtenus dans \cite{Ayadi}. Ensuite, on valide num\'eriquement ces r\'esultats th\'eoriques. Pour cela, on utilise une m\'ethode de diff\'erences finies d'ordre $4$. Ainsi la solution num\'erique obtenue permet de valider la d\'ecroissance de l'\'energie dans plusieurs cas selon la valeur du param\`etre de stabilit\'e.