Convergence of a finite volume scheme for a parabolic system with a free boundary modeling concrete carbonation

Chainais-Hillairet, Claire; Merlet, Benoît; Zurek, Antoine

doi:10.1051/m2an/2018002

Cited by 6 publications

(18 citation statements)

References 18 publications

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“…The matrices

{double-struckM}_{\overset{u}{true}}

and

{double-struckM}_{\overset{v}{true}}

are tridiagonal. Moreover,

{double-struckM}_{\overset{u}{true}}

and

{double-struckM}_{\overset{v}{true}}

are M‐matrices and thus invertible and monotone, see . Since,

b_{\overset{u}{true}} \geq 0

and

b_{\overset{v}{true}} \geq 0

, we deduce thanks to the induction hypothesis that

\overset{true^}{U} \geq 0, \overset{true^}{V} \geq 0

and by (23) we conclude that

{\overset{v}{true}}_{l + 1} \geq 0

.…”

Section: Existence Of a Solution To The Schemementioning

confidence: 73%

“…Finally, following the proof of , we show that for all i ∈ {1, … , l + 1} we have

{\overset{u}{true}}_{i} \leq g^{*} and {\overset{v}{true}}_{i} \leq r^{*} .

Thus, T n stabilizes the set

K

and then, thanks to the Brouwer's fixed‐point theorem, T n has a fixed‐point in

K

, denoted by ( u n + 1 , v n + 1 ). Eventually, we construct s n + 1 by

s^{n + 1} = s^{n} + Δ italict α {((), u_{l + 1}^{n + 1})}^{p} .

Hence, we deduce the existence of ( s n + 1 , u n + 1 , v n + 1 ) solution to ( S ) such that u n + 1 , v n + 1 and s n + 1 satisfy (19).…”

Section: Existence Of a Solution To The Schemementioning

confidence: 75%

“…To this end, we introduce the application

T^{n} : K \to ℝ^{l + 2} \times ℝ^{l + 2},

such that

T^{n} (u, v) = (\overset{u}{true}, \overset{v}{true})

. The definition of T n is based on the linear scheme proposed in and defined in two steps. First, for

(u, v) \in K

we define the element

\overset{true^}{s} = s^{n} + Δ italict α {((), u_{l + 1})}^{p} .

…”

Section: Existence Of a Solution To The Schemementioning

confidence: 99%

“…For numerical reasons, it is convenient to rewrite this model on a fixed domain. To this end, we use a change of variables . Then, in the new coordinate system we consider

Q (T) = \{(t, x) : 0 < x < 1, 0 < t < T\} .

So that, we can rewrite the model in as a convection–diffusion–reaction system defined by:

s (t) \partial_{t} (s (t) u) + \partial_{x} J_{u} = s^{2} (t) f (u, v) in Q (T),

s (t) \partial_{t} (s (t) v) + \partial_{x} J_{v} = - s^{2} (t) f (u, v) in Q (T),

s^{'} (t) = ψ (u (1, t)) for 0 < t < T,

u (0, t) = g (t) for 0 < t < T,

…”

Section: Introductionmentioning

confidence: 99%

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Numerical approximation of a concrete carbonation model: Study of the ‐law of propagation

Zurek

2019

Numerical Methods Partial

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In this paper, we are interested in the long time behavior of approximate solutions to a free boundary model which appears in the modeling of concrete carbonation [1]. In particular, we study the long time regime of the moving interface. The numerical solutions are obtained by an implicit in time and finite volume in space scheme. We show the existence of solutions to the scheme and, following [2‐3], we prove that the approximate free boundary increases in time following a t‐law. Finally, we supplement the study through numerical experiments.

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“…The matrices

{double-struckM}_{\overset{u}{true}}

and

{double-struckM}_{\overset{v}{true}}

are tridiagonal. Moreover,

{double-struckM}_{\overset{u}{true}}

and

{double-struckM}_{\overset{v}{true}}

are M‐matrices and thus invertible and monotone, see . Since,

b_{\overset{u}{true}} \geq 0

and

b_{\overset{v}{true}} \geq 0

, we deduce thanks to the induction hypothesis that

\overset{true^}{U} \geq 0, \overset{true^}{V} \geq 0

and by (23) we conclude that

{\overset{v}{true}}_{l + 1} \geq 0

.…”

Section: Existence Of a Solution To The Schemementioning

confidence: 73%

“…Finally, following the proof of , we show that for all i ∈ {1, … , l + 1} we have

{\overset{u}{true}}_{i} \leq g^{*} and {\overset{v}{true}}_{i} \leq r^{*} .

