Convergence and long time behavior of a Finite Volume scheme for an isotropic seawater intrusion model with a sharp-diffuse interface in a confined aquifer.

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Convergence and long time behavior of a Finite Volume scheme for an isotropic seawater intrusion model with a

sharp-diffuse interface in a confined aquifer.

Ahmed Ait Hammou Oulhaj, David Maltese, Naoufel Staïli

To cite this version:

Ahmed Ait Hammou Oulhaj, David Maltese, Naoufel Staïli. Convergence and long time behavior of

a Finite Volume scheme for an isotropic seawater intrusion model with a sharp-diffuse interface in a

confined aquifer.. 2020. �hal-02865698�

VOLUME SCHEME FOR AN ISOTROPIC SEAWATER INTRUSION MODEL WITH A SHARP-DIFFUSE INTERFACE IN

A CONFINED AQUIFER

AHMED AIT HAMMOU OULHAJ, DAVID MALTESE AND NAOUFEL STAÏLI

Abstract. We study and simulate a sharp-diffuse interface model in the con- text of seawater intrusion in an isotropic confined aquifer. It is a strongly coupled system of partial differential equations which include elliptic and par- abolic equations modelling the flow of fresh and saltwater. We study a finite volume scheme. This scheme ensures a discrete maximum principle for the discrete solution without any restriction on the transmissibility coefficients.

Moreover it also provides a control on the entropy. The existence of a discrete solution and the convergence of this scheme are obtained, based on nonlinear stability results. The temporal decay rates for convergence of the discrete so- lution to the homogeneous steady state is shown using a discrete logarithmic Sobolev inequality. Finally, we propose a numerical simulations to illustrate the behavior of the model and of the scheme.

Keywords. Isotropic porous media, cross-diffusion, seawater intrusion, confined aquifer, nonlinear discretization, control of the entropy, numerical analysis

AMS subjects classification. 65M12, 65M08, 76S05

1. Introduction

1.1. Presentation of the continuous problem. In this paper, we are interested in a strongly coupled systems of partial differential equations which include elliptic and parabolic equations modelling the so-called seawater intrusion (we refer to the textbooks [12, 13,14] for general information about seawater intrusion problems).

The model presented here is derived in [25]. The basis of the modeling is the mass conservation law for each species (freshwater and saltwater) combined with the classical Darcy law for porous media and a sharp interface approach. This approach is based on the assumption that the two fluids are immiscible. More precisely that each fluid is confined to a well defined portion of the flow domain with a smooth interface separating them called sharp interface. No mass transfer occurs between the fresh and the salt area and capillary pressure’s type effects are neglected. This approximation is often reasonable. Of course, this type of model does not describe the behavior of the real transition zone but gives information concerning the movement of the saltwater front. This abrupt interface approach is combined in [25] with a phase field approach for re-including the existence of a diffuse interface between fresh and salt water where mass exchanges occur. This approach combines the advantage of respecting the physics of the problem and the computational efficiency (see [25] for numerical simulations). Moreover from a theoretical point of view this approach has also advantages resulting from the

1

addition of diffuse areas (see [24] and [26]). The authors consider the Dupuit approximation considering that the hydraulic head is constant along each vertical direction. Using a vertical integration this yields a 2D reduced model obtained from a full 3D model where the unknowns are the heights of the fluid layers. We refer to [12, 13, 14, 69, 68, 8] for more details about this approach. In the literature, there exists other modeling approaches (see [22, 65, 66, 23, 47]). Note also that this approach is very popular for instance in the community working on carbon dioxide sequestration [45,44,63,56,61,67]. The model proposed in [25] describes the evolution of the freshwater and saltwater in an isotropic porous media. We distinguish two important case. The case of unconfined aquifer and the one of confined aquifer. In these two cases, the aquifer is bounded by two layers, the lower layer is always supposed to be impermeable. For the confined aquifer, the upper surface of the aquifer is impermeable and for the unconfined, the upper surface is a permeable. In this article we are interested to confined case and the aquifer is then represented by

A= Ω×(0, H),

where Ω is a bounded domain ofR²andHis the thickness of the aquifer (see Figure 1). We denote by hthe thickness of the saltwater which satisfies 0≤h(t,x)≤H and we denote byf the freshwater hydraulic head. With the assumption that the viscosity the same for the saltwater and freshwater, the evolution of the thickness of the saltwater and the freshwater hydraulic head are given by a nonlinear strongly coupled system of partial differential equations which include elliptic and parabolic equations of the form

(1)

(φ∂th− ∇. αKh∇f

− ∇ ·(αKh∇h)−δ∇ ·(φ∇h) = 0 in (0, T)×Ω, H∇ ·(αK∇f) +∇. αKh∇h

= 0 in (0, T)×Ω, where Ω⊂R² is an open bounded polygonal connected subset. The parameterα is given by

(2) α=ρs−ρf

ρf

∈R^?+,

where ρf (resp. ρs) is the specific weight of the freshwater (resp. saltwater) (as- sumed to be constant with 0< ρf < ρs). This parameter characterizes the densities contrast. The matrix K is the hydraulic conductivity. It expresses the ability of the ground to conduct water. For the numerical analysis we need to assume the medium isotropic and uniformly elliptic meaning thatK=κI2 whereκ: Ω→Ris a given function such that there exists (κ, κ)∈R^∗+×R^∗+ satisfying

κ≤κ(x)≤κ, for a.e. x∈Ω.

