Christophe Berthon
email: christophe.berthon@univ-nantes.fr

Meissa M'baye
email: meissa.mbaye@univ-nantes.fr

Minh Hoang Le

Diaraf Seck
email: diaraf.seck@ucad.edu.sn

A well-defined moving steady states capturing Godunov-type scheme for Shallow-water model

Keywords: Hyperbolic conservation laws, Balance laws, Fully Wellbalanced schemes, Godunov-type schemes, entropy inequalities

The present work concerns the derivation of a well-balanced scheme to approximate the weak solutions of the shallow-water model. Here, the numerical scheme exactly captures all the smooth steady solutions with nonvanishing velocities. To address such an issue, a Godunov-type scheme is adopted. A particular attention is paid on the derivation of the intermediate states within the approximate Riemann solver. Indeed, because of the moving steady states, the intermediate states may be ill-defined. Here, we introduce a suitable correction in order to get a fully well-defined finite volume scheme. In addition, the numerical method is established to be positive preserving and to satisfy a discrete entropy inequality up to small perturbations. Several numerical experiments, including wet/dry transition, illustrate the relevance of the designed scheme.

Introduction

The present works is devoted to the numerical approximation of the solutions of the well-known shallow-water system given as follows:

     ∂ t h + ∂ x hu = 0,
x ∈ R, t > 0,

∂ t hu + ∂ x hu 2 + g h 2 2 = -gh∂ x z, (1)
where h(x, t) ≥ 0 stands for the water height and u(x, t) ∈ R denotes the water velocity. Here, g > 0 is the gravitational constant and z(x) is a given smooth function to represent the bottom topography. For the sake of simplicity in the forthcoming notations, we set

w = h hu , f (w) =   hu hu 2 + g h 2 2
  and S(w, z) = 0 -gh∂ x z .

In addition, we introduce Ω ⊂ R 2 the set of physical admissible states given by Ω = {w ∈ R 2 ; h > 0, u ∈ R}.

At this level, we do not consider the dry areas and we have imposed h > 0. However, the derived scheme will be seen relevant to deal with wet/dry transitions. Since the first-order extracted system is knows to be hyperbolic, in a finite time, the solution may contain discontinuities governed by the Rankine-Hugoniot conditions [START_REF] Godlewski | Numerical approximation of hyperbolic systems of conservation laws[END_REF][START_REF] Serre | Systems of conservation laws[END_REF][START_REF] Toro | Riemann solvers and numerical methods for fluid dynamics[END_REF].

In order to rule out nonphysical solutions, the system (1) must be endowed with an entropy inequality defined by

∂ t η(w) + ∂ x G(w) ≤ -ghu∂ x z, (2)
where both entropy function η : Ω → R and entropy flux function G : Ω → R are given by

η(w) = h u 2 2 + g h 2 2 and G(w) = u 2 2 + gh hu. (3)
Now, considering the numerical approximation of the solution of the system (1), after the pioneer works by Bermudez [START_REF] Bermudez | Upwind methods for hyperbolic conservation laws with source terms[END_REF] and Greenberg-LeRoux [START_REF] Greenberg | A well-balanced scheme for the numerical processing of source terms in hyperbolic equations[END_REF] (for instance, see also Gosse [START_REF] Gosse | A well-balanced flux-vector splitting scheme designed for hyperbolic systems of conservation laws with source terms[END_REF] or Jin [START_REF] Jin | A steady-state capturing method for hyperbolic systems with geometrical source terms[END_REF]), the steady solutions must be accurately approximate to ensure the relevance of the designed scheme. These solutions of particular interest are governed by the following system:

     ∂ x (hu) = 0, ∂ x hu 2 + g h 2 2 = -gh∂ x z. (4)
After direct computations, the smooth steady solutions are defined by

     ∂ x (hu) = 0, ∂ x u 2 2 + g(h + z) = 0.
International Journal on Finite Volumes Figure 1: Numerical approximation of the lake at rest obtained by adopting an HLL first-order scheme [START_REF] Harten | On upstream differencing and Godunovtype schemes for hyperbolic conservation laws[END_REF] supplemented with a centred discretization of the source term.

As a consequence, for any given constant pair (Q, B) ∈ R 2 , the smooth steady solutions are characterized by the two following relations:

   hu = Q, u 2 2 + g(h + z) = B. (5)
For the sake of simplicity in the forthcoming developments, we set

B(w, z) = u 2 2 + g(h + z). (6)
During the two last decades, numerous works are devoted to the derivation of numerical schemes to approximate the weak solutions of (1) and able to accurately capture the steady solutions. Indeed, after [START_REF] Bermudez | Upwind methods for hyperbolic conservation laws with source terms[END_REF][START_REF] Goutal | Dam-break wave simulation[END_REF][START_REF] Greenberg | A well-balanced scheme for the numerical processing of source terms in hyperbolic equations[END_REF][START_REF] Greenberg | Analysis and approximation of conservation laws with source terms[END_REF], if steady solutions are not correctly captured, the approximation may involve huge errors for large time (see Figure 1 for an illustration). Since [START_REF] Berthon | A positive preserving high order VFRoe scheme for shallow water equations: a class of relaxation schemes[END_REF] turns out to be nonlinear, several works only consider the lake at rest steady states defined by u = 0 and a constant free surface h + z. For instance the reader is referred to the well-known hydrostatic reconstruction introduce by Audusse and al. [START_REF] Audusse | A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows[END_REF] (see also [START_REF] Bouchut | Nonlinear stability of finite volume methods for hyperbolic conservation laws and well-balanced schemes for sources[END_REF]). Such an easy technique has been adopted in lot of situations and the preservation of the lake at rest is now very usual to develop well-balanced scheme (see [START_REF] Berthon | A positive preserving high order VFRoe scheme for shallow water equations: a class of relaxation schemes[END_REF][START_REF] Castro | High order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products. Applications to shallow-water systems[END_REF][START_REF] Chen | A new hydrostatic reconstruction scheme based on subcell reconstructions[END_REF][START_REF] Delestre | A limitation of the hydrostatic reconstruction technique for shallow water equations[END_REF] for a non exhaustive list of references). Since [START_REF] Berthon | A positive preserving high order VFRoe scheme for shallow water equations: a class of relaxation schemes[END_REF] may admit one, two or three solutions according to the choice of the pair (Q, B) ∈ R × R + , it is clear that the derivation of a scheme able to capture all smooth steady solutions is a very challenging. In [START_REF] Gosse | A well-balanced flux-vector splitting scheme designed for hyperbolic systems of conservation laws with source terms[END_REF], the nonlinear system ( 5) is solved at each interface to get a fully well-balanced scheme able to exactly capture all the smooth steady solutions (moving and non moving). Next, in [START_REF] Castro | Well-balanced numerical schemes based on a generalized hydrostatic reconstruction technique[END_REF], the authors proposed a suitable extension of the hydrostatic reconstruction in order to deal with moving steady solutions. Unfortunately, this generalized hydrostatic reconstruction does not preserve the water height non negative. Several works by Xing [START_REF] Noelle | High-order well-balanced finite volume WENO schemes for shallow water equation with moving water[END_REF][START_REF] Xing | Exactly well-balanced discontinuous Galerkin methods for the shallow water equations with moving water equilibrium[END_REF][START_REF] Xing | On the advantage of well-balanced schemes for moving-water equilibria of the shallow water equations[END_REF] (see also [START_REF] Russo | High order well balanced schemes for systems of balance laws[END_REF][START_REF] Russo | High order well-balanced schemes based on numerical reconstruction of the equilibrium variables[END_REF]) proposed relevant extensions of the high order discontinuous Galerkin method to preserve all the steady states solutions of (5).

More recently, in [START_REF] Berthon | A fully well-balanced, positive and entropysatisfying godunov-type method for the shallow-water equations[END_REF], a Godunov-type scheme is derived in order to introduce the source term within the approximate Riemann solver. They got a positive and entropy preserving scheme which exactly capture all the smooth steady solutions. Unfortunately, once again, a strongly non linear equation must be solved at each interface.