Thus, T n stabilizes the set

K

and then, thanks to the Brouwer's fixed‐point theorem, T n has a fixed‐point in

K

, denoted by ( u n + 1 , v n + 1 ). Eventually, we construct s n + 1 by

s^{n + 1} = s^{n} + Δ italict α {((), u_{l + 1}^{n + 1})}^{p} .

Hence, we deduce the existence of ( s n + 1 , u n + 1 , v n + 1 ) solution to ( S ) such that u n + 1 , v n + 1 and s n + 1 satisfy (19).…”

Section: Existence Of a Solution To The Schemementioning

confidence: 75%

“…To this end, we introduce the application

T^{n} : K \to ℝ^{l + 2} \times ℝ^{l + 2},

such that

T^{n} (u, v) = (\overset{u}{true}, \overset{v}{true})

. The definition of T n is based on the linear scheme proposed in and defined in two steps. First, for

(u, v) \in K

we define the element

\overset{true^}{s} = s^{n} + Δ italict α {((), u_{l + 1})}^{p} .

…”

Section: Existence Of a Solution To The Schemementioning

confidence: 99%

“…For numerical reasons, it is convenient to rewrite this model on a fixed domain. To this end, we use a change of variables . Then, in the new coordinate system we consider

Q (T) = \{(t, x) : 0 < x < 1, 0 < t < T\} .

So that, we can rewrite the model in as a convection–diffusion–reaction system defined by:

s (t) \partial_{t} (s (t) u) + \partial_{x} J_{u} = s^{2} (t) f (u, v) in Q (T),

s (t) \partial_{t} (s (t) v) + \partial_{x} J_{v} = - s^{2} (t) f (u, v) in Q (T),

s^{'} (t) = ψ (u (1, t)) for 0 < t < T,

u (0, t) = g (t) for 0 < t < T,

…”

Section: Introductionmentioning

confidence: 99%

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Numerical approximation of a concrete carbonation model: Study of the ‐law of propagation

Zurek

2019

Numerical Methods Partial

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“…In this short paper, we just give sketch of the convergence proof. Full details will be provided in a forthcoming paper [8].…”

Section: Introductionmentioning

confidence: 99%

Design and Analysis of a Finite Volume Scheme for a Concrete Carbonation Model

Chainais-Hillairet

Merlet

Zurek

2017

Springer Proceedings in Mathematics &Amp; Statistics

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. Design and analysis of a finite volume scheme for a concrete carbonation model. Design and analysis of a finite volume scheme for a concrete carbonation model Claire Chainais-Hillairet, Benoît Merlet and Antoine Zurek Abstract In this paper we introduce a finite volume scheme for a concrete carbonation model proposed by Aiki and Muntean in [1]. It consists in a Euler discretisation in time and a Scharfetter-Gummel discretisation in space. We give here some hints for the proof of the convergence of the scheme and show numerical experiments.

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A free boundary problem describing migration into rubbers – Quest for the large time behavior

2022

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In many industrial applications, rubber‐based materials are routinely used in conjunction with various penetrants or diluents in gaseous or liquid form. It is of interest to estimate theoretically the penetration depth as well as the amount of diffusants stored inside the material. In this framework, we prove the global solvability and explore the large time‐behavior of solutions to a one‐phase free boundary problem with nonlinear kinetic condition that is able to describe the migration of diffusants into rubber. The key idea in the proof of the large time behavior is to benefit from a contradiction argument, since it is difficult to obtain uniform estimates for the growth rate of the free boundary due to the use of a Robin boundary condition posed at the fixed boundary.

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Convergence of a finite volume scheme for a parabolic system with a free boundary modeling concrete carbonation

Cited by 6 publications

References 18 publications

Numerical approximation of a concrete carbonation model: Study of the ‐law of propagation

Numerical approximation of a concrete carbonation model: Study of the ‐law of propagation

Design and Analysis of a Finite Volume Scheme for a Concrete Carbonation Model

A free boundary problem describing migration into rubbers – Quest for the large time behavior

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