Numerical Methods for the Simulation of a Corrosion Model with Moving Oxide Layer $

Numerical Methods for the Simulation of a Corrosion Model with Moving Oxide Layer

C. Bataillon^a, F. Bouchon^b, C. Chainais-Hillairet^c,1, J. Fuhrmann^d,∗, E. Hoarau^e, R. Touzani^b

aCEA/DEN/DANS/DPC/SCCME/LECA, F-91191 Gif sur Yvette Cedex

bUniversit´e Blaise Pascal, Laboratoire de Math´ematiques, BP10448, F-63000 Clermont-Ferrand.

CNRS, UMR 6620, Laboratoire de Math´ematiques, F-63177 Aubi`ere

cLaboratoire P. Painlev´e, CNRS UMR 8524, Universit´e Lille 1, 59655 Villeneuve d’Ascq Cedex

dWeierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, 10117 Berlin

eANDRA/DS, 92298 Chˆatenay-Malabry Cedex

Abstract

In this paper, we design numerical methods for a PDE system arising in corrosion modelling. This system describes the evolution of a dense oxide layer. It is based on a drift-diffusion system and includes moving boundary equations. The choice of the numerical methods is justified by a stability analysis and by the study of their numerical performance.

Finally, numerical experiments with real-life data shows the efficiency of the developed methods.

Keywords: Corrosion modelling, drift-diffusion system, moving boundaries, adaptive time stepping.

2000 MSC:65M08, 65M12

1. Introduction

1.1. General framework of the study

The concept for long term storage of high-level radioactive waste in France under study is based on an underground repository. The waste shall be confined in a glass matrix and then placed into cylindrical steel canisters. These containers shall be placed into micro-tunnels in the highly impermeable Callovo-Oxfordian claystone layer at a depth of several hundred meters. At the request of the French nuclear waste management agency ANDRA, investigations are conducted to optimize and finalize this repository concept with the aim to ensure its long-term safety and its reversibility. In particular, the repository concept requires a minimum containment time of 1000 years.

The long-term safety assessment of the geological repository has to take into account the degradation of the carbon steel used for the waste overpacks and the cell disposal liners, which are in contact with the claystone formation. This degradation is mainly caused by generalized corrosion processes which form a passive layer on the metal surface consisting of a dense oxide inner layer and a porous hydroxide outer layer in contact with the groundwater in the pore space of the claystones. The processes take place under anaerobic conditions, since the groundwater is anoxic.

The neighboring geochemical environment (pH, concentrations) and groundwater flow trends induce changes of the corrosion conditions, which, in turn, influence the geochemical and thermo-hydro-mechanics of the claystones.

The temperature also affects the corrosion conditions. Indeed, the waste canisters are a source of heat. At the beginning of the repository development, the temperature is estimated to be about 90^◦C. Later, it is supposed to reach its steady- state value near 40^◦C.

As a tool to investigate the corrosion processes at the surface of the carbon steel canisters, the Diffusion Poisson Coupled Model (DPCM) for corrosion has been developed by Bataillonet al.[1].

IThis work has been supported by the Nuclear Waste Management Agency ANDRA

∗Corresponding author. Phone:+49 30 20372 560, Fax:+49 30 2044975

Email addresses:[email protected](C. Bataillon),[email protected](F. Bouchon), [email protected](C. Chainais-Hillairet),[email protected](J. Fuhrmann)

1Partially supported from the ANR VFSitCom and from the GDR MOMAS CNRS/PACEN

The model focuses on the development of the dense oxide layer in the region of contact between the claystones and the metal. In order to reduce complexity, in the present paper, the porous hydroxide layer is not taken into account. The system claystones – oxide layer – metal is described by a coupled system of electromigration – diffusion equations for the transport of the charge carriers in the oxide layer, and a Poisson equation for the electric potential. The interaction between the oxide layer and the adjacent structures are described in terms of Robin boundary conditions for the electrochemical reactions and the potential drops. The system includes moving boundary equations based on the Pilling-Bedworth ratio. The system evolution can be investigated in both potentiostatic and galvanostatic situations.

As the oxide layer is very thin compared to the waste overpack size, it is sufficient to consider the model in one space dimension.

The model allows to assess the evolution of the carbon steel corrosion rate, the chemical species release and the characteristic time of these processes. These data shall be used to estimate the lifetime of the carbon steel overpack and the pressure rise resulting from hydrogen release.

Carbon steel is expected to exhibit a corrosion rate between 1µm·year⁻¹and 10µm·year⁻¹. Due to the formation of the oxide layer, we expect a reduction of the corrosion rate.

1.2. Presentation of the corrosion model

We recall here the DPCM model introduced by Bataillon et al [1]. The domain under study is the oxide layer whose interfaces are moving. The moving domain is denoted by (X₀(t),X₁(t)). In the oxide layer, three charge carriers are taken into account: electrons, cations (Fe³⁺) and oxygen vacancies (VO¨). The densities of these charge carriers are respectively denoted byN,PandC. The corresponding current densities are denoted byJ_N,J_PandJ_C. These current densities contain a drift part and a diffusion part, so that the equations forN,PandCare linear convection-diffusion equations. They are coupled with a Poisson equation for the electrical potentialΨ.