Next, in [START_REF] Michel-Dansac | A well-balanced scheme for the shallow-water equations with topography[END_REF][START_REF] Michel-Dansac | A well-balanced scheme for the shallow-water equations with topography or manning friction[END_REF], a relevant linearization of the Godunov-type scheme is designed to get a positive preserving fully well-balanced scheme. From now on, we underline that this numerical technique exactly capture all the steady solutions without solving [START_REF] Berthon | A positive preserving high order VFRoe scheme for shallow water equations: a class of relaxation schemes[END_REF], which make very attractive such a scheme. However, the obtained topography source term discretization involves some difficulties. Indeed, in [START_REF] Michel-Dansac | A well-balanced scheme for the shallow-water equations with topography[END_REF][START_REF] Michel-Dansac | A well-balanced scheme for the shallow-water equations with topography or manning friction[END_REF], the source term is approximated at each interface as follows:

(h∂ x z) n i+1/2 = h n i h n i+1 (h n i + h n i+1 )/2 × z i+1 -z i ∆x + 1 2(h n i + h n i+1 ) × [h] 3 i+1/2 ∆x , (7)
where

[h] i+1/2 = h n i+1 -h n i if |h n i+1 -h n i | ≤ C∆x, C∆x otherwise. (8)
Here, the definition of the constant C remains open. Moreover, we remark that, with a flat topography, the source term is given by O(∆x 2 ) instead of zero. In addition, the adopted linearization introduces a perturbation within the definition of the numerical flux function given by

(h∂ x z) n i+1/2 g h n i +h n i+1 2 - (q 2 i +q 2 i+1 )/2 h n i h n i+1 ∆x. ( 9
)
Such a perturbation is necessary to recover the exact capture of the moving steady states. But we note that this needed above perturbation may be undefined as soon as we have

g h n i + h n i+1 2 = (q n i ) 2 + (q n i+1 ) 2 2h n i h n i+1 . ( 10
)
This failure is underlined in [START_REF] Michel-Dansac | A well-balanced scheme for the shallow-water equations with topography[END_REF][START_REF] Michel-Dansac | A well-balanced scheme for the shallow-water equations with topography or manning friction[END_REF], but the authors claimed that the equality (10) never holds during the numerical experiments. However, it seems that this lack of consistency may produce numerical errors as illustrated in Figure 2.

The objective of the present paper is twofold. First, we propose a new source term discretization which is naturally consistent without involving some cutoff technique as introduce in ( 7)- [START_REF] Castro | Well-balanced numerical schemes based on a generalized hydrostatic reconstruction technique[END_REF]. Next, the fully well-balanced perturbation [START_REF] Chen | A new hydrostatic reconstruction scheme based on subcell reconstructions[END_REF] is improved in order to get a well-defined numerical method. In addition, the resulting fully well-balanced scheme must be water positive preserving and entropy preserving in a sense to be prescribed.

To address such an issue, the paper is organized as follows. In the next section, we design the Godunov-type scheme with source term. Such a scheme comes form a suitable definition of the approximate Riemann solver. After [START_REF] Berthon | A fully well-balanced, positive and entropysatisfying godunov-type method for the shallow-water equations[END_REF][START_REF] Michel-Dansac | A well-balanced scheme for the shallow-water equations with topography[END_REF][START_REF] Michel-Dansac | A well-balanced scheme for the shallow-water equations with topography or manning friction[END_REF], the mains novelty here stays in the introduction of the source term in the definition of the approximate Riemann solver. The next section, is devoted to design a relevant approximation of the source term to be put within the approximate Riemann solver.

P = g h n i +h n i+1 2 - (q n i ) 2 +(q n i+1 ) 2 2h n i h n i+1
. In order to illustrate the failure of this method, in (c) and (d), we respectively present the behavior of the water height and ill-posed quantity P. We remark that P vanishes and produces the spurious perturbation.

Here, we impose to the source term approximation to be fully consistent in order to avoid some cutoff technique. The fourth section concerns the complete derivation of the approximate Riemann solver. A suitable improvement is adopted to get a well defined fully well-balanced scheme. Next, in Section 5, we exhibit the main properties satisfied by the derived scheme. In the last section, several numerical approximations, including wet/dry transition, illustrate the relevance of the proposed scheme.

Godunov-type schemes for shallow-water with source term

To approximate the solution of (1), we first introduce a space discretization made of cells (x i-1/2 , x i+1/2 ) with a constant size ∆x. Then, we have x i+1/2 = x i-1/2 + ∆x for all i ∈ Z and we denote x i = x i-1/2 + ∆x/2 the cell center. Considering the time discretization, we set t n+1 = t n + ∆t where the time increments ∆t must be restricted according to a CFL-like condition. At time t n , we assume known a constant piecewise approximation of the solution of (1) as follows:

w n ∆ (x, t = 0) = w n i if x ∈ (x i-1/2 , x i+1/2 ).
We research for an updated approximation of the solution at time t n+1 = t n + ∆t.

To address such an issue, after [START_REF] Godunov | A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics[END_REF][START_REF] Harten | On upstream differencing and Godunovtype schemes for hyperbolic conservation laws[END_REF][START_REF] Toro | Riemann solvers and numerical methods for fluid dynamics[END_REF], the updated state w n+1 i is given by

w n+1 i = 1 ∆x x i+1/2 x i-1/2 w n ∆ (x, ∆t) dx, (11)
where w n ∆ (x, t) stands for the juxtaposition of non-interacting approximate Riemann solvers stated at each interface x i+1/2 as follows:

w n ∆ (x, t) = w R x -x i+1/2 t ; w n i , w n i+1 , z i , z i+1 if (x, t) ∈ (x i , x i+1 ) × (0, ∆t).
Here, w R x t ; w L , w R , z L , z R denotes a suitable approximation of Riemann solution of (1) associated with the initial data

(w, z)(x, t = 0) = (w L , z L ) if x < 0, (w R , z R ) if x > 0. ( 12
)
According to [START_REF] Harten | On upstream differencing and Godunovtype schemes for hyperbolic conservation laws[END_REF], the approximate Riemann solver has to satisfy the following integral consistency condition:

1 ∆x ∆x/2 -∆x/2 w R ( x t ; w L , w R , z L , z R ) dx = 1 ∆x ∆x/2 -∆x/2 w R ( x t ; w L , w R , z L , z R ) dx, (13)
International Journal on Finite Volumes where w R ( x t ; w L , w R , z L , z R ) is nothing but the exact solution of the Riemann problem for (1) associated with the initial data [START_REF] Gallice | Positive and entropy stable godunov-type schemes for gas dynamics and mhd equations in lagrangian or eulerian coordinates[END_REF]. Now, since the system (1) under consideration contains a source term, the average of the exact Riemann solution is not reachable. Indeed, as long as ∆t is small enough such that

w R -∆x/2 ∆t ; w L , w R , z L , z R = w L and w R ∆x/2 ∆t ; w L , w R , z L , z R = w R ,
the integration of (1) over the domain (-∆x/2, ∆x/2)×(0, ∆t), after straightforward computation, reads

1 ∆x ∆x/2 -∆x/2 w R ( x t ; w L , w R , z L , z R ) dx = 1 2 (w L + w R ) - ∆t ∆x (f (w R ) -f (w L )) - g ∆x ∆t 0 ∆x/2 -∆x/2 (h∂z) R ( x ∆t , w L , w R , z L , z R ) dx. (14)
Because of the source term integral, which must be defined in a prescribed sense, the expected average of the exact Riemann solution is, a priori, not correctly defined.

As a consequence, we suggest to introduce SLR to approximate this integral (for instance, see [START_REF] Gallice | Schémas de type godunov entropiques et positifs pour la dynamique des gaz et la magnétohydrodynamique lagrangiennes[END_REF][START_REF] Gallice | Positive and entropy stable godunov-type schemes for gas dynamics and mhd equations in lagrangian or eulerian coordinates[END_REF]). Hence, we impose the approximate Riemann solver to satisfy, instead of (13), the following approximated consistency condition:

1 ∆x ∆x/2 -∆x/2 w R ( x t ; w L , w R , z L , z R ) dx = 1 2 (w L + w R ) - ∆t ∆x (f (w R ) -f (w L )) + ∆t SLR , (15)
where SLR = SLR (w L , w R , z L , z R ) must be defined according to a consistent discretization of (0, -gh∂ x z) T . Now equipped with the approximate Riemann solver w R ( x ∆t ; w L , w R , z L , z R ), the updated state w n+1 i , can be reformulate in a standard finite volume formulation. Indeed, from [START_REF] Gallice | Schémas de type godunov entropiques et positifs pour la dynamique des gaz et la magnétohydrodynamique lagrangiennes[END_REF] we easily get

w n+1 i = 1 ∆x x i x i-1/2 wR x -x i-1/2 ∆t ; w n i-1 , w n i , z i-1 , z i dx+ 1 ∆x x i+1/2 x i wR x -x i+1/2 ∆t ; w n i , w n i+1 , z i , z i+1 dx.
According to the interface location x i+1/2 , we adopt a change of variable, given by x → x -x i+1/2 , to obtain 1 ∆x

x i+1/2 x i wR x -x i+1/2 ∆t ; w n i , w n i+1 , z i , z i+1 dx = 1 ∆x 0 -∆x/2 wR x ∆t ; w n i , w n i+1 , z i , z i+1 dx 1 ∆x x i+1 x i+1/2 wR x -x i+1/2 ∆t ; w n i , w n i+1 , z i , z i+1 dx = 1 ∆x ∆x/2 0 wR x ∆t ; w n i , w n i+1 , z i , z i+1 dx.
For the sake of simplicity in the notations, we set