As previously mentioned the approach developed in [25] includes an existence of miscible zone, taking the form of diffuse interface of characteristic thicknessδ be- tween freshwater and saltwater. We obtain the term−δ∇ ·(φ∇h) in the equation governing the evolution of the saltwater thickness. At the continuous level this term allows to get more regularity on the solution but also to ensure a maximum principle naturally satisfied by the solution (see [26] for a comparison between sharp and sharp-diffuse interfaces). This term will be also useful at the discrete level to obtain a discrete maximum principle, stability properties and long time behavior of the numerical scheme. In the following we takeαK(x) =I2for anyx∈Ω. From

a mathematical point of view, these assumptions do not change the complexity of the analysis but rather avoid complicated computations (we refer to [59], [36] for a suitable discretization of the tensor permeability). We assume that the porosity is constant in the aquifer and we choose φ = 1 in Ω. Indeed, in the field envisaged here, the effects due to variations in φ are negligible compared with those due to density contrasts. With these assumptions the system (1) can be written as follows (3)

(∂th− ∇. h∇f

− ∇ ·(h∇h)−δ∆h = 0 in (0, T)×Ω, H∆f +∇. h∇h

= 0 in (0, T)×Ω.

h H

z=0 z=H

saltwater

freshwater

p

Figure 1. Seawater intrusion in a confined aquifer

The thickness of the saltwater satisfies

(4) 0≤h(t,x)≤H, for a.e (t,x)∈(0, T)×Ω.

Without loss of generality we can assume that (5)

Z

Ω

f(t,x) dx= 0, for a.et∈(0, T).

The system (3) is supplemented by no-flux boundary conditions on (0, T)×∂Ω (6) (h∇f+h∇h+δ∇h)·n= 0, (H∇f +h∇h)·n= 0,

wherenis the unit normal to the boundary∂Ω. Initial data are

(7) h_|t=0=h0,

whereh0 is a measurable function such that

(8) 0≤h₀(x)≤H, for a.ex∈Ω.

Let us mention that the system (1), have been the object of several studies. The authors in [24, 26] studied the weak solutions of the system (1) under different

assumptions. Moreover, the uniqueness of weak solutions is established under dif- ferent assumptions in [27].

1.2. Entropy of the system. We recall ([32, 52, 53]) the definition of entropy functional :

(9) E(h) =

Z

Ω

Γ(h) dx,

where Γ(s) = slogs−s+ 1. We multiply (formally) the first equation of (3) by loghand integrating over Ω gives

d dtE(f) +

Z

Ω

∇h· ∇fdx+ Z

Ω

|∇h|²dx+ 4δ Z

Ω

|∇√

h|²dx= 0.

We multiply (formally) the second equation of (3) byf gives Hk∇fk²_L2(Ω)+

Z

Ω

h∇h· ∇fdx= 0.

Using the Hölder’s inequality and (4) one has

(10) k∇fkL²(Ω)≤ k∇hkL²(Ω).

Consequently using the previous identities we obtain the entropy/dissipation prop- erty :

(11) d

dtE(h) + 4δ Z

Ω

|∇√

h|²dx≤0.

One has by virtue of (4) 1

4Hk∇hk²_L2(Ω)≤ k∇√

hk²_L2(Ω). This last identity with (4) gives the following estimation

(12) d

dtE(h) + δ H

Z

Ω

|∇h|²dx≤0.

and in particular (13)

Z T 0

Z

Ω

|∇h|²dxdt≤ H δE(h0).

1.3. Long-time behavior. Let us now focus on the long time behavior of the model. The long time behavior of the system is based on the method of the relative entropy. This method has been intensively developed for the study of the long time behavior of different systems of partial differential equations (see [1] to the survey paper for the entropy method for nonlinear diffusion equations). Let us also mention the founding papers [20] and [46]. Similar techniques were already used in [39] and [40] for the linear and nonlinear drift-diffusion systems of equations arising in semiconductor devices modeling. The entropy method has been further successfully applied to the study of the long-time behavior of reaction-diffusion equations [31] and [43] or cross-diffusion systems of equations [55] and [48] for instance. We also refer to the recent book [49] and the references therein. Let us denote byh^eq: Ω→Rthe constant function defined by

h^eq(x) = 1 mΩ

Z

Ω

h0(x) dxfor anyx∈Ω.

We introduce the relative entropy of the system

E(t) =E(h(t))−E(h^eq) for anyt≥0.

Note that by virtue of the boundary condition (6) we have (14)

Z

Ω

h(t,x) dx= Z

Ω

h0(x) dxfor anyt≥0, and the relative entropy can be written as follows

E(t) = Z

Ω

h(t,x) logh(t,x) h^eq

dx.

This last identity combined with the Sobolev logarithmic inequality E(t)≤LΩ

Z

Ω

|∇√ h|²dx, gives the following differential inequality

d

dtE(t) + 4δ

L_ΩE(t)≤0.

Using Gronwall’s inequality we obtain the following exponential decay to the steady state in term of the relative entropy

(15) E(t)≤E(h0)e^−L4δΩtfor any t≥0.

Using the fact that for any 0≤t≤1 we have

(16) tlog(t)≥t−1 +1

2(t−1)² and

(17) kh(t,·)−h^eqkL¹(Ω)= 2 Z

{h(t,·)≤h^eq}

|h(t,·)−h^eq|dx.

We obtain (which is a particular case of the Csiszár–Kullback inequality [28, 51]) theL¹ decay ofh−h^eqthat is

(18) kh(t,·)−h^eqk_L1(Ω)≤2q

2kh₀k_L1(Ω)E(h₀)e

− 4δ

2L_Ω^tfor anyt≥0.