Charge carriers are created and consumed at both interfaces: x=X0(t) is the outer interface (oxide/solution) and x=X1(t) is the inner interface (oxide/metal). The kinetics of the electrochemical reactions at interfaces leads to Robin boundary conditions onN,PandC.

The boundary conditions for the Poisson equation take into account that the metal and the solution can be charged because they are respectively electronic and ionic conductors. Such an accumulation of charges induces a field given by the Gauss law. These accumulations of charges depend on the voltage drop at the interface by the usual Helmholtz law which links the charge to the voltage drop through a capacitance. The parameters∆Ψ₁^pzcand∆Ψ^pzc₀ are the voltage drop corresponding to no accumulation of charges respectively in the metal and in the solution.

The DPCM model takes into account the growth of the oxide host lattice atx=X1(t) and its dissolution atx=X0(t) leading to moving boundary equations. In order to properly reflect the mass balance an the moving interfaces, in the boundary conditions, we need to take into account the interface velocities. This is a consequence e.g. of the pillbox lemma [2, 3]. The dimensionless model writes fort≥0:

•Equation and boundary conditions for the density of cationsP:

∂_tP+∂_xJP=0, JP =−∂_xP−3P∂_xΨ, x∈(X0(t),X1(t)), (1.1a) JP+PX⁰₀(t)=m⁰_P(P^m−P)e^−3b0PΨ−k⁰_PPe^3a0PΨ, x=X0(t), (1.1b) JP+PX₁⁰(t)=m¹_PPe^{−3b1P(V−Ψ)}−k¹_P(P^m−P)e^3a1P(V−Ψ), x=X1(t). (1.1c)

•Equation and boundary conditions for the density of electronsN:

D1

D2

∂tN+∂xJN =0, JN=−∂xN+N∂xΨ, x∈(X0(t),X1(t)), (1.2a) JN+NX⁰₀(t)=m⁰_Ne^b0NΨ−k⁰_NNe^−a0NΨ+n⁰_Ne^a0rΨ−p⁰_NNe^−b0rΨ, x=X0(t), (1.2b) J_N+NX₁⁰(t)=m¹_NN−k¹_NN_metallog(1+e^−(V−Ψ)), x=X1(t). (1.2c)

•Equation and boundary conditions for the density of oxygen vacanciesC:

D1

D3

∂tC+∂xJC=0, JC=−∂xC−2C∂xΨ, x∈(X0(t),X1(t)), (1.3a)

JC+CX₀⁰(t)=m⁰_C

1−C 4

e^−2b0CΨ−k_C⁰C

4e^2a0CΨ, x=X0(t), (1.3b) JC+CX₁⁰(t)=m¹_CC

4e^{−3b1C(V−Ψ)}−k¹_C

1−C 4

e^3aC1(V−Ψ) x=X1(t). (1.3c)

•Equation on the electrical potentialΨ:

−λ²∂²_xxΨ =3P−N+2C+ρhl, x∈(X0(t),X1(t)), (1.4a) Ψ−α0∂xΨ = ∆Ψ^pzc₀ , x=X0(t), (1.4b) Ψ +α1∂xΨ =V−∆Ψ₁^pzc, x=X1(t). (1.4c)

•Moving boundary equations

dX0

dt =v⁰_d(t)+dX1

dt (1− Ωox

mΩFe

), (1.5a)

dX1

dt =− D3

4D1

ΩFe

Ωox

JC(X1)+CX⁰₁(t), (1.5b)

withv⁰_d(t)=k⁰_de^{−5a0dΨ(X0(t))}. (1.5c) The system is supplemented with initial conditions:

N(x,0)=N⁰(x), P(x,0)=P⁰(x),C(x,0)=C⁰(x),x∈(0,1), (1.6a)

X0(0)=0, X1(0)=1. (1.6b)

We shortly explain the parameters of the model:

• D1,D2andD3are respectively the mobility or diffusion coefficients of cations, electrons and oxygen vacancies.

D1 andD3have the same order of magnitude, butD1 D2 due to the difference of size between cations and electrons and the resulting difference of mobilities.

• (mⁱ_P,kⁱ_P)i=0,1, (mⁱ_N,kⁱ_N)i=0,1, (mⁱ_C,kⁱ_C)i=0,1, (n⁰_N,p⁰_N), k⁰_d are interface kinetic functions. We assume that these functions are constant and strictly positive.

• (a⁰_u,b⁰_u) foru=P,N,C,r,dand (a¹_u,b¹_u) foru=P,Care positive transfer coefficients.

• P^mis the maximum occupancy for octahedral cations in the host lattice.

• N_metalis the electron density of state in the metal (Friedel model).

• ρ_hlis the net charge density of the ionic species in the host lattice. We assume thatρ_hlis homogeneous.

• Ωoxis the molar volume of the oxide.