I - i+1/2 = 1 ∆x 0 -∆x/2 w R x ∆t ; w n i , w n i+1 , z i , z i+1 dx, (16)
I + i+1/2 = 1 ∆x ∆x/2 0 w R x ∆t ; w n i , w n i+1 , z i , z i+1 dx, (17)
so that the updated state w n+1 i immediately rewrites as follows:

w n+1 i = I + i-1/2 + I - i+1/2 , = 1 2 I - i-1/2 + I + i-1/2 + 1 2 I - i+1/2 + I + i+1/2 + 1 2 I + i-1/2 -I - i-1/2 + 1 2 I - i+1/2 -I + i+1/2 .
Since, for all i ∈ Z

1 2 I - i+1/2 + I + i+1/2 = 1 ∆x ∆x/2 -∆x/2 wR x ∆t ; w n i , w n i+1 , z i , z i+1 dx,
arguing the consistency condition (15), we obtain

w n+1 i = w n i - ∆t ∆x f ∆ (w n i , w n i+1 , z i , z i+1 ) -f ∆ (w n i-1 , w n i , z i-1 , z i ) + ∆t 2 Si+1/2 + Si-1/2 , (18)
with the numerical flux function given by

f ∆ (w R , w L , z L , z R ) = 1 2 (f (w R ) + f (w L )) - ∆x 4∆t (w R -w L ) + ∆x 2∆t I + LR -I - LR ,
where

I + LR = 1 ∆x ∆x/2 0 w R x ∆t ; w L , w R , z L , z R dx, I - LR = 1 ∆x 0 -∆x/2 w R x ∆t ; w L , w R , z L , z R dx.
Concerning the source term discretization, with clear notations, we have

Si+1/2 := SLR (w n i , w n i+1 , z i , z i+1 ),
where SLR is introduce in [START_REF] Godunov | A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics[END_REF]. The presentation of the Godunov-type scheme with source term is then achieved. Now, the full scheme derivation will be completed as soon as the approximate Riemann solver w R (x/∆t; w L , w R , z L , z R ) and the source term approximation SLR (w L , w R , z L , z R ) are characterized.

Discretization of the source term

In this section, the local approximation of the source term SLR (w L , w R , z L , z R ), stated at each interface, is defined in order to preserve the stationary solutions. First, according to (5), a locally smooth steady solution is defined as follows:

h L u L = h R u R , B(w L , z L ) = B(w R , z R ) and w R -w L = o(|z R -z L |), (19)
where B(w, z) is given by ( 6). As a consequence, SLR must be fixed such that w R (x/∆t; w L , w R , z L , z R ) remains stationary; namely

w R (x/∆t, w L , w R , z L , z R ) = w L if x < 0, w R if x > 0. ( 20
)
as soon as (w L , w R , z L , z R ) satisfies [START_REF] Greenberg | Analysis and approximation of conservation laws with source terms[END_REF]. Now, let us underline that the integral consistency condition [START_REF] Godunov | A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics[END_REF], for smooth steady solution in the form [START_REF] Harten | On upstream differencing and Godunovtype schemes for hyperbolic conservation laws[END_REF], rewrites as follows:

∆x SLR = f (w R ) -f (w L ). Since (w L , w R , z L , z R ) are given according (19), with SLR = ( Sh LR , Sq LR ) T , we get ∆x Sh LR = 0, ∆x Sq LR = (h R u 2 R + g h 2 R 2 ) -(h L u 2 L + g h 2 L 2 ). ( 21
)
Since the evolution equation of the water height does not contain source term, we fix Sh LR = 0. We focus our attention on Sq LR and we have to design this quantity in order to be consistent with the topography source term -gh∂ x z and such that (21) holds as soon as (w L , w R , z L , z R ) defines a local steady solution [START_REF] Greenberg | Analysis and approximation of conservation laws with source terms[END_REF]. To address such an issue, we now state a necessary condition to be satisfied by Sq LR .

Lemma 3.1 Let us introduce

F 2 r = q 2 0 h gh 2 L h 2 R , where h = 1 2 (h L + h R ) and q 0 = h L u L = h R u R , (22)
an approximation of the well-known Froude number. As soon as Fr = 1 and (w L , w R , z L , z R ) are steady states according to [START_REF] Greenberg | Analysis and approximation of conservation laws with source terms[END_REF], the identity (21) reformulates as follows:

∆x Sq LR = -g h(z R -z L ) + q 2 0 4h 2 L h 2 R × (h R -h L )(z R -z L ) 2 (1 -F 2 r ) 2 . ( 23
)
From now on, we remark that [START_REF] Michel-Dansac | A well-balanced scheme for the shallow-water equations with topography or manning friction[END_REF] immediately gives a consistent definition of Sq LR . Indeed, for smooth enough topography function, we have

lim ∆x→0 (z R -z L ) 2 ∆x = 0,
to ensure that Sq LR is consistent with -gh∂ x z. However, this relevant definition of Sq LR must be improved in order to deal with Fr = 1. Now, we establish the above lemma.

Proof With

q 0 = h L u L = h R u R , the relation (21) easily rewrites ∆x Sq LR = q 0 1 h R - 1 h L + g h(h R -h L ), to get ∆x Sq LR = g h - q 2 0 h L h R (h R -h L ). ( 24
)
Now, since we have B(w L , z L ) = B(w L , z L ) with B(w, z) given by ( 6), we have

q 2 0 2h 2 L + g(h L + z L ) = q 2 0 2h 2 R + g(h R + z R ), to obtain (h R -h L ) + (z R -z L ) = q 2 0 2g × h 2 R -h 2 L h 2 L h 2 R .
We introduce Fr , defined by [START_REF] Michel-Dansac | A well-balanced scheme for the shallow-water equations with topography[END_REF], to write the above relation as follows:

h R -h L = - z R -z L 1 -F 2 r . (25)
Next, we plug the above water height jump into [START_REF] Noelle | High-order well-balanced finite volume WENO schemes for shallow water equation with moving water[END_REF] to have

∆x SLR = -g h × 1 - q 2 0 gh L h R h 1 -F 2 r × (z R -z L ), (26)
which rewrites as follows:

∆x SLR = -g h(z R -z L ) - q 2 0 4h 2 L h 2 R × (h L -h R ) 2 1 -F 2 r ∆z. ( 27
)
The expected identity ( 23) is obtained by introducing ( 25) into [START_REF] Serre | Systems of conservation laws[END_REF]. The proof is then completed.

In order to propose a suitable correction within [START_REF] Michel-Dansac | A well-balanced scheme for the shallow-water equations with topography or manning friction[END_REF] to get a well-defined discretization of the source term, we exhibit the behavior of (23) in the limit of Fr to 1.

Lemma 3.2 Let us consider (w L , w R , z R , z L ) according to [START_REF] Greenberg | Analysis and approximation of conservation laws with source terms[END_REF] so that Sq LR is given by [START_REF] Michel-Dansac | A well-balanced scheme for the shallow-water equations with topography or manning friction[END_REF]. Then we have lim

Fr→1 Sq LR = O(∆x 2 ).
Proof Since (w L , w R , z L , z R ) is given according [START_REF] Greenberg | Analysis and approximation of conservation laws with source terms[END_REF], then the relation ( 25) holds and we easily reformulate [START_REF] Michel-Dansac | A well-balanced scheme for the shallow-water equations with topography or manning friction[END_REF] as follows:

∆x Sq LR = g h(h R -h L )(1 -F 2 r ) + g F 2 r 4 h (h R -h L ) 3 ,
to immediately obtain the following limit lim

F 2 r →1 Sq LR = g 4 h (h R -h L ) 3 ∆x . ( 28
)
First, as soon as the water height is a smooth solution, we have h R -h L = O(∆x) and the proof is achieved. Next let us show that, in the limit of Fr to 1, necessarily the approximated water height is smooth and then h R -h L = O(∆x). Indeed, let us assume w L and w R be connected by a stationary shock solution. Because of the Rankine-Hugoniot relations (see [START_REF] Serre | Systems of conservation laws[END_REF]), since the shock velocity is equal to 0, we have

q L = q R = q 0 , q 2 R h R + g h 2 R 2 - q 2 L h L + g h 2 L 2 = 0, so that we obtain (h R -h L )g h 1 - q 2 0 g hh L h R = 0, ( 29
)
where h is defined by [START_REF] Michel-Dansac | A well-balanced scheme for the shallow-water equations with topography[END_REF]. With h L = h R , from [START_REF] Toro | Restoration of the contact surface in the HLL-Riemann solver[END_REF], it comes

q 2 0 g hh L h R = 1.
In the limit of F r to 1, with Fr given by ( 22), we get

q 2 0 g hh L h R = q 2 0 h gh 2 L h 2 R , to obtain h L = h R in contradiction with h L = h R .
As a consequence, in the limit of Fr to 1, we always have h R -h L = O(∆x). The proof is then completed.