This motivates the following notion of weak solution to (3)-(6)

Definition 1.1. Let Ω be a bounded domain of R². Let h₀ ∈ L^∞(Ω) such that 0 ≤h₀ ≤H a.e in Ω. We shall say that the couple of functions (h, f) is a weak solution to (3)-(6) if h∈ L²(0, T;H¹(Ω)) with 0 ≤h ≤H a.e in (0, T)×Ω and f ∈L²(0, T;H¹(Ω)/R) and

(19) Z T

0

Z

Ω

(h∂tψ−h∇f· ∇ψ−h∇h· ∇ψ−δ∇h· ∇ψ) dxdt+ Z

Ω

h0ψ(0,·) dx= 0,

(20)

Z T 0

Z

Ω

(H∇f · ∇ψ+h∇h· ∇ψ) dxdt= 0, for all test functions ψ∈C₀^∞([0, T)×Ω).

There are several finite-volume schemes for other cross-diffusion systems in the mathematical literature. In [37], a finite volume scheme, is analyzed for the Patlak- Keller-Segel (PKS) chemotaxis model. In [17], the author proposes and analyzes a finite-volume scheme for a PKS system with additional cross diffusion. The analysis of a finite volume method for a cross diffusion model in population dynamics is pre- sented in [10]. In [59,36] the author propose and analyze a finite volume scheme for two-phase immiscible flow in porous media, used in petroleum engineering. For nu- merical schemes in the context of seawater intrusion and in particular for this model, we refer to [60,57,3,2]. More precisely in [3] a finite element method and a finite volume method on rectangular meshes are proposed for the model (3). Let us cite [4,5, 7] where upstream mobility finite volume and control volume finite elements schemes for a seawater intrusion cross-diffusion model are studied. For a degenerate cross-diffusion model describing the ion transport through biological membranes, a finite volume scheme is analyzed in [18]. An implicit Euler finite-volume scheme with mobilities given by arithmetic means for a cross-diffusion system modeling biofilm growth is analyzed in [29]. In [19] the authors study a two-point flux ap- proximation finite volume scheme for a cross-diffusion system with mobilities given by logarithmic mean. In [6], the authors studied the large time behavior for a degenerate parabolic system modeling the flow of fresh and saltwater in a porous medium in the context of seawater intrusion.

In this work, we propose a finite volume scheme for the problem (3). This scheme is based on a two-point flux approximation with upwind mobilities. It is designed in order to preserve at the discrete level the main features of the continuous problem that is the control of the thickness of the saltwater (4), the control of the entropy and its dissipation (11) and the long-time behaviour of the model (18). Based on this control and on compactness arguments, we show that this scheme converges towards a weak solution to the problem. Finally, we show that the discrete solution converges for large times to the homogeneous steady state.

2. The numerical scheme 2.1. Space-time discretizations of (0, T)×Ω.

2.1.1. Discretizations of Ω. We describe two different space discretizations of Ω : aprimal mesh and adual mesh.

Let Ω be an open bounded polygonal connected subset of R². An admissible finite volume mesh of Ω, denoted byT, is given by a family of “control volumes”, which are open polygonal convex subsets of Ω, a family of subsets of Ω contained in hyperplanes of R², denoted by E (these are the edges of the control volumes), with strictly positive 1-dimensional Lebesgue measure, and a family of points of Ω denoted byP satisfying the following properties (in fact, we shall denote, somewhat incorrectly, byT the family of control volumes):

(1) The closure of the union of all the control volumes is Ω;

(2) For any K ∈ T, there exists a subset E_K of E such that ∂K =K\K =

∪_σ∈EKσ. Furthermore,E=∪_K∈EKEK.

(3) For any (K, L)∈ T² withK6=L, either the 1-dimensional Lebesgue mea- sure ofK∩Lis 0 orK∩L=σfor someσ∈ E, which will then be denoted byK|L.

(4) The family P = (xK)K∈T is such thatxK ∈ K (for all K ∈ T) and, if σ=K|L, the straight line DK,L going through xK and xL is orthogonal toK|L.

In the sequel, the following notations are used. For any K ∈ T and σ ∈ E, mK

is the 2-dimensional Lebesgue measure of K andmσ the 1-dimensional Lebesgue measure measure of σ. For K ∈ T, we denote by diam(K) the diameter of the control volumeK. For σ=K|L we denote bydσ represents the distance between xK and xL. Then, we define the mesh diameter diam(T) and the mesh regularity reg(T) by

diam(T) = max

K∈Tdiam(K), reg(T) = max

K∈T

X

σ∈E(K), σ=K|L

m_σd_σ mK

+diam(K) dσ

.

Once theprimal mesh has been built, we can define a dual meshD as follows.

For allK∈ T andσ∈ E(K) we denote byDσ,K ⊂Kthe open triangle with vertex xK and basisσthat is

Dσ,K ={txK+ (1−t)y, t∈(0,1),y∈σ}.

Finally forσ=K|Lwe denoteD_σ=D_σ,K∪D_σ,Land forσ∈ E_int∩E(K) we denote D_σ =D_σ,K. Note that Ω = S

σ∈ED_σ,K. We refer to Figure 2 for an illustration of the primary and dual meshes. The 2-dimensional Lebesgue measure of D_σ is denoted bymD_σ.