• ΩFeis the molar volume of the metal.

• mis the number of moles of iron per mole of oxide (m=3 for magnetite).

• ∆Ψ₀^pzc,∆Ψ₁^pzcare respectively the outer and the inner voltages of zero charge.

• λ²,α0,α1are positive dimensionless parameters.

In the system (1.1)–(1.5),Vcan either be considered as an applied potential (“potentiostatic case”) or be given by another equation ensuring the electron charge balance at the inner interface (“galvanostatic case”):

−3

JP+PX₁⁰(t)+ D3

4D1

(JC+CX₁⁰(t)) +D2

D1

(JN+NX⁰₁(t))=J,˜ x=X1(t) (1.7) If ˜J=0, we speak of free corrosion, in this case,Vis called “free corrosion potential”.

Remark 1.1. The equations for the carrier densities with their boundary conditions(1.1),(1.2),(1.3)have the same form. They can all be written :

ε_u∂_tu+∂_xJ_u=0, J_u=−∂_xu−z_uu∂_xΨin(X₀(t),X₁(t)),∀t≥0, (1.8a)

−J_u−uX₀⁰(t)=r⁰_u(u(X0(t)),Ψ(X0(t)))on x=X0(t),∀t≥0, (1.8b) Ju+uX₁⁰(t)=r¹_u(u(X1(t)),Ψ(X1(t)),V)on x=X1(t),∀t≥0. (1.8c) For u = P,N,C, the charge numbers of the carriers are respectively zu = 3,−1,2 and we respectively haveεu = 1,D₁

D2

,D₁ D3

. We also note that both functions r_u⁰and r¹_uare linear and monotonically increasing with respect to their first argument. More precisely, the functions r⁰_uand r¹_uhave the following form:

r⁰_u(s,x)=β⁰_u(x)s−γ_u⁰(x), (1.9a)

r_u¹(s,x,V)=β¹_u(V−x)s−γ¹_u(V−x), (1.9b) whereβ⁰_u, β¹_u, γ⁰_u, γ¹_uare smooth positive functions.

1.3. Main results

The system of equations (1.1a), (1.2a), (1.3a), (1.4a) is close to the van Roosbroeck’s drift-diffusion system arising in semiconductor modelling (with three charge carriers instead of two). In the semiconductor framework, the drift- diffusion system is defined on a fixed domain and is usually supplemented with Dirichlet, Neumann or Robin boundary conditions. Furthermore, an additional coupling is given by reaction terms in the interior of the domain which describe the generation and recombination of the charge carriers. It has been shown that the van Roosbroek system exhibits global existence and uniqueness results under natural assumptions (see [4, 5, 6, 7, 8]). Existence results for the case of more than two species motivated by dopant diffusion in semiconductors can be found in [9].

In the present paper, no reactions take place in the interior of the domain, and the boundary conditions supplement- ing the drift-diffusion equations are given by Butler-Volmer-like laws at the interfaces. They lead to nonlinear Robin boundary conditions which induce an additional coupling in the system. Moreover, the system includes equations for the position of the moving boundary. Theoretical results (existence, uniqueness, long-time behavior) for (1.1)–(1.7) have not been proven yet. These questions can possibly be studied by combining the techniques developed in the semiconductor framework cited above and ideas by Aiki, B¨ohm and Muntean [10, 3, 11, 12] for somewhat simpler systems with one moving interface.

While these questions are worth to be studied, due to the significant interest in the ability to use this model in simulations within the framework described in the introduction, the aim of the present paper is rather the numerical approximation of the corrosion model. We want to propose an efficient numerical method in order to solve the full DPCM model (1.1)–(1.7).

In Section 2 we focus on the discretization of the system of convection-diffusion equations coupled with the Poisson equation (1.1)–(1.4). Therefore, we study a simplified two-species model on a fixed domain discretized by the finite volume method and study two different choices for the time discretization. The stability analysis of both schemes will show that a fully implicit time discretization must be chosen.

Then, in Section 3, we consider the full system in the potentiostatic case with the moving boundary equations. We introduce a change of variables in order to rewrite the system on a fixed domain and adapt the scheme designed in Section 2. We will study two different possibilities for the time discretization of the interface equations. We will also consider the galvanostatic case by taking into account equation (1.7).

In Section 4, we will give some details on the practical implementation of the numerical methods proposed in Section 3. Some improvements, like adaptive time stepping, will also be proposed. The performance of the designed numerical methods will be studied in Section 5. Finally, Section 6 is devoted to the presentation of some numerical experiments in a real-life context.

2. Study of a simplified model, choice for the time discretization

In this Section, we focus on a two-species model in order to fix the choice of the time discretization for our scheme.

We just consider the case with electrons and cationsFe³⁺. Without oxygen vacancies, there will be no evolution of the oxide layer: the domain will be considered as fixed. Furthermore, we consider here the potentiostatic case :Vis an applied potential.