In fact, the above result states that the discretization ( 23) is well-defined for all values of Fr as long as the local equilibrium condition [START_REF] Greenberg | Analysis and approximation of conservation laws with source terms[END_REF] is satisfied. As a consequence, [START_REF] Michel-Dansac | A well-balanced scheme for the shallow-water equations with topography or manning friction[END_REF] must be improved in order to be well-defined for all values of Fr with left and right states not verifying a local steady solution [START_REF] Greenberg | Analysis and approximation of conservation laws with source terms[END_REF]. We now suggest an improvement in order to get a formula well-defined for all values of Fr . In the present work, we propose the following discretization of the topography source term

∆x Sq LR = -g h(z R -z L ) + q 2 4h 2 L h 2 R × (h R -h L )(z R -z L ) 2 (1 -F 2 r ) 2 + ε LR ∆x k , (30)
where k > 0 is a parameter to be fixed and

q 2 = |q L q R |, ε LR = |B(w R , z R ) -B(w L , z L )| + |q R -q L |. (31)
From now on, we note that ε LR = 0 if and only if (w L , w R , z L , z R ) define a local steady solution according to [START_REF] Greenberg | Analysis and approximation of conservation laws with source terms[END_REF]. As a consequence, the suggested discretization (30) satisfy the necessary condition [START_REF] Michel-Dansac | A well-balanced scheme for the shallow-water equations with topography or manning friction[END_REF] for local steady solution. Moreover, the discretization ( 30) is well-defined for all values of Fr . Indeed, as long as, the local equilibrium, given by [START_REF] Greenberg | Analysis and approximation of conservation laws with source terms[END_REF], is satisfied, arguing Lemma 3.2 the source term discretization is well-defined as expected. Next, for non-steady states, since 30) is well-defined for all values of Fr . Of course, since the correction is given by ε LR ∆x k , the consistency property of Sq LR is preserved. Moreover, according to the continuous extension argument given Lemma 3.2, we impose Sq LR = 0 if we have simultaneously Fr = 1 and ε LR = 0. Now, concerning the value of k > 0, let us underline a specific behavior of Sq LR with Fr = 1 but far away from local equilibrium. Indeed, enforcing Fr = 1, we have

(1 -F 2 r ) 2 + ε LR ∆x k = 0, once again (
Sq LR = -g h (z R -z L ) ∆x + q 2 4h 2 L h 2 R × (h R -h L )(z R -z L ) 2 ε LR ∆x -1-k ,
since the topography function is smooth, we have z R -z L = O(∆x) so that

Sq LR = -g h (z R -z L ) ∆x + (h R -h L ) ε LR O(∆x 1-k ).
Next, we have ε LR = O(∆x 1/2 ) with smooth enough solutions. Then we obtain

Sq LR = -g h (z R -z L ) ∆x + O(∆x 3 2 -k ).
As a consequence, we fix k = 1 2 .

Approximate Riemann solver

To complete the characterisation of the numerical scheme [START_REF] Greenberg | A well-balanced scheme for the numerical processing of source terms in hyperbolic equations[END_REF], we now have to define a suitable approximate Riemann solver w R (x/t; w L , w R , z L , z R ) according to the integral consistency condition [START_REF] Godunov | A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics[END_REF] where the source term approximation SLR = (0, Sq LR ) T is defined by [START_REF] Xing | Exactly well-balanced discontinuous Galerkin methods for the shallow water equations with moving water equilibrium[END_REF]. In the present work, we adopt an approximate Riemann solver made of two intermediate constant states, w * L and w * R , as follows:

w R (x/t; v L , v R ) =        w L if x t < λ L , w * L if λ L < x t < 0, w * R if 0 < x t < λ R , w R if x t > λ R , (32)
where λ L < 0 and λ R > 0 denote the discontinuity propagation speed. As usual (see [START_REF] Harten | On upstream differencing and Godunovtype schemes for hyperbolic conservation laws[END_REF][START_REF] Toro | Restoration of the contact surface in the HLL-Riemann solver[END_REF]), we impose

λ L ≤ min(-|u L | -gh L , -|u R | -gh R ), λ R ≥ max(|u L | + gh L , |u R | + gh R ),
so that the cone defined by λ L ≤ x t ≤ λ R contains the exact cone coming from the exact Riemann solution associated with the initial data [START_REF] Gallice | Positive and entropy stable godunov-type schemes for gas dynamics and mhd equations in lagrangian or eulerian coordinates[END_REF]. Now, the integral consistency condition (15) reads as follows:

λ L (h * L -h L ) + λ R (h R -h * R ) = q R -q L , (33)
λ L (q * L -q L ) + λ R (q R -q * R ) = h R u 2 R + 1 2 gh 2 R -h L u 2 L + 1 2 gh 2 L + ∆x Sq LR . ( 34
)
For the sake of simplicity in the notations, let us introduce (see [START_REF] Harten | On upstream differencing and Godunovtype schemes for hyperbolic conservation laws[END_REF])

h HLL = λ R h R -λ L h L λ R -λ L - q R -q L λ R -λ L , (35)
q HLL = λ R q R -λ L q L λ R -λ L - 1 λ R -λ L h R u 2 R + 1 2 gh 2 R -h L u 2 L + 1 2 gh 2 L . (36)
Then, both relations (33) and (34) reformulate as follows:

λ R h * R -λ L h * L = (λ R -λ L )h HLL , ( 37
) λ R q * R -λ L q * L = (λ R -λ L )q HLL + ∆x Sq LR . (38)
Let us emphasize that the full determination of w R is achieved as soon as the two intermediate states, (h * L , q * L ) and (h * R , q * R ), are fixed. As a consequence, we have to define four values while, at this level, we have two equations (37) and (38). Clearly, two relations are missing. We must also preserve the steady states. Hence, we formally adopt the local steady state definition [START_REF] Greenberg | Analysis and approximation of conservation laws with source terms[END_REF] to propose suitable closure relations. First, we note that the discharge must be constant to define a steady state. Then, concerning the intermediate discharge we adopt the following equalities:

q * L = q * R = q * . ( 39
)
Because of (38), the intermediate discharge is then given by

q * = q HLL + ∆x Sq LR (λ R -λ L ) . (40)
Now, concerning the evaluation of the intermediate water heights, h * L and h * L , we have the equation (37) to be supplemented with an additional relation such that the pair (h * L , h * R ) is solution of (2 × 2) system. Our objective is to propose a relevant additional equation such that h * L = h L and h * R = h R is the unique solution as soon as (w L , w R , z L , z R ) verifies [START_REF] Greenberg | Analysis and approximation of conservation laws with source terms[END_REF]. For instance, in [START_REF] Berthon | A fully well-balanced, positive and entropysatisfying godunov-type method for the shallow-water equations[END_REF], the expected second equation is given by B(w * L , z L ) = B(w * R , z R ). But such a choice involves strong nonlinearities. After [START_REF] Michel-Dansac | A well-balanced scheme for the shallow-water equations with topography[END_REF][START_REF] Michel-Dansac | A well-balanced scheme for the shallow-water equations with topography or manning friction[END_REF], this nonlinear system can be substituted by a linear system which preserve all the required properties. In [START_REF] Michel-Dansac | A well-balanced scheme for the shallow-water equations with topography[END_REF][START_REF] Michel-Dansac | A well-balanced scheme for the shallow-water equations with topography or manning friction[END_REF], the authors adopt the following additional relation:

h * R -h * L = ∆x Sq LR α LR , (41)
where

α LR = g h - q 2 h L h R , (42)
where q 2 is defined by [START_REF] Xing | On the advantage of well-balanced schemes for moving-water equilibria of the shallow water equations[END_REF]. The 2 × 2 linear system made of (37) and (41) does not admit a unique solution as long as q 2 = g hh L h R . As a consequence, we suggest the following improvement:

h * R -h * L = α LR ∆x Sq LR α 2 LR + ε LR ∆x k . ( 43
)
International Journal on Finite Volumes