Let us now introduce two discrete functional spaces. The spaceX_T(Ω)⊂L²(Ω) is made of piecewise constant functions on the primal mesh, i.e.,

X_T(Ω) ={u: Ω→Rmeasurable |u_|K is constant, ∀K∈ T }.

The spaceX_T(Ω)/R is made of piecewise constant functions on the primal mesh with zero mean value, i.e.,

X_T(Ω)/R={u∈X_T(Ω) | Z

Ω

u(x) dx= 0}.

Given u∈ X_T(Ω), we denote by u_K the value of the function uover the control volumeK∈ T. Foru∈X_T(Ω), define the discreteH¹(Ω) seminorm by

(21) kuk²_1,T = X

σ∈Eint

σ=K|L

τ_σ(u_K−u_L)²,

where for any σ =K|L the transmibility coefficient τ_σ is given by τ_σ = ^m_d^σ

σ and where dσ represents the distance between xK and xL. We recall the discrete Poincare inequality for the function X_T(Ω) with a zero mean value (see [35] for a proof).

Proposition 2.1. LetΩbe a be an open bounded polygonal connected subset ofR². Let T be an admissible mesh ofΩ. Then there existsC1 only depending onΩsuch that for anyu∈X_T(Ω)/R one has

(22) kuk_L2(Ω)≤C1kuk_1,T.

The spaceX_D(Ω)⊂L²(Ω) is made of piecewise constant functions on the dual mesh, i.e.,

X_D(Ω) ={u: Ω→Rmeasurable |u_|

Dσ is constant, ∀σ∈ E}.

Given u∈XT(Ω) we define the projection of uover the dual mesh denoteduD ∈ XD(Ω) and defined by

u_D= X

σ∈Eint

σ=K|L

mD_σ,K

m_Dσ u_K+mD_σ,L

m_Dσ u_L χ_Dσ.

We define the discrete gradient denoted∇^D :X_T(Ω)→X_D(Ω)² by

∇^Du= X

σ∈Eint

σ=K|L

2 dKL

(uL−uK)nKLχD_σ,

where for any σ ∈ Eint the function χD_σ : Ω → R is the characteristic function of the dual control volume Dσ and for any σ = K|L the vector nKL is the unit normal vector to the edgeσoutward to the control volumeK. Note that by virtue of the boundary conditions (6) the value of these functions is zero on a dual control volumeD_σ satisfyingσ∈∂Ω. Note that for anyu∈X_T(Ω) one has

kuk²_1,T = 1

2k∇^Duk²_L2(Ω,R²).

Given a vector (u_K)_K∈T ∈ R^T, there exists a unique u ∈ X_T(Ω) (in the sense almost everywhere in Ω) such that for allK∈ T one hasu(x) =uK for allx∈K.

Let us note thatu=P

K∈TuKχK.

K

L

xK

xL

σ=K|L

n_K,L Dσ

Dσ,K

∂Ω

Figure 2. admissible mesh

2.1.2. Space-time discretizations. In the numerical analysis, we restrict our study to the case of a uniform time discretization of (0, T).

LetN be a nonnegative integer, then we define ∆t= tf

N+ 1, and tn =n∆tfor alln∈ {0, ..., N+ 1}, so thatt⁰= 0, andt^N+1=T.

We define the space and time discrete spaces X_T,∆t and X_D,∆t as the set of piecewise constant functions in time with values inXT andXD respectively:

X_T,∆t(ΩT) ={u: (0, T)×Ω→Rmeasurable |u_|(tn ,tn+1 )×K is constant,

∀K∈ T and∀n∈ {0, ..., N}}.

X_D,∆t(Ω_T) ={u: (0, T)×Ω→R²measurable |u_|(tn ,tn+1 )×Dσ is constant,

∀σ∈ E and∀n∈ {0, ..., N}}.

Givenu∈X_T,∆t(Ω) we denote byuⁿ⁺¹_K the value of the functionuover (tⁿ, tⁿ⁺¹)×

K for any n ∈ {0, ..., N} and for any K ∈ T and we denote by uⁿ the function of X_T(Ω) such that for all K ∈ T one has uⁿ(x) = uⁿ_K for all x ∈ K. Given u∈XT,∆twe define the discrete spatial gradient∇^D,∆tu∈XD,∆t(ΩT) by

∇^D,∆tu=

N

X

n=0

X

σ∈Eint

σ=K|L

2 dKL

(uⁿ⁺¹_L −uⁿ⁺¹_K )nKLχD_σ(x)χ_(tn,tⁿ⁺¹)(t),

where for anyn∈ {0, ..., N} the functionχ_(tn,tⁿ⁺¹) is the characteristic function of the time interval (tⁿ, tⁿ⁺¹). For a given (uⁿ⁺¹_K )n∈{0,...,N},K∈T ∈ R^(N+1)#T, there exists a unique u ∈ X_T,∆t(ΩT) (in the sense almost everywhere in Ω) such that u(t,x) =uⁿ⁺¹_K for allt∈(tⁿ, tⁿ⁺¹) andK∈ T.

Proposition 2.2. LetΩbe a be an open bounded polygonal subset ofR². Consider a sequence (Tm)m≥1 of admissible meshes ofΩsuch that

(23) diam(Tm) = max

K∈Tm

diam(K) −→

m→∞0,

Let(Nm)_m≥1be an increasing sequence of integers, then we define the corresponding sequence of time steps ∆tm= _N^tf

m+1 tending to 0 asm tends to ∞. Let (um)_m≥1 be a sequence of fonction where u_m ∈X_Tm,∆tm(Ω_T) for anym≥1 and such that for exists C independant ofm≥1 such that

(24) ∆tm N_m

X

n=0

kuⁿ⁺¹k²_L2(Ω)+ ∆tm N_m

X

n=0

kuⁿ⁺¹k²_1,Tm≤C for any m≥1.