2.1. Presentation of the simplified model and discretization

The simplified system is obtained from (1.1), (1.2), (1.4) by settingC =0 andX0(t) =0,X1(t)= 1 for allt. It consists of two drift-diffusion equations for the charge densities, coupled with a Poisson equation for the electrostatic potential. The boundary conditions are Robin boundary conditions.

For the sake of simplicity, we also assume that the boundary conditions onPandNhave exactly the same form.

It means that the functionsβ⁰_u,γ⁰_u,β¹_uandγ¹_udefined in (1.9) have the following form for bothu=Poru=N:

β⁰_u(x)=m⁰_ue^−zub0ux+k⁰_ue^zua0ux, γ⁰_u(x)=m⁰_uu^me^−zub0ux, (2.1a) β¹_u(x)=m¹_ue^−zub1ux+k¹_ue^zua1ux, γ¹_u(x)=k¹_uu^me^zua1ux, (2.1b) with the following hypotheses on the transfer coefficients:

a⁰_u,b⁰_u,a¹_u,b¹_u∈[0,1] foru=N,P, (2.2) and on the interface kinetic constants:

m⁰_u,k⁰_u,m¹_u,k¹_u>0. (2.3)

Moreover, we assume that

3P^m−N^m+ρ_hl=0. (2.4)

Remark 2.1. In the applications, we haveρ_hl =−5, P^m=2and N^m=1, so that hypothesis(2.4)is satisfied. Then it is expected that the solution to the corrosion model verifies0 ≤P≤P^mand0≤N ≤N^m. Therefore, it is crucial to design schemes that satisfy these stability properties.

In order to write a finite volume scheme for this system, we first introduce notations concerning the mesh and the time step. We consider a mesh for the domain [0,1], which is not necessarily uniform,i.ea family of given points (xi)0≤i≤I+1satisfying

x0=0<x1<x2< . . . <xI <xI+1=1.

Then, for 1≤i≤I−1, we definex_i+¹

2 = xi+xi+1

2 and we setx¹

2 =x0=0,x_I+¹

2 =xI+1=1. The cells of the mesh are the intervals (x_i−¹

2,x_i+¹

2) for 1≤i≤I. Let us set hi=x_i+¹

2

−x_i−¹

2, for 1≤i≤I, h_i+1

2 =xi+1−xi, for 0≤i≤I andh=max{hi, 1≤i≤I}is the size of the mesh. The time step is denoted by∆t.

The numerical scheme will be Euler implicit in time and finite volume in space. The choice of Euler implicit discretization in time is sensible for convection-diffusion equations in order to avoid a restrictive condition linking time step and mesh size. However, for the discretization of equations (1.1a), (1.2a), different choices can be made for the time approximation of the electrical field∂_xΨ. In the sequel, we propose two possibilities leading to two different schemes.

A decoupled scheme The scheme writes:

−λ²(dΨⁿ_i+1 2

−dΨⁿ_i−1 2

)=hi(3Pⁿ_i −N_iⁿ+ρhl), 1≤i≤I, (2.5a) ε_uhi

uⁿ_i⁺¹−uⁿ_i

∆t +Fⁿ⁺¹

u,i+¹2

− Fⁿ⁺¹

u,i−¹₂ =0, 1≤i≤I, foru=P,N, (2.5b) with the numerical fluxes

dΨⁿ_i+1

2 = Ψⁿ_i+1−Ψⁿ_i h_i+1

2

, 0≤i≤I, (2.6a)

Fⁿ⁺¹

u,i+¹₂ = B(zuh_i+¹

2dΨⁿ_i+1 2

)uⁿ_i⁺¹−B(−zuh_i+¹

2dΨⁿ_i+1 2

)uⁿ_i+⁺₁¹ h_i+1

2

, 0≤i≤I, foru=P,N, (2.6b) where the functionBis the Bernoulli function, leading to Scharfetter-Gummel fluxes:

B(x)= x

e^x−1 forx,0, B(0)=1. (2.7)

The scheme must be supplemented with the discretization of the boundary conditions and of the initial conditions.

For the discretization of the boundary conditions (1.4b), (1.4c), (1.8b), (1.8c), we write:

Ψⁿ0−α0dΨⁿ1

2 = ∆Ψ^pzc₀ , (2.8a)

Ψⁿ_I+1+α₁dΨⁿ_I+1 2

=V−∆Ψ^pzc₁ , (2.8b)

−Fⁿ⁺¹

u,¹₂ =r_u⁰(uⁿ₀⁺¹,Ψⁿ0)=β⁰_u(Ψⁿ0)uⁿ₀⁺¹−γ⁰_u(Ψⁿ0), foru=P,N, (2.8c) F_u,Iⁿ⁺¹

+¹₂ =r¹_u(uⁿ_I+⁺₁¹,ΨⁿI+1,V)=β¹_u(V−ΨⁿI+1)uⁿ_I+⁺₁¹−γ¹_u(V−ΨⁿI+1), foru=P,N (2.8d) and for the initial condition

u⁰_i = 1 h_i

Z x_i+1 2

xi−1 2

u⁰(x)dx, foru=P,N. (2.9)

The scheme (2.5)–(2.9) will be denoted by (Sdec) in all the sequel. It is decoupled in the following sense:

• Starting from (Pⁿ,Nⁿ)=(Pⁿ_i,N_iⁿ)0≤i≤I+1,Ψⁿ=(Ψⁿ_i)0≤i≤I+1is defined as the solution of the linear system (2.5a), (2.6a), (2.8a), (2.8b).