At the main discrepancy with (41), we remark that the right hand side in (43) is well-defined. Indeed, if α 2 LR + ε LR ∆x k = 0, necessarily we have α LR = 0 and ε LR = 0. But ε LR = 0 holds if and only if ( 19) is satisfied. Then, according to (24), we have ∆x Sq LR = α LR (h R -h L ), and as a consequence, we get lim

α LR ,ε LR →0 α LR ∆x Sq LR α 2 LR + ε LR ∆x k = h R -h L . (44)
To conclude, h * L and h * R are given as solution of the 2 × 2 linear system made of (37) and ( 43). The solution is directly given by

h * L = h HLL + λ R ∆x Sq LR α LR (λ R -λ L )(α 2 LR + ε LR ∆x k ) , (45)
h * R = h HLL + λ L ∆x Sq LR α LR (λ R -λ L )(α 2 LR + ε LR ∆x k ) . ( 46
)
However, such an evaluation of the intermediate water heights does not ensure the required positiveness of h * L and h * R . Here, we adopt a positive cutoff introduce in [START_REF] Berthon | A fully well-balanced, positive and entropysatisfying godunov-type method for the shallow-water equations[END_REF] (see also [START_REF] Berthon | A well-balanced finite volume scheme for a mixed hyperbolic/parabolic system to model chemotaxis[END_REF][START_REF] Michel-Dansac | A well-balanced scheme for the shallow-water equations with topography[END_REF][START_REF] Michel-Dansac | A well-balanced scheme for the shallow-water equations with topography or manning friction[END_REF]). As a consequence, h * L and h * R are corrected as follows:

h * L = min max h HLL + λ R ∆x Sq LR α LR (λ R -λ L )(α 2 LR + ε LR ∆x k ) , σ , 1 - λ R λ L h HLL + λ R λ L σ (47) h * R = min max h HLL + λ L ∆x Sq LR α LR (λ R -λ L )(α 2 LR + ε LR ∆x k ) , σ , 1 - λ L λ R h HLL + λ L λ R σ (48)
where σ = min(h L , h R , h HLL ). Now, let us remark that h HLL > 0 as long as the exact water height Riemann solution h R remains positive. Moreover, by definition of σ, we have σ -h HLL ≤ 0. Since λ L < 0 < λ R then we get

1 - λ R λ L h HLL + λ R λ L σ = h HLL + λ R λ L σ -h HLL > 0, 1 - λ L λ R h HLL + λ L λ R σ = h HLL + λ L λ R σ -h HLL > 0.
As a consequence, we immediately get h * L > 0 and h * R > 0. In addition to this positiveness statement, it is worth noticing that the proposed water height cut-off correction preserves the conservation since the relation (37), or equivalently (33), always is satisfied. To establish (37), for the sake of simplicity in the notations, let us set

Σ LR = λ L λ R ∆x Sq LR α LR (λ R -λ L )(α 2 LR + ε LR ∆x k )
, International Journal on Finite Volumes so that we have

λ R h R -λ L h L = min max λ R h HLL + Σ LR , λ R σ , (λ R -λ L )h HLL + λ L σ + min max -λ L h HLL -Σ LR , -λ L σ , (λ R -λ L )h HLL -λ R σ .
The above quantity is easily evaluated by comparing Σ LR , λ R (σ -h HLL ) and λ L (σh HLL ). Indeed, let us first assume

λ R (σ -h HLL ) < Σ LR < λ L (σ -h HLL ). ( 49
)
Then we get

max λ R h HLL + Σ LR , λ R σ = λ R h HLL + Σ LR , max -λ L h HLL -Σ LR , -λ L σ = -λ L h HLL -Σ LR .
Moreover, we easily notice that the assumption (49) imposes

λ R h HLL + Σ LR ≤ (λ R -λ L )h HLL + λ L σ, -λ L h HLL -Σ LR ≤ (λ R -λ L )h HLL -λ R σ,
to recover the conservation relation (37). Now, let us assume

Σ LR < λ R (σ -h HLL ) < λ L (σ -h HLL ), (50)
to get

max λ R h HLL + Σ LR , λ R σ = λ R σ, max -λ L h HLL -Σ LR , -λ L σ = -λ L h HLL -Σ LR .
In addition, according to (50), from a direct evaluation, we obtain

λ R σ ≤ (λ R -λ L )h HLL + λ L σ, -λ L h HLL -Σ LR ≥ (λ R -λ L )h HLL -λ R σ,
and once again we recover the expected relation (37). We here skip the last assumption λ R (σ -h HLL ) < λ L (σ -h HLL ) < Σ LR since the establishment of (37) is similar to the above case with (50). The definition of the approximate Riemann solver (32) is now achieved. To conclude this section, we establish that the approximate Riemann solver preserves the steady solutions. Lemma 4.1 Let w L and w R be given in Ω. Consider the approximate Riemann solver in the form (32) where the intermediate states are given by (40), ( 47) and (48). If (w L , w R , z L , z R ) satisfy the local steady state condition [START_REF] Greenberg | Analysis and approximation of conservation laws with source terms[END_REF] with Fr = 1, then the approximate Riemann solver stays at rest, namely we have

w R (x/∆t; w L , w R , z L , z R ) = w L if x < 0 w R if x > 0. ( 51
)
Proof Since the local steady condition ( 19) is satisfied, the topography source term, given by ( 30), reads

∆x Sq LR = -g h(z R -z L ) - q 2 4h 2 L h 2 R × (h L -h R )(z R -z L ) 2 (1 -F 2 r ) 2
, where q = q L = q R . As long as Fr = 1, arguing Lemma 3.1, Sq LR is now given by ( 21). Then, from (40), straightforward computations gives q * = q, and then the discharge is stationary. Concerning the intermediate water height, since the local steady condition ( 5) is verified, we have the following equality:

h HLL - λ L,R λ R -λ L ∆x Sq LR α LR α 2 LR + ε LR ∆x k = 1 2 (h L + h R ) - λ L,R λ R -λ L ∆x Sq LR α LR ,
where α LR is given by (42). For the sake of simplicity in the notations, we have set λ L,R = λ L or λ R . With α LR = 0, since the relation ( 24) holds, we directly obtain

h HLL - λ L,R λ R -λ L ∆x Sq LR α LR α 2 LR + ε LR ∆x k = h L,R .
Then according to (47) and (48), we easily obtain h * L = h L and h * R = h R , and the proof is achieved.

Main properties of the scheme

Equipped with the approximate Riemann solver (32), where the intermediate states are given by ( 40), ( 47) and (48), supplemented with the source term discretization [START_REF] Xing | Exactly well-balanced discontinuous Galerkin methods for the shallow water equations with moving water equilibrium[END_REF], then the scheme ( 18) is fully determined. Now, we state the main properties satisfied by the derived numerical approximation.

Theorem 5.1 Assume the following CFL-like restriction:

∆t ∆x max i∈Z |λ L i+1/2 |, |λ R i+1/2 | ≤ 1 2 . ( 52
)
The scheme ( 18) is consistent with [START_REF] Audusse | A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows[END_REF]. Let w n i be in Ω for all i ∈ Z. The updated states w n+1 i given by the scheme [START_REF] Greenberg | A well-balanced scheme for the numerical processing of source terms in hyperbolic equations[END_REF], satisfy the following properties:

1. Positiveness preservation: h n+1 i > 0 for all i ∈ Z, 2. Preservation of all smooth steady states: w n+1 i = w n i for all i ∈ Z as soon as (w n i ) i∈Z verifies

q n i = Q and (q n i ) 2 2(h n i ) 2 + g(h n i + z i ) = B for all i ∈ Z, (53)
for given constants Q and B such that two successive states satisfy [START_REF] Greenberg | Analysis and approximation of conservation laws with source terms[END_REF].