Then there exists u∈ L²(0, T;H¹(Ω)) such that passing to a subsequence the fol- lowing convergence holds

um→uweakly inL²((0, T)×Ω),

∇^Dm,∆tmu_m→ ∇uweakly inL²((0, T)×Ω)².

2.2. The nonlinear Finite Volume scheme. We explicit, in this section, the discretization of the problem (3). The time discretization relies on backward Euler method. The space discretization relies on finite volume approach (see e.g [35]), with two-point flux approximation and a suitable upstream mobility scheme.

The discretization h⁰ ∈ XT(Ω) of the initial data h0 ∈ L^∞(Ω) satisfying 0≤ h0≤H a.e in Ω is defined by

(25) h⁰_K = 1

m_K Z

K

h0(x) dx, ∀K∈ T. Note that we have 0≤h⁰_K ≤H for anyK∈ T.

We will introduce now the scheme. For all n ∈ {0, ..., N}, a solution hⁿ⁺¹ ∈ X_T(Ω) andfⁿ⁺¹∈X_T(Ω)/Rto the scheme at the time step n+ 1 have to satisfy the following equations : for allK∈ T,

hⁿ⁺¹_K −hⁿ_K

∆t m_K+ X

σ∈E(K), σ=K|L

τ_σhⁿ⁺¹_σ (f_Kⁿ⁺¹−f_Lⁿ⁺¹) (26a)

+ X

σ∈E(K), σ=K|L

τσh˜ⁿ⁺¹_σ (hⁿ⁺¹_K −hⁿ⁺¹_L ) +δ X

σ∈E(K), σ=K|L

τσ(hⁿ⁺¹_K −hⁿ⁺¹_L ) = 0, (26b)

H X

σ∈E(K), σ=K|L

τ_σ(f_Kⁿ⁺¹−f_Lⁿ⁺¹) + X

σ∈E(K), σ=K|L

˜hⁿ⁺¹_σ (hⁿ⁺¹_K −hⁿ⁺¹_L ) = 0, (26c)

hⁿ⁺¹_σ =

(hⁿ⁺¹_K if f_Kⁿ⁺¹−f_Lⁿ⁺¹≥0, hⁿ⁺¹_L if f_Kⁿ⁺¹−f_Lⁿ⁺¹<0, (26d)

˜hⁿ⁺¹_σ =

(hⁿ⁺¹_K if hⁿ⁺¹_K −hⁿ⁺¹_L ≥0, hⁿ⁺¹_L if hⁿ⁺¹_K −hⁿ⁺¹_L <0.

(26e) Denoting

F_σ,Kⁿ⁺¹=τσhⁿ⁺¹_σ (f_Kⁿ⁺¹−f_Lⁿ⁺¹) +τσh˜ⁿ⁺¹_σ (hⁿ⁺¹_K −hⁿ⁺¹_L ) +δτσ(hⁿ⁺¹_K −hⁿ⁺¹_L ).

We notice that the discrete equation (26b) rewrites under the locally conservative form on each control volumeK∈ T :

(27)











F_σ,Kⁿ⁺¹+F_σ,Lⁿ⁺¹= 0, for all σ=K|L∈ E hⁿ⁺¹_K −hⁿ_K

∆t mK+ X

σ∈E(K), σ=K|L

F_σ,Kⁿ⁺¹= 0, for allK∈ T.

As a consequence, we can state that the scheme (26) is globally conservative, i.e.,

(28) X

K∈T

m_Khⁿ_K= X

K∈T

m_Kh⁰_K = Z

Ω

h₀(x)dx, ∀n∈ {0, ..., N+ 1}.

2.3. Main results. Let us introduce the discrete counterpart (Eⁿ)_n≥0 of the en- tropy (9) defined as follows :

Eⁿ := X

K∈T

m_KΓ(hⁿ_K).

The numerical analysis of the scheme (26) strongly relies on a discrete version of relation (11). The first main result of our paper is the existence of a solution to the nonlinear scheme (26) satisfying the maximum principle (4) and the stability in terms of the discrete entropy.

Theorem 2.3. There exists (at least) one solution(h, f)∈X_T,∆t(ΩT)×XT,∆t(ΩT)/R to the scheme(26). Moreover any solution satisfies0≤hⁿ_K≤H for allK∈ T and for all n∈ {0, . . . , N + 1} and there exists C depending only on Ω, f0, g0, ρ, tf,Λ andb such that

sup

n∈{0,...,N}

Eⁿ⁺¹+ 4δ

N

X

n=0

∆tk√

hⁿ⁺¹k²_1,T ≤C.

Once we have the discrete solution (h, f)∈X_T,∆t(Ω_T)×X_T,∆t(Ω_T)/Rat hand for all meshes and all time discretizations, then we can study the convergence of the scheme when the discretization parameters tend to 0. More precisely, consider a sequence (Tm)_m≥1 of admissible meshes of Ω such that

(29) diam(Tm) = max

K∈Tm

diam(K) −→

m→∞0, and such that there existsθ^?>0 such that

(30) reg(T_m)≤θ^?, ∀m≥1.

