Stability criterion to explicit finite difference applied to the Bresse system

Abstract

In this work, we show that the stability criterion of the explicit time integration method applied to the Bresse system is given by

$$\begin{aligned} \Delta t\le \displaystyle \frac{2\epsilon }{\sqrt{ \bigg (12 +\displaystyle \frac{\epsilon ^2}{R^2}\bigg )}\displaystyle \frac{k G}{\rho }}, \end{aligned}$$

where the thickness $\epsilon$ constitutes a limitation to compute the numerical solutions. This restriction to the stability criterion is not standard (is not CFL condition) and if $\epsilon <<1$ it is very restrictive to numerical computations. To overcome this restriction, we use the technics performed by Wright [Commun Appl Numer Methods 3:181–185 (1987), Commun Numer Methods Eng 14:81–86 (1998)] to minimize the influence of $\epsilon$ on stability criterion such that the CFL condition prevails.