| Abstract: |
| In this talk, we consider a free-boundary model in one space dimension which appears in the modeling of concrete carbonation. The unknowns $u$ and $v$ represent the mass concentration of $\mathrm{CO}_2$ respectively in aqueous and gaseous phase and $s$ represents the penetration depth which measures the size of the carbonated zone. The model consists in a system of two weakly coupled diffusion-reaction equations in a varying domain $(0, s(t))$ where $s$ solves an ordinary differential equation.\
It has been shown that the interface of the domain follows a $\sqrt{t}$-law of propagation (see Aiki and Muntean, Commun. Pure Appl. Anal. 2010 and Interfaces Free Bound. 2013). Our goal, is to define a numerical scheme which ensures that the approximate penetration depth behaves like $\sqrt{t}$.\
To this end, we propose a scheme obtained by an implicit Euler discretization in time and a finite volume discretization in space. We first prove the existence of a solution to the scheme. Then, we establish some estimates satisfied by the approximate solutions. Finally, we show that the approximate penetration depth follows a $\sqrt{t}$-law of propagation. |
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