A sequence of dual meshes (D_m)_m≥1corresponding to the primal meshes (T_m)_m≥1 is built as in §2.1.1. Let (N_m)_m≥1 be an increasing sequence of integers, then we define the corresponding sequence of time steps ∆t_m = _N^tf

m+1 tending to 0 as m tends to∞.

Theorem 2.4. Let(Tm)_m≥1be a sequence of admissible meshes ofΩsuch that(29) and(30)hold. Let(h_Tm,∆t_m, f_Tm,∆t_m)_m≥1be a sequence of solutions to the scheme(26).

Then there existsh: (0, T)×Ω→Randf : (0, T)×Ω→Rtwo mesurable functions such that (h, f) ∈L²(0, T;H¹(Ω))∩L²(0, T;H¹(Ω)/R) with 0 ≤ h(t,x)≤ H for a.e(t,x)∈(0, T)×Ωand such that, up a subsequence

h_Tm,∆tm −→h a.e in(0, T)×Ω, f_Tm,∆tm −→f weaklyL²((0, T)×Ω),

∇^Dm,Tmh_Tm,∆t_m −→ ∇h weakly inL²((0, T)×Ω)²,

∇^Dm,Tmf_Tm,∆tm −→ ∇f weakly inL²((0, T)×Ω)²,

where the couple(h, f)is a weak solution to (3)-(6) in the sense of Definition 1.1.

The goal of this paper is to prove Theorems 2.3 and 2.4. It is organized as follows : In Section3 the existence of nonnegative solution is shown. In Section4 discrete counterpart of the entropy/entropy-dissipation (11) is established . Section 5 is devoted to the convergence proof of a subsequence of discrete solutions to the weak solution. This proof is based first on the compactness of the sequence of approximate solutions and then on the identification of the limit. The long-time behavior of the discrete solution is investigated in Section6. Finally, in Section 7 we present numerical simulation to illustrate the behaviour of the model and of the scheme.

3. Existence of a nonnegative discrete solution

In this section we prove the nonnegativity of the discrete solutions. We need this estimate to prove the existence of a solution to the scheme (26).

Proposition 3.1. For all K∈ T, n∈ {0, ..., N+ 1},

(31) 0≤hⁿ_K≤H.

Proof. The property hⁿ_K ≥ 0 clearly holds for n = 0 thanks to the assumptions of the initial data. Assume now that hⁿ_K ≥ 0 holds at time step n. With the identificationX_T(Ω)'R^d whered= card(T) we can write

A_T(fⁿ⁺¹, hⁿ⁺¹)hⁿ⁺¹=hⁿ, whereAT(fⁿ⁺¹, hⁿ⁺¹) = (aKL)_(K,L)∈T2 ∈ Md(R) is given by

aKK = 1 + ∆t mK

X

σ∈E(K), σ=K|L

τσ

(h_Kⁿ⁺¹−hⁿ⁺¹_L )⁺+ (f_Kⁿ⁺¹−f_Lⁿ⁺¹)⁺+δ ,

a_KL= ( ∆t

mKτ_σ

(h_Kⁿ⁺¹−hⁿ⁺¹_L )⁻+ (f_Kⁿ⁺¹−f_Lⁿ⁺¹)⁻−δ

ifσ=K|L∈ E(K), 0 else.

The matrixA_T(fⁿ⁺¹, hⁿ⁺¹) satisfies

aKL≤0 for any (K, L)∈ T² such thatK6=L, aKK+ X

L∈T,K6=L

aKL>0.

Consequently A_T(fⁿ⁺¹, hⁿ⁺¹) is regular and satisfies A_T(fⁿ⁺¹, hⁿ⁺¹)⁻¹ ≥ 0 (see [62]). Sincehⁿ ≥0 we obtain thathⁿ⁺¹≥0. Consequently

hⁿ⁺¹_K ≥0, ∀K∈ T,∀n∈ {0, ..., N}.

Let us now denoteH_Kⁿ =hⁿ_K−H for anyK∈ T and for anyn∈ {0, ...N+ 1}. The property H_Kⁿ ≤0 clearly holds forn= 0 thanks to the assumptions of the initial data. Assume now thatH_Kⁿ ≤0 holds at time step n. Then (26b) can be written in term ofH_Kⁿ as follows

m_KH_Kⁿ⁺¹−H_Kⁿ

∆t + X

σ∈E(K), σ=K|L

τ_σhⁿ⁺¹_σ (f_Kⁿ⁺¹−f_Lⁿ⁺¹)

+ X

σ∈E(K), σ=K|L

τσ˜hⁿ⁺¹_σ (hⁿ⁺¹_K −hⁿ⁺¹_L ) +δ X

σ∈E(K), σ=K|L

τσ(H_Kⁿ⁺¹−H_Lⁿ⁺¹) = 0,

and the equation (26c) can be written in term ofH_Kⁿ as follows

H X

σ∈E(K), σ=K|L

τσ(f_Kⁿ⁺¹−f_Lⁿ⁺¹) + X

σ∈E(K), σ=K|L

τσ˜hⁿ⁺¹_σ (H_Kⁿ⁺¹−H_Lⁿ⁺¹) = 0.

Subtracting (26c) into the previous identity gives m_KH_Kⁿ⁺¹−H_Kⁿ

∆t + X

σ∈E(K), σ=K|L

τ_σ(hⁿ⁺¹_σ −H)(f_Kⁿ⁺¹−f_Lⁿ⁺¹)+δ X

σ∈E(K), σ=K|L

τ_σ(H_Kⁿ⁺¹−H_Lⁿ⁺¹) = 0.