3. An incomplete entropy inequality in the form:

η n+1 i -η n i ∆t + G i+1/2 -G i-1/2 ∆x ≤ -Sη i + O(∆x), (54)
is satisfied as long as

h HLL i+1/2 + λ L i+1/2 ∆x Sq i+1/2 α i+1/2 α 2 i+1/2 + ε i+1/2 ∆x k > 0 and h HLL i+1/2 + λ R i+1/2 ∆x Sq i+1/2 α i+1/2 α 2 i+1/2 + ε i+1/2 ∆x k > 0, ( 55
) where we have set

G i+1/2 = 1 2 (G(w n i+1 ) + G(w n i )) + ∆x 4∆t (η(w n i+1 ) -η(w n i )) + ∆x 2∆t (I η + i+1/2 -I η - i-1/2 ), ( 56
)
with

I η + i+1/2 = 1 ∆x ∆x/2 0 η( w R ( x ∆t ; w n i , w n i+1 , z i , z i+1 )) dx, (57)
I η - i+1/2 = 1 ∆x 0 -∆x/2 η( w R ( x ∆t ; w n i , w n i+1 , z i , z i+1 )) dx, (58)
and

Sη i = 1 2
h n i-1 + h n i 2 q HLL i-1/2 h HLL i-1/2 ( z i -z i-1 ∆x ) + h n i + h n i+1 2 q HLL i+1/2 h HLL i+1/2 ( z i+1 -z i ∆x ) . (59)
Before we establish the above result, let us comment the discrete entropy inequality (54). Indeed, it is worth noticing that the residue, given by O(∆x), is governed by the smoothness of the topography function z and never depends on the unknowns (eventually discontinuous). As a consequence, the well-known Lax-Wendroff Theorem can be applied so that the converged solution turns out to be an entropy preserving weak solution of (1).

In addition, let us highlight the assumption (55) to get the discrete entropy inequality (54). In fact, (55) enforces the intermediate water heights to be given by ( 45) and ( 46) so that the cut-off corrections (47) and (48) stay inactive. Put in other words, we here present discrete entropy inequalities far away from the dry areas and wet/dry transitions. Now, in order to establish the above result, we need the following technical lemma where we state the behavior of the entropy at each interface. Lemma 5.1 Let w L and w R be in Ω such that h * L and h * R , defined by ( 47) and ( 48), coincide with (45) and (46). Let η and G be given by [START_REF] Berthon | A fully well-balanced, positive and entropysatisfying godunov-type method for the shallow-water equations[END_REF]. Assume a CFL-like restriction in the form ∆t ∆x max

(|λ L |, |λ R |) ≤ 1 2 . ( 60
)
The approximate Riemann solver (32) with the intermediate states given by (40), (47) and (48) verifies the following inequality:

1 ∆x ∆x/2 -∆x/2 η( w R (x/∆t; w L , w R , z L , z R )) dx ≤ 1 2 (η(w L ) + η(w R )) - ∆t ∆x (G(w R ) -G(w L )) -∆t Sη LR + ∆t ∆x O(∆x 2 ), (61)
where

Sη LR (∆x, w L , w R , z L , z R ) = g hq HLL h HLL × z R -z L ∆x . ( 62
)
Proof With an approximate Riemann solver given by (32), under the CFL-like condition (60) we immediately have

1 ∆x ∆x/2 -∆x/2 η( w R (x/∆t; w L , w R , z L , z R )) dx = 1 2 -λ L ∆t ∆x η(w L ) + ∆t ∆x (λ R η(w * R ) -λ L η(w * L )) + 1 2 -λ R ∆t ∆x η(w R ). (63)
Now, we have

η(w * L ) = (q * ) 2 2h * L + g (h * L ) 2 2 and η(w * R ) = (q * ) 2 2h * R + g (h * R ) 2 2 ,
where q * and h * L,R are given by ( 40), ( 45) and ( 46). Since the topography function is smooth, then z R -z L = O(∆x). As a consequence, as long as α LR = 0, we have

q * = q HLL - g h λ R -λ L (z R -z L ) + O(∆x 2 ), h * R = q HLL + λ L λ R -λ L g h α LR (z R -z L ) + O(∆x 2 ), h * L = q HLL + λ R λ R -λ L g h α LR (z R -z L ) + O(∆x 2 ).
Then, straightforward computations give

λ R η(w * R ) -λ L η(w * L ) = (λ R -λ L )η(w HLL ) -g hq HLL h HLL (z R -z L ) + O(∆x 2 ). ( 64
)
Moreover, after [START_REF] Harten | On upstream differencing and Godunovtype schemes for hyperbolic conservation laws[END_REF], we have

η(w HLL ) ≤ λ R λ R -λ L η(w R ) - λ L λ R -λ L η(w L ) - 1 λ R -λ L (G(w R ) -G(w L )), so that λ R η(w * R ) -λ L η(w * L ) ≤ λ R η(w R ) -λ L η(w L ) -(G(w R ) -G(w L )) -g hq HLL h HLL (z R -z L ) + O(∆x 2 ). ( 65
)
The expected inequality (61) comes combining (63) with the above estimation. To complete the proof, let us underline that h * L = h * R = h HLL if α LR = 0 so that the estimation (64) immediately holds. The proof is then achieved.

Equipped with the above result, we are able to establish Theorem 5.1.

Proof First, we establish the consistency of the derived scheme [START_REF] Greenberg | A well-balanced scheme for the numerical processing of source terms in hyperbolic equations[END_REF]. To address such an issue, first we notice that the CFL-like condition (52) ensures that the sequence of approximate Riemann solver are non-interacting. Next, we show the consistency of both numerical flux and source term. Concerning the numerical flux, we have

f ∆ (w, w, z L , z R ) = f (w) + ∆x 2∆t (I + LR -I + LR ),
where I + LR and I - LR are defined by ( 17) and ( 16). Since the approximate Riemann solver is in the form (32), then we get

f ∆ (w, w, z L , z R ) = f (w) + λ R 2 (w * R -w) + λ L 2 (w * L -w).
Arguing the definition of the intermediate states, with w L = w R = w ∈ Ω, enforcing ∆x to be small enough, we get

q * = q - 1 λ R -λ L gh(z R -z L ), h * L = h + λ R λ R -λ L gh(z R -z L ) gh -q 2 /h 2 (gh -q 2 /h 2 ) 2 + ∆x k |z R -z L | , h * L = h + λ R λ R -λ R gh(z R -z L ) gh -q 2 /h 2 (gh -q 2 /h 2 ) 2 + ∆x k |z R -z L | ,
to immediately obtain the expected consistency of the numerical flux as follows:

lim z L ,z R →z f ∆ (w, w, z L , z R ) = f (w).
Next, concerning the consistency of the discrete source term, given by ( 30), as soon as w L = w R = w we have

Sq LR (w, w, z L , z R ) = -gh (z R -z L ) ∆x .
With a smooth topography function, let us fix z L = z(x -∆x 2 ) and z R = z(x + ∆x 2 ) so that we obtain lim ∆x→0 Sq LR (w, w, z L , z R ) = -gh∂ x z, and the scheme ( 18) is established to be consistent. Now, we show the positiveness of h n+1 i . After [START_REF] Gallice | Schémas de type godunov entropiques et positifs pour la dynamique des gaz et la magnétohydrodynamique lagrangiennes[END_REF], since the intermediate water height, defined by ( 47)-(48), involved within the approximate Riemann solver, are positive, then h n+1 i is nothing but the integral of the positive quantities. As a consequence, we get h n+1 i > 0. Next, concerning the well-balanced property, since (w n i ) i∈Z verify (53), at each interface the local steady condition [START_REF] Greenberg | Analysis and approximation of conservation laws with source terms[END_REF] is satisfied. As a consequence of Lemma 4.1, all the non interacting approximate Riemann solution are steady so that we get

w R x -x i+1/2 ∆t ; w n i , w n i+1 , z i , z i+1 = w n i if x i+1/2 < 0, w n i+1 if x i+1/2 > 0. ( 66
)
Then we have the following sequence of equalities

w n+1 i = 1 ∆x x i x i-1/2 w R ( x -x i-1/2 ∆t ; w n i-1 , w n i , z i-1 , z i ) dx + 1 ∆x x i+1/2 x i w R ( x -x i+1/2 ∆t ; w n i , w n i+1 , z i , z i+1 ) dx, = w n i .
To conclude the proof, we establish the entropy inequality (54). Since the entropy function η is convex, the well-known Jensen inequality gives

η(w n+1 i ) ≤ 1 ∆x x i+1/2 x i-1/2 η(w ∆ R (x, t n + ∆t)) dx, ≤ 1 ∆x ∆x/2 0 η( wR ( x ∆t ; w n i-1 , w n i , z i-1 , z i )) dx + 1 ∆x 0 -∆x/2 η( wR ( x ∆t ; w n i , w n i+1 , z i , z i+1 )) dx,
so that, arguing (57) and (58), we obtain

η(w n+1 i ) ≤ 1 2 (I η - i-1/2 + I η + i-1/2 ) + 1 2 (I η + i-1/2 -I η - i-1/2 ) + 1 2 (I η - i+1/2 + I η + i+1/2 ) - 1 2 (I η + i+1/2 -I η - i+1/2 ). (67)
Next, because of (55), necessarily the intermediate water heights, given by ( 45) and ( 46), are positive. Since the positive water height cut-off corrections (47) and ( 48) is not active then Lemma 5.1 can be applied. As a consequence, from (61), we deduce

I η - i+1/2 + I η + i+1/2 ≤ 1 2 (η(w n i )) + η(w n i+1 )) - ∆t ∆x (G(w n i+1 )) -G(w n i ) -∆t Sη LR + ∆t ∆x O(∆x 2 ),
where, according to (59), we have set Sη i+1/2 = Sη LR (∆x, w n i , w n i+1 , z i , z i+1 ). Now, by introducing the above estimation in (67), after straightforward computation, we obtain the expected discrete entropy inequality (54). The proof is then achieved.