• Then, knowingΨⁿ, (Pⁿ⁺¹,Nⁿ⁺¹) is defined as the solution of the linear system (2.5b), (2.6b), (2.7), (2.8c), (2.8d).

In Section 2.2, we will prove the invertibility of the involved linear systems and study the stability of (Sdec).

Remark 2.2. If we want to replace the Scharfetter-Gummel fluxes by classical upwind fluxes, we just need to change the B function and choose:

B(x)=1+x⁻, where x⁻=max(−x,0).

In fact, the Bernoulli function could be replaced by any function satisfying the three following properties: B(0) =1, B(x) ≥0for all x ∈ Rand B(x)−B(−x) =−x for all x ∈ R(see for instance Ref. [13]). But, with the Bernoulli function, the scheme may be second order in space [14, 15].

A fully implicit scheme

The only difference in the fully implicit scheme is in the definition of the numerical fluxes and of the boundary conditions forPandN. Indeed, we replace (2.6b) by

Fⁿ⁺¹

u,i+¹2 = B(zuh_i+¹

2dΨⁿ_i+⁺¹1 2

)uⁿ_i⁺¹−B(−zuh_i+¹

2dΨⁿ_i+⁺¹1 2

)uⁿ_i+⁺₁¹ h_i+¹

2

,0≤i≤I, foru=P,N (2.10) and (2.8c)-(2.8d) by

−Fⁿ⁺¹

u,¹₂ =r⁰_u(uⁿ₀⁺¹,Ψⁿ₀⁺¹), foru=P,N, (2.11a) Fⁿ⁺¹

u,I+¹₂ =r¹_u(uⁿ_I+⁺₁¹,Ψⁿ_I+⁺₁¹,V), foru=P,N. (2.11b) The new scheme will be denoted in all the sequel by (Sf i).

At each time step, the vector of unknowns (Pⁿ⁺¹,Nⁿ⁺¹,Ψⁿ⁺¹)=(Pⁿ_i⁺¹,N_iⁿ⁺¹,Ψⁿ_i⁺¹)0≤i≤I+1is defined as the solution of a nonlinear system of equations. In Section 2.3, we will prove the existence of a solution to this nonlinear system and investigate the stability of (S_{f i}).

2.2. Stability analysis of a decoupled scheme

Proposition 2.3. Under the hypotheses(2.1)–(2.3), the scheme(S_dec)admits a unique solution(Pⁿ,Nⁿ,Ψⁿ)n≥0. More- over, if P⁰,N⁰ ∈ L^∞(0,1)satisfy P⁰,N⁰ ≥0almost everywhere on(0,1), then Pⁿ_i,N_iⁿ ≥0for all0 ≤i ≤I+1and n≥0.

Proof. Let us first note that (P⁰,N⁰) is well defined by (2.9) and satisfies the nonnegativity condition. Assume now that for anyn≥0, (Pⁿ,Nⁿ) is given and satisfies the nonnegativity assumption. ThenΨⁿis defined as the solution of the linear system (2.5a), (2.6a), (2.8a), (2.8b). The matrix of this linear system is a tridiagonal matrix. Asα₀andα₁ are strictly positive, this matrix is semistrictly diagonally dominant and thus invertible.

KnowingΨⁿ,Pⁿ⁺¹andNⁿ⁺¹ are also defined as solutions of some linear systems. Foru =P,Nthe linear system defined by (2.5b), (2.6b), (2.8c), (2.8d) rewrites:

Mⁿuuⁿ⁺¹=Sⁿ_u. (2.12)

The matrixMⁿu∈ M_I+2(R) is a tridiagonal matrix, whose nonzero terms are defined by:

(Mⁿu)_ii=ε_uh_i

∆t+B(zuh_i+1 2dΨⁿ_i+1

2

) h_i+¹

2

+B(−zuh_i−1 2dΨⁿ_i−1

2

) h_i−¹

2

, ∀1≤i≤I,

(Mⁿu)ii+1 =−

B(−zuh_i+1 2dΨⁿ_i+1

2

) h_i+1

2

, ∀1≤i≤I,

(Mⁿu)ii−1=− B(z_uh_i−¹

2dΨⁿ_i−1 2

) h_i−1

2

, ∀1≤i≤I,

(Mⁿu)00 = B(zuh¹

2dΨⁿ1 2

) h1

2

+β⁰_u(Ψⁿ0),

(Mⁿu)₀₁ =−

B(−zuh1 2dΨⁿ1

2

) h¹

2

,

(Mⁿu)I+1I+1= B(−zuh_I+1 2dΨⁿ_I+1

2

) h_I+1

2

+β¹_u(V−Ψⁿ_I+1),

(Mⁿu)I+1I =−

B(z_uh_I+1 2dΨⁿ_I+1

2

) h_I+1

2

.