In particular we can write

B_T(fⁿ⁺¹, hⁿ⁺¹)Hⁿ⁺¹=Hⁿ, whereB_T(fⁿ⁺¹) = (b_KL)_(K,L)∈T2 ∈ M_d(R) is given by

aKK = 1 + ∆t mK

X

σ∈E(K), σ=K|L

τσ

(f_Kⁿ⁺¹−f_Lⁿ⁺¹)⁺+δ

aKL= ( ∆t

mKτσ

(f_Kⁿ⁺¹−f_Lⁿ⁺¹)⁻−δ

ifσ=K|L∈ E(K) 0 else

The matrixBT(fⁿ⁺¹) satisfies

bKL≤0 for any (K, L)∈ T²such thatK6=L, bKK+ X

L∈T,K6=L

bKL>0.

ConsequentlyB_T(fⁿ⁺¹, hⁿ⁺¹) is regular and satisfiesB_T(fⁿ⁺¹, hⁿ⁺¹)⁻¹≥0. Since Hⁿ≤0 we obtain thatHⁿ⁺¹≤0. Consequently

hⁿ⁺¹_K ≤H, ∀K∈ T,∀n∈ {0, ..., N},

which concludes the proof of this proposition.

Now, one can apply the same strategy as in [5, Proposition 2.2] for proving the existence of a solution to the scheme (26).

Proposition 3.2. Givenh^?∈X_T such that 0≤h^?≤H a.e inΩ. There exists (at least) one solution(h, f)∈X_T ×X_0,T to the scheme (26). Moreover 0≤h≤H a.e in Ω.

Proof. We follow the methodology proposed in [34], using a topological degree argument [30, 54] to prove the existence of a discrete solution. Let h^? ∈ XT

such that 0 ≤ h^? ≤ H a.e in Ω. We consider the finite dimensional space E = XT(Ω)×XT(Ω)/R with the norm k(h, f)kE = khkL^∞(Ω)+kfk1,T. Define the functional

(32) F:

(X_T(Ω)×X_T(Ω)/R×[0,1]→X_T(Ω)×X_T(Ω)/R (h, f, t)7→ ˜h,f˜

where ˜h∈X_T is given for anyK∈ T by mK˜hK =h_K−h^?_K

∆t mK−t X

σ∈E(K), σ=K|L

τσhσ(fK−fL)

−t X

σ∈E(K), σ=K|L

τσhσ(hK−hL)−tδ X

σ∈E(K), σ=K|L

τσ(hK−hL),

and where ˜f ∈XT(Ω)/Ris given for anyK∈ T by m_Kf˜_K =H X

σ∈E(K), σ=K|L

τ_σ(f_K−f_L) +t X

σ∈E(K), σ=K|L

h_σ(h_K−h_L).

Let (h, f, t)∈XTXT(Ω)/R×[0,1] such thatF(h, f, t) = 0. Then we have hK−h^?_K

∆t mK+t X

σ∈E(K), σ=K|L

τσhσ(fK−fL)

+t X

σ∈E(K), σ=K|L

τ_σh_σ(h_K−h_L) +tδ X

σ∈E(K), σ=K|L

τ_σ(h_K−h_L) = 0

H X

σ∈E(K), σ=K|L

τσ(fK−fL) +t X

σ∈E(K), σ=K|L

hσ(hK−hL) = 0.

Following the proof to obtain (31) and using the fact that 0≤h^?≤H a.e in Ω one has

h_K≥0 for anyK∈ T, and in particular

khkL¹(Ω)= Z

Ω

h(x) dx= Z

Ω

h^?(x) dx≤mΩH.

Consequently

khk_L∞(Ω)≤ 1

minK∈T m_Km_ΩH.

We then obtain the following estimation sincet∈[0,1]

kfk1,T ≤ 1 minK∈T mK

m_ΩHkhk1,T.

Using the equivalence of the norm onX_T there existsC₂only depending the mesh such that

khk1,T +khk_L2(Ω)≤C2khk_L^∞_(Ω), which gives

kfk1,T ≤ C₂ min_K∈T mK

mΩH.

Consequently we obtain

(33) k(h, f)kE ≤ C2+ 1 min_K∈T mK

mΩH.

For t = 0 the equation F(h, f,0) = 0 gives h = h^? and f = 0. Let us define relatively compact open set

U ={(h, f)∈X_T(Ω)×X_T(Ω)/Rsuch thatkhkE < C₃+ 1}, where the constant is given by C3 = _min^C2+1

K∈TmKmΩH +mΩH. For t = 0, F(·,0) turns out to be affine, and the system F(h, f,0) = 0 admits (h^?,0) as a unique solution. The topological degree corresponding toF(·,0) and U is equal to 1 since (h^?,0) belongs to U. Moreover by virtue of (33) any solution of F(h, f, t) = 0 necessarily belongs toU. Therefore, the topological degree corresponding toF(·, t) and U does not depend on t ∈[0,1]. In particular, it is also equal to 1 for t = 1 which gives the existence of at least one solution to the scheme.

4. Entropy estimate

In this section we will establish discrete counterpart to the entropy/entropy- dissipation estimate (11) that appears to be sufficient to establish Theorem2.4. In what follows, (f_Kⁿ, g_Kⁿ)_K∈T,n∈{0,...,N+1} denotes a solution to the scheme (26). We recall the discrete version of entropy functional :

Eⁿ= X

K∈T

m_KΓ(f_Kⁿ).