Numerical experiments

In this section, we perform several numerical experiments to illustrate the behavior of the designed fully well-balanced corrected scheme (FWBC). A comparison with the well-known hydrostatic reconstruction (HR) method [START_REF] Audusse | A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows[END_REF], applied to the HLL scheme [START_REF] Harten | On upstream differencing and Godunovtype schemes for hyperbolic conservation laws[END_REF], and the well-balanced scheme (FWB) proposed in [START_REF] Michel-Dansac | A well-balanced scheme for the shallow-water equations with topography[END_REF][START_REF] Michel-Dansac | A well-balanced scheme for the shallow-water equations with topography or manning friction[END_REF], is displayed for all the simulations. Concerning the FWB-scheme, according to the source term discretization given by ( 7) and ( 8), a parameter denoted C must be fix. In the present work, all the displayed numerical simulations are performed considering C ∈ [1, 2.5] according to the realized simulations. We emphasize that spurious oscillations appear with C ≥ 5 while the approximate solutions blow-up with C ≥ 10. Once again, we underline that the proposed correction of the FWB-scheme is now free from this parameter C.

As long as the exact solution is known, the behavior of the obtained approximate solutions is exhibited by evaluating the L 1 , L 2 and L ∞ errors. Such errors are given by

||w n -w ex (., t n )|| L 1 = i∈Z |w n i -w ex (x i , t n )|∆x, (68)
||w n -w ex (., t n )|| L 2 = i∈Z (w n i -w ex (x i , t n )) 2 ∆x, (69)
||w n -w ex (., t n )|| L ∞ = max i∈Z |w n i -w ex (x i , t n )|, (70)
where (w n i ) i∈Z is defined by the adopted numerical scheme while w ex (x, t) denotes the expected exact solution.

The subcritical flow

This test is used to see the ability of the scheme to capture steady states. The domain is [0, 25] and the topography is defined by z(x) = (0, 0.2 -0.05(x -10) 2 ) + where x + = max(0, x). The initial conditions are h(x, 0) = 2 and q(x, 0) = 4.42. On the left boundary condition, the water height satisfies a homogeneous Neumann condition and the discharge is equal to q 0 = 4.42. On the right boundary, the water height is equal to h 0 = 2 and the discharge follows a homogeneous Neumann boundary condition.

h + z q L 1 error L 2 error L ∞ error L 1 error L 2 error L ∞ error HR-scheme
1.32e-02 1.36e-02 1.59e-02 2.89e-03 2.89e-03 2.89e-02 FWB-scheme 1.15e-13 1.20e-13 1.51e-13 6.63e-14 6.99e-14 8.26e-14 FWBC-scheme 1.10e-13 1.17e-13 1.44e-13 6.57e-14 6.84e-14 8.21e-14 Table 1: Free surface and discharge errors for the subcritical flow.

The domain is here discretized with 1000 cells. At times t = 100, so that the steady regime is researched, the approximate solutions are displayed in Figure 3. In Table 1, we present the numerical errors. We note that the approximate solution given by the fully well-balanced scheme ( 18) is in very good agreements with the exact solution since such a scheme exactly preserves this solution.

The transcritical flow without shock

We consider a computational domain given by [0, 25] and the topography is defined by z(x) = (0, 0.2 -0.05(x -10) 2 ) + . We consider the initial data given by h(x, 0) = 0.66 and q(x, 0) = 1.53. On the left boundary condition, the water height satisfies a homogeneous Neumann condition and the discharge is equal to q 0 = 1.53. On the right boundary, the water height is equal to h 0 = 0.66 and the discharge follows a homogeneous Neumann boundary condition. The obtained numerical simulation are displayed Figure 4 at time t = 500 with 1000 cells.

h + z q L 1 error L 2 error L ∞ error L 1 error L 2 error L ∞
error HR-scheme 4.79e-02 6.07e-02 8.12e-02 8.28e-04 3.30e-03 1.82e-02 FWB-scheme 1.67e-14 2.13e-14 4.26e-14 1.47e-14 1.58e-14 2.04e-14 FWBC-scheme 1.27e-14 1.67e-14 3.14e-14 1.27e-14 1.48e-14 1.98e-14 Table 2: Free surface and discharge errors for the transcritical flow without shock.

Once again, the fully well-balanced corrected scheme gives very good agreement with the exact solution in the steady regime which is exactly captured by the here derived scheme. The very good behavior of the numerical solution is highlight by Table 2 where the numerical errors are given.

The perturbed transcritical flow without shock

In this numerical simulation, we test the behavior of the derived scheme to deal with a perturbed stationary solution. Here, the initial data is given by h(x, 0) = h 0 (x) + h δ (x) and q(x, 0) = q 0 (x) + q δ (x) where (h 0 , q 0 ) stands for the stationary solution of a transcritical flow without shock according to the numerical experiment International Journal on Finite Volumes in [START_REF] Michel-Dansac | A well-balanced scheme for the shallow-water equations with topography[END_REF][START_REF] Michel-Dansac | A well-balanced scheme for the shallow-water equations with topography or manning friction[END_REF] and illustrated Figure 2, is now corrected. Indeed, in Figure 8, we notice that the spurious perturbation no longer stays. The good behavior of the scheme is highlight by the errors evaluations given Table 4.

Wet/dry transition flow

According to [START_REF] Michel-Dansac | A well-balanced scheme for the shallow-water equations with topography[END_REF][START_REF] Michel-Dansac | A well-balanced scheme for the shallow-water equations with topography or manning friction[END_REF], we impose a vanishing velocity as soon as the water height vanishes. In addition, we have enforced u h = 0 in the limit of h to zero. As a consequence, in a wet/dry transition interface, with h L = 0 or h R = 0, we get ∆x Sq

LR = -g h(z R -z L ) and α LR = g h, (71)
so that all required properties, stated Theorem 5.1, are preserved.

Dam-break on a dry domain

This numerical experiment illustrates the ability of the scheme to correctly locate and treat the wet/dry transition. The domain is [0, 10] with a flat topography. The initial data is made of a vanishing discharge in the whole domain while the water height is defined by

h(x, 0) = 0.005 if 0 < x < 5, 0 if 5 < x < 10.
We adopt homogeneous Neumann boundary conditions.

h + z q L 1 error L 2 error L ∞ error L 1 error L 2 error L ∞
error HR-scheme 7.06e-5 5.20e-5 1.33e-4 1.33e-5 1.15e-5 2.92e-5 FWB-scheme 1.50e-4 9.54e-5 1.92e-4 2.62e-5 2.11e-5 4.17e-5 FWBC-scheme 1.51e-4 9.54e-5 1.92e-4 2.62e-5 2.11e-5 4.18e-5 that this simulation contains an emerged bottom. Concerning the initial discharge, we impose a vanishing velocity in the whole domain. We consider the homogeneous Neumann boundary conditions. 6: Free surface and discharge errors for the lake at rest with an emerged bottom.

h + z q L 1 error L 2 error L ∞ error L 1 error L
In Figure 12 we display the approximate solution with 500 cells at time t = 100. According to the errors presented Table 6, the solution is exactly captured. Figure 13: Both exact and approximate free surface (13a) and both exact and approximate discharge (13b) for the Thacker test case.