Asε_u>0 and the functionsβ⁰_u,β¹_uandBare nonnegative, the matrixMⁿuhas positive diagonal terms and nonpos- itive offdiagonal terms. Moreover, it is strictly diagonally dominant with respect to its columns. Therefore,Mⁿuis an M-matrix: it is invertible and (Mⁿu)⁻¹≥0. It yields the existence of a unique solution to (2.12).

The right hand side of the linear system (2.12),Sⁿ_u, is defined by:

(Sⁿ_u)i=εu

hi

∆tuⁿ_i,∀1≤i≤I, (Sⁿ_u)₀ =γ⁰_u(Ψⁿ₀), (Sⁿ_u)_I+1 =γ¹_u(V−Ψⁿ_I+1).

Asuⁿ_i ≥0 for 1≤i≤Iandγ⁰_u,γ¹_uare nonnegative functions,Sⁿ_uis a positive-valued vector and as (Mⁿu)⁻¹≥0, we get thatuⁿ_i⁺¹≥0 for all 0≤i≤I+1. It concludes the proof of Proposition 2.3 by induction.

Proposition 2.4. Assume(2.1)–(2.4)and that P⁰,N⁰ ∈ L^∞(0,1) satisfy0 ≤ P⁰ ≤ P^mand 0 ≤ N⁰ ≤ N^malmost everywhere on(0,1). Then, if∆Ψ₀^pzcand∆Ψ₁^pzcfulfill:

− 1

3a⁰_P(1+log(α0a⁰_Pk⁰_P))≤∆Ψ₀^pzc≤ 1

a⁰_N(1+log(α0a⁰_Nk⁰_N)), (2.13a)

−1

b¹_N(1+log(α1b¹_Nm¹_N))≤∆Ψ₁^pzc≤ 1

3b¹_P(1+log(α1b¹_Pm¹_P)), (2.13b) and if the following inequality on the time step holds:

∆t≤min( λ² 9P^m,D1

D2

λ²

N^m), (2.14)

then the solution to the scheme(S_dec)satisfies

0≤Pⁿ_i ≤P^mand0≤N_iⁿ≤N^m ∀0≤i≤I+1,∀n≥0. (2.15) Proof. First, we note that the hypotheses on the initial conditions ensure that (2.15) is satisfied forn =0. We now assume that (2.15) is verified forn ≥0 and prove that it also holds forn+1. The nonnegativity has already been proven.

Let us first compute the productMⁿuu^m, whereu^mis the constant vector ofR^I+2, whose components are all equal to the same valueu^m. As the Bernoulli function defined by (2.7) satisfiesB(x)−B(−x)=−xfor allx∈R, we have

(Mⁿuu^m)_i=u^m ε_uh_i

∆t −zu(dΨⁿ_i+1 2

−dΨⁿ_i−1 2

)

!

,∀1≤i≤I, (Mⁿuu^m)0=u^m

−zudΨⁿ1

2+β⁰_u(Ψⁿ0) , (Mⁿuu^m)I+1=u^m

zudΨⁿ_I+1

2 +β¹_u(V−ΨⁿI+1) .

Using (2.5a), (2.8a), (2.8b), it rewrites

(Mⁿuu^m)_i=u^m ε_uh_i

∆t+z_uh_i

λ² (3Pⁿ_i −N_iⁿ+ρ_hl)

!

, ∀1≤i≤I, (Mⁿuu^m)₀ =u^m

−zu

α₀(Ψⁿ₀−∆Ψ₀^pzc)+β⁰_u(Ψⁿ₀) , (Mⁿuu^m)I+1 =u^mzu

α1

(V−∆Ψ₁^pzc−Ψⁿ_I+1)+β¹_u(V−Ψⁿ_I+1) .

Using (2.4), it yields

(Mⁿu(uⁿ⁺¹−u^m))_i=ε_uh_i

∆t(uⁿ_i −u^m)−z_uh_i

λ² u^m(3(Pⁿ_i −P^m)−(N_iⁿ−N^m))

∀1≤i≤I, (Mⁿu(uⁿ⁺¹−u^m))0=γ⁰_u(Ψⁿ0)−u^m

−zu

α0

(Ψⁿ0−∆Ψ₀^pzc)+β⁰_u(Ψⁿ0) , (Mⁿu(uⁿ⁺¹−u^m))_I+1=γ¹_u(V−Ψⁿ_I+1)−

u^mz_u α1

(V−∆Ψ₁^pzc−Ψⁿ_I+1)+β¹_u(V−Ψⁿ_I+1) .

We now apply these results tou =PandN, with respectivelyu^m =P^m,u^m=P^m, andu^m=N^m,u^m=N^m. For the inner terms, we get∀1≤i≤I:

(MⁿP(Pⁿ⁺¹−P^m))i= hi

∆t Pⁿ_i −P^m 1−9∆t λ² P^m

!