Proposition 4.1. For all n∈ {0, ..., N}, one has

(34) kfⁿ⁺¹k_1,T ≤ khⁿ⁺¹k_1,T

For all n∈ {0, ..., N}, one has

Eⁿ⁺¹−Eⁿ+ 4δ∆tk√

hⁿ⁺¹k²_1,T ≤0.

(35)

Proof. Let us proof (34). Multiplying (26c) byf_Kⁿ⁺¹ and summing overK∈ T we obtain

H X

σ=K|L∈E

τσ(f_Kⁿ⁺¹−f_Lⁿ⁺¹)²+ X

σ=K|L∈E

τσ˜hⁿ⁺¹_σ (hⁿ⁺¹_K −hⁿ⁺¹_L )(f_Kⁿ⁺¹−f_Lⁿ⁺¹) Using Hölder’s inequality and (31) we obtain that

Hkfⁿ⁺¹k²_1,T ≤Hkfⁿ⁺¹k1,Tkhⁿ⁺¹k1,T,

which gives the expected result. Let us assume momentarly thathⁿ⁺¹_K >0 for any K∈ T. We multiply (26b) by ∆tloghⁿ⁺¹_K and summing overK∈ T, provides that :

A+D=B+C, where

A= X

K∈T

m_K(hⁿ⁺¹_K −hⁿ_K) loghⁿ⁺¹_K , B=−∆t X

K∈T

X

σ∈E(K), σ=K|L

τσhⁿ⁺¹_σ

f_Kⁿ⁺¹−f_Lⁿ⁺¹

loghⁿ⁺¹_K ,

C=−∆t X

K∈T

X

σ∈E(K), σ=K|L

τσh˜ⁿ⁺¹_σ

hⁿ⁺¹_K −hⁿ⁺¹_L

loghⁿ⁺¹_K ,

D=δ∆t X

K∈T

X

σ∈E(K), σ=K|L

τσ

hⁿ⁺¹_K −hⁿ⁺¹_L

loghⁿ⁺¹_K .

Since Γ is convex, we find that A≥ X

K∈T

m_K(Γ(hⁿ⁺¹_K )−Γ(hⁿ_K)) =Eⁿ⁺¹−Eⁿ.

We perform a summation by parts : B=−∆t X

σ=K|L∈E

τσhⁿ⁺¹_σ (f_Kⁿ⁺¹−f_Lⁿ⁺¹)(loghⁿ⁺¹_K −loghⁿ⁺¹_L )

=−∆t X

σ=K|L∈E

τσ hⁿ⁺¹_K [(f_Kⁿ⁺¹−f_Lⁿ⁺¹)]⁺−hⁿ⁺¹_L [(f_Kⁿ⁺¹−f_Lⁿ⁺¹)]⁻

×(loghⁿ⁺¹_K −loghⁿ⁺¹_L ).

The Taylor expansion aroundhⁿ⁺¹_K shows that

(36) hⁿ⁺¹_KL(loghⁿ⁺¹_K −loghⁿ⁺¹_L ) =hⁿ⁺¹_K −hⁿ⁺¹_L ,

where hⁿ⁺¹_KL =tKLhⁿ⁺¹_K + (1−tKL)hⁿ⁺¹_L for sometKL∈(0,1). It is shown in [37, p. 468] that

B≤ −∆t X

σ=K|L∈E

τσ(f_Kⁿ⁺¹−f_Lⁿ⁺¹)h_KLⁿ⁺¹(loghⁿ⁺¹_K −loghⁿ⁺¹_L ).

Using (36), one has

B≤ −∆t X

σ=K|L∈E

τσ(hⁿ⁺¹_K −h_Lⁿ⁺¹)(f_Kⁿ⁺¹−f_Lⁿ⁺¹).

In the same way, we have C≤ −∆t X

σ=K|L∈E

τσ(hⁿ⁺¹_K −hⁿ⁺¹_L )(h_Kⁿ⁺¹−hⁿ⁺¹_L ) =−∆tkhⁿ⁺¹k²_1,T. Consequently using (34) one has

B+C≤∆tkhⁿ⁺¹k_1,Tkf_Tⁿ⁺¹k_1,T −∆tkhⁿ⁺¹k²_1,T ≤0, One has

D=δ∆t X

σ=K|L∈E

τσ(hⁿ⁺¹_K −hⁿ⁺¹_L )(log(hⁿ⁺¹_K )−log(hⁿ⁺¹_L )).

From the inequality 2(a−1)≤(a+ 1) log(a) for anya≥1 we obtain

(37) 4δ∆tk√

hⁿ⁺¹k²_1,T ≤D, which gives

Eⁿ⁺¹−Eⁿ+ 4δ∆tk√

hⁿ⁺¹k²_1,T ≤0.

which is the expected result. In the general case that ishⁿ⁺¹_K ≥0 for anyK ∈ T and for anyn∈ {0, ..., N+ 1}we introducef_K^n,=f_Kⁿ +. The equation (26b) can be written for anyK∈ T and for anyn∈ {0, ..., N}

h^n+1,_K −h^n,_K

∆t mK+ X

σ∈E(K), σ=K|L

τσh^n+1,_σ (f_Kⁿ⁺¹−f_Lⁿ⁺¹)

+ X

σ∈E(K), σ=K|L

τσ˜h^n+1,_σ (h^n+1,_K −h^n+1,_L ) +δ X

σ∈E(K), σ=K|L

τσ(h^n+1,_K −h^n+1,_L ) =R^n+1,_K ,