+3P^mhi

λ² (N_iⁿ−N^m), (MⁿN(Nⁿ⁺¹−N^m))_i= h_i

∆t(N_iⁿ−N^m) D1

D2

−∆t λ²N^m

!

+3N^mh_i

λ² (Pⁿ_i −P^m).

Therefore, if∆tverifies (2.14), we have (MⁿP(Pⁿ⁺¹−P^m))_i≤0 and (MⁿN(Nⁿ⁺¹−N^m))_i≤0 for all 1≤i≤I. It remains to prove that (Mⁿu(uⁿ⁺¹−u^m))₀ ≤ 0, (Mⁿu(uⁿ⁺¹−u^m))_I+1 ≤ 0 foru = P,N. But, (Mⁿu(uⁿ⁺¹ −u^m))₀ = ξ_u⁰(Ψⁿ₀) and (Mⁿu(uⁿ⁺¹−u^m))_I+1=ξ¹_u(V−Ψⁿ_I+1), with

ξ⁰_u(x)=γ⁰_u(x)−u^mβ⁰_u(x)+u^mz_u α0

x−u^mz_u α0

∆Ψ₀^pzc, ξ¹_u(x)=γ¹_u(x)−u^mβ¹_u(x)−u^mzu

α1

x+u^mzu

α1∆Ψ₁^pzc. Using the definitions (2.1), we get

ξ_u⁰(x)=−u^mk⁰_ue^zua0ux+u^mz_u

α₀x−u^mz_u α₀∆Ψ^pzc₀ , ξ_u¹(x)=−u^mm¹_ue^−zub1ux−u^mzu

α1

x+u^mzu

α1∆Ψ₁^pzc.

The study of the variations ofξ⁰_uandξ¹_uforu=N,Pshows that these functions are always increasing on an interval (−∞,x] and then decreasing on [ ¯¯ x,+∞). Therefore they reach a maximum value and the hypotheses (2.13) ensure that the maximum value is always nonpositive. Thenξ⁰_P,ξ¹_P,ξ⁰_N,ξ¹_Nare nonpositive functions onR.

Finally, we get that (Mⁿ_P(Pⁿ⁺¹−P^m))i ≤0 and (Mⁿ_N(Nⁿ⁺¹−N^m))i≤0 for all 0≤i≤I+1. AsMⁿ_PandMⁿ_N are M-matrices, it concludes the proof of Proposition 2.4.

In the applications, we haveλ²≈10⁻⁶and ^D_D¹

2 ≈10⁻¹⁴, so that the hypothesis on∆t, (2.14), imposes∆t≈10⁻²⁰. It makes the decoupled scheme unusable in practice and motivates the study of the fully implicit scheme.

2.3. Stability analysis of a fully implicit scheme

The scheme (Sf i) is defined as a set of nonlinear equations at each time step. We need to prove the existence of a solution to the nonlinear system of equations and to study the stability of the scheme.

Proposition 2.5. Assume(2.1)–(2.4)and that P⁰,N⁰∈L^∞(0,1)satisfy0≤P⁰≤P^mand0≤N⁰≤N^malmost every- where on(0,1). Then, there exists a solution to the fully implicit scheme(S_{f i}):((Pⁿ,Nⁿ)n≥0,(Ψⁿ)n≥1). Furthermore, if

∆Ψ₀^pzcand∆Ψ₁^pzcverify(2.13), the solution to the scheme(S_{f i})satisfies the stability property(2.15).

Proof. In order to prove Proposition 2.5, we adapt some ideas developed by A. Prohl and M. Schmuck in [16].

First, we note that (P⁰,N⁰) is well defined by (2.9) and satisfies (2.15). Assume now that for anyn≥0, (Pⁿ,Nⁿ,Ψⁿ) is given and satisfies (2.15). Then, we will prove that there exists a solution (Pⁿ⁺¹,Nⁿ⁺¹,Ψⁿ⁺¹) to (S_{f i}) and that

0≤Pⁿ_i⁺¹ ≤P^mand 0≤N_iⁿ⁺¹ ≤N^m ∀0≤i≤I+1.

Therefore, let us consider the mapping

T_n^µ: R^I+2× R^I+2 → R^I+2× R^I+2 (P, N) 7→ (eP, eN) defined, forµ >0, by

−λ²(dΨi+¹₂−dΨi−¹₂)=h_i(3P_i−N_i+ρ_hl), 1≤i≤I, Ψ0−α₀dΨ¹

2 = ∆Ψ₀^pzc, ΨI+1+α₁dΨI+¹₂ =V−∆Ψ₁^pzc, εu

hi

∆t (1+ µ

εuλ²)eui− µ

εuλ²ui−uⁿ_i

! +F_u,i+1

2

− F_u,i−1 2 =0, 1≤i≤I, foru=P,N,

F_u,i+1

2 = B(zuh_i+¹

2dΨi+¹2)eui−B(−zuh_i+¹

2dΨi+¹2)eui+1

h_i+1 2

, 0≤i≤I, foru=P,N,

−F_u,1

2 =r⁰_u(eu0,Ψ0)=β⁰_u(Ψ0)euⁿ₀⁺¹−γ⁰_u(Ψ0), foru=P,N, F_u,I+1

2 =r¹_u(eu_I+1,ΨI+1,V)=β¹_u(V−ΨI+1)eu_I+1−γ¹_u(V−ΨI+1),foru=P,N.

This mappingT_n^µis defined by two successive steps:

• the first step defines an intermediate vectorΨas the solution of a linear system of equations,

• the second step definesePandeNas solutions of linear systems of equations, very close to the systems involved in the decoupled scheme (there is an additional term on the diagonal of the matrices and the right-hand-sides are slightly modified).

Therefore, it is a continuous mapping fromR^I+2×R^I+2to itself. Applying the proof of stability of Proposition 2.4, we obtain thatT_n^µpreserves the set:

K ={(P,N)∈R^I+2×R^I+2; 0≤Pi≤P^m,0≤Ni≤N^m,∀0≤i≤I+1}

as long as (2.13) is satisfied and∆tverifies :

∆t≤µmin 1 9P^m, 1

N^m

! .

Then, thanks to Brouwer’s theorem, we conclude thatT_n^µhas a fixed point inK. Moreover, this fixed point, with the correspondingΨ, is a solution to the scheme (S_{f i}). It shows the existence of a solution to the scheme and the stability properties (2.15). As this result holds for anyµ, the scheme is unconditionnally stable.

Remark 2.6. With the Brouwer fixed point theorem, we get the existence of one solution to the scheme(Sf i)but we do not get any information on uniqueness. This question of uniqueness is relevant but still open. Therefore, in the sequel, the solution to the numerical scheme will be the one selected by Newton’s method used to solve the nonlinear system of equations(Sf i). During the numerical experiments we did not encounter situations hinting at multiple solutions.

3. Numerical schemes for the complete corrosion model 3.1. Change of variables

The corrosion model involves moving boundaries through (1.5). The system of equations is defined on the space- time domainS

0≤t≤T[X₀(t),X₁(t)]× {t}. The size of the domain at timetisL(t)=X₁(t)−X₀(t). An Eulerian description is chosen in order to handle the motion of the boundaries; the physical spatial domain is substituted by a computational one. Therefore, we introduce the following change of variables:

(x,t)7→(ξ(x,t)= x−X0(t) X1(t)−X0(t),t) [

0≤t≤T

[X0(t),X1(t)]× {t} →[0,1]×[0,T] and to every functionu(respectivelyΨ) defined onS

0≤t≤T[X0(t),X1(t)]× {t}we associate a function ¯u(respectively Ψ¯) defined on [0,1]×[0,T] such thatu(x,t)=u(ξ(x,¯ t),t) (respectivelyΨ(x,t)=Ψ¯(ξ(x,t),t)). Then, we have:

∂tu=∂tu¯− 1

L(t)(X₀⁰(t)+ξL⁰(t))∂ξu¯ Ju= 1

L(t)(−∂ξu¯−zuu∂¯ ξΨ¯)= 1 L(t)Ju¯. But, using the fact thatξ∂_ξu¯=∂_ξ(ξ¯u)−u, we get¯

∂_tu= 1 L(t)

∂_t L(t) ¯u−∂_ξ (X₀⁰(t)+ξL⁰(t)) ¯u .

Finally, forgetting the bars, the convection-diffusion equation with boundary conditions (1.8) rewrites in the new variables:

εuL(t)∂t(L(t)u)+∂ξJu−εu∂ξ

L(t)(X₀⁰(t)+ξL⁰(t))u

=0,

Ju=−∂ξu−zuu∂ξΨ, ξ∈(0,1) (3.1a)

−J_u−ε_uuL(t)X₀⁰(t)=L(t)r⁰_u(u,Ψ), ξ=0 (3.1b) Ju+εuuL(t)X₁⁰(t)=L(t)r¹_u(u,Ψ,V), ξ=1. (3.1c) We note that in the new variables, the convection-diffusion equation (3.1a) contains an additional convection term coming from the displacement of the interfaces. Therefore, (3.1) is equivalent to the following formulation:

ε_uL(t)∂_t(L(t)u)+∂_ξJˆu=0,

Jˆu=−∂ξu− zu∂ξΨ +εuL(t)(X₀⁰(t)+ξL⁰(t))u, ξ∈(0,1), (3.2a)

−Jˆu=L(t)r⁰_u(u,Ψ), ξ=0, (3.2b)

Jˆu=L(t)r¹_u(u,Ψ,V), ξ=1. (3.2c)

The equation on the electrostatic potential (1.4) rewrites in the new variables:

− λ²

L(t)²∂²_ξξΨ =3P−N+2C+ρhl, ξ∈(0,1), (3.3a) Ψ− α0

L(t)∂ξΨ = ∆Ψ₀^pzc, ξ=0, (3.3b) Ψ + α₁

L(t)∂_ξΨ =V−∆Ψ₁^pzc, x=1. (3.3c)