Thacker test case

The computational domain is [0, L] and the topography is defined by

z(x) = h 0 1 a 2 x - L 2 2 -1 ,
where h 0 , a and L are parameters to be fixed. The exact solution is an oscillation of the free surface between the points where w = √ 2gh 0 /a, b = w/2 and the period T = 2π/w. The initial conditions are given by q(x, 0) = 0 and

h(x, 0) =      - h 0 a 2 (x - L 2 ) + b √ 2gh 0 2 -a 2 if x 1 (0) < x < x 2 (0), 0 otherwise.
The exact water height is given by Concerning the parameters, we fix a = 1, L = 4 and h 0 = 1/2. In Figure 13, the approximate solution is displayed at time t = 10.0303 with 1000 cells.

h(x, t) =      - h 0 a 2 (x - L 2 ) + b √ 2gh 0 cos(wt)
FWBC-scheme q cells L 1 error L Concerning the water height, displayed in Figure 13a, we obtain a good approximation of the wet/dry transition. Now, the approximation of the discharge is not good as expected. However, we recall that usual method like the hydrostatic reconstruction cannot give good approximation of the discharge. In Tables 7, 8 and 9, we present the obtained errors for q. We notice that the here designed FWBC-scheme produces better approximation than the hydrostatic reconstruction scheme.

Drain on a non-flat bottom

This numerical experiment is proposed in [START_REF] Gallouët | Some approximate godunov schemes to compute shallow-water equations with topography[END_REF], where the topography is defined by z(x) = (0.2 -0.05(x -10) 2 ) + over a computational domain given by [0,[START_REF] Russo | High order well balanced schemes for systems of balance laws[END_REF]. The initial data is fixed as follows: h(x, 0) + z(x) = 0.5 and q(x, 0) = 0.

Concerning the boundary conditions, according in [START_REF] Gallouët | Some approximate godunov schemes to compute shallow-water equations with topography[END_REF], the left boundary condition is given by h L = h n 1 and q L = 0, International Journal on Finite Volumes The domain discretization is made of 500 cells. In Figure 14, we displays the numerical approximation obtained at time t end = 0, 10, 20, 100, 1000. Once again, we notice a very good behavior of the approximate solution.

Figure 2 :

 2 Figure 2: Figure (a) displays the free surface for a transcritical flow with shock test case obtained by the scheme introduced in [22, 23]. In (b), we present a zoom of the spurious perturbation of the water height produced by the ill-posed quantity

Figure 3 :

 3 Figure 3: Free surface (3a) and discharge (3b) for the simulation of a subcritical flow.

Figure 4 :

 4 Figure 4: Free surface (4a) and discharge (4b) for the simulation of a transcritical flow without shock.

Figure 9 :

 9 Figure 9: Free surface (9a) and discharge (9b) for the dam-break on a dry domain.

Figure 11 :

 11 Figure 11: Evolution of the time step ∆t versus time during the simulation of the dam-break test case on steps topography.

Figure 12 :

 12 Figure 12: Free surface (12a) and discharge (12b) for the lake at rest with emerged bottom.

x 1 (

 1

Figure 14 :

 14 Figure 14: Free surface (14a) and discharge (14b) for the simulation of drain on a non-flat bottom

Table 5 :

 5 Free surface and discharge errors for the dam-break on a dry domain.

International Journal on Finite Volumes

Table 7 :

 7 Discharge errors for the Thacker test case for different cells using FWBCscheme.

		2 error	L ∞ error
	1600 3.74e-2	2.76e-2	2.64e-2
	2000 2.66e-2	1.97e-2	1.94e-2
	4000 1.15e-2	8.56e-3	9.51e-3
		FWB-scheme
		q
	cells L 1 error	L 2 error	L ∞ error
	1600 1.24e-2	1.31e-2	3.30e-2
	2000 1.10e-2	1.16e-2	2.99e-2
	4000 7.98e-3	7.88e-3	2.19e-2

Table 8 :

 8 Discharge errors for the Thacker test case for different cells using FWBscheme.

		HR-scheme
		q
	cells L 1 error	L 2 error	L ∞ error
	1600 7.29e-2	5.59e-2	5.36e-2
	2000 5.83e-2	4.47e-2	4.48e-2
	4000 2.91e-2	2.23e-2	2.13e-2

Table 9 :

 9 Discharge errors for the Thacker test case for different cells using HRscheme.

International Journal on Finite Volumes

Acknowledgements

Christophe Berthon and Meissa M'baye thank the ANR project MUFFIN ANR-19-CE46-0004 for financial supports.

presented Section 6.2 and (h δ , q δ ) given by h δ (x) = 1 if 10 < x < 11, 0 otherwise, and

This initial data is displayed Figure 5.

The resulting numerical approximations obtained with a mesh made of 200 cells for the HR-scheme, the FWB-scheme and FWBC-scheme at time t = 500 are displayed Figure 6. After a long time, the stationary solution is recovered according to the scheme accuracy (see also Table 3).

error HR-scheme 4.89e-02 6.17e-02 8.22e-02 8.38e-04 3.43e-03 1.98e-02 FWB-scheme 2.79e-14 2.85e-14 4.20e-14 1.62e-14 1.85e-14 2.18e-14 FWBC-scheme 2.77e-14 2.87e-14 4.14e-14 1.58e-14 1.75e-14 2.08e-14 Table 3: Free surface and discharge errors for the perturbed transcritical flow without shock.

with respect to the made correction.

Transcritical flow with shock

In this simulation, we show the behavior of an hydraulic jump approximation. The domain of the computation is [0, 25] and the topography fixed by z(x) = (0, 0.2 -0.05(x -10) 2 ) + . The initial data are given by h(x, 0) = 0.33 and q(x, 0) = 0.18.

On the left boundary condition, the water height satisfies a homogeneous Neumann condition and the discharge is equal to q 0 = 0.18. On the right boundary, the water height is equal to h 0 = 0.33 and the discharge follows a homogeneous Neumann boundary condition.

1.16e-03 1.82e-03 5.12e-02 1.54e-04 1.53e-03 4.00e-02 FWB-scheme 2.81e-03 3.01e-03 9.85e-02 2.94e-04 3.35e-03 5.39e-02 FWBC-scheme 2.49e-03 2.40e-03 8.84e-02 2.851e-04 3.65e-03 5.08e-02 Table 4: Free surface and discharge errors for the transcritical flow with shock.

The numerical simulation obtained at time t = 1000 with the fully well-balanced scheme are displayed Figure 7. It is worth noting that the failure of scheme detected The domain is here discritized with 500 cells and the obtained approximate solution at time t = 6 is displayed Figure 9. We notice the ability of the FWBC-scheme to deal with wet/dry transition and the here proposed correction preserves the relevance of the numerical method (see Table 5).

Dam-break on a step topography

This numerical simulation is devoted to illustrate the relevance of the scheme approximating the solution when the topography function is discontinuous. Here, in the domain [-10, 10], the topography is made of three successive steps as follows:

The initial discharge is vanishing in the whole domain and the initial water height is defined by

We consider homogeneous Neumann boundary conditions. The approximate solution for the free surface and the discharge are displayed Figure 10. We notice the good behaviors of both HR-scheme and FWBC-scheme while the discharge of the original FWB-scheme presents spurious values located at the discontinuities of the topography function. In Figure 11, we display the behavior of the time step ∆t throughout the simulation. We remark that the derived correction does not drastically modify the behavior of ∆t.

Lake at rest with an emerged bottom

In the computational domain [0, 25], the bottom is defined by z(x) = (0, 0.2 -0.05(x -10) 2 ) + . For the initial water height, we adopt h(x, 0) = (0.15 -z(x)) + so


Camille Baulant

Gemma Davies

The Role of Networks in Helping Firms and Countries Invent New Competitive Strategies Adapted to the World Knowledge Economy

Keywords: competitive advantage, information competitiveness, learning networks, lobbying network, meso-institution JEL Classification Code: F40, L14, L16 1

For the last twenty years, the world economy has evolved at a great speed. Every good, capital asset, and knowledge is mobile and induces more competition. Innovation in commodities is a complex process that requires more cooperation. To innovate in the knowledge economy, firms nowadays must establish "win-win situations" for individuals in creating networks. These networks are useful for firms in order to come up with innovative strategies. The building of networks enables the interactions between agents, the environment, and institutions. The interdependence of agents and institutions is not new to evolutionary theory (Commons 1931; Veblen 1898). However, I argue that institutions must be more flexible than ever before in order to help agents adapt to the modern knowledge economy. On the basis of the role of meso-networks, I propose new long-run specialization and short-run competitiveness that will promote greater efficiency and equality around the world in relation to firms and countries exporting industrial goods into world markets. Within the innovative networks, I analyze the role of two different actors: (i) the "economic leader" who has a long-run strategy and (ii) the "go-between leader" who knows how to diffuse "useful information" to actors to help them innovate in new products, services, or processes.

For the last twenty years, the world economy has been evolving with a great speed. Every good, capital asset, and knowledge is mobile and induces rising competition. The knowledge economy stimulates more cooperation with the division of cognitive labor (DCL) process. The interdependence between individuals and institutions is an old question to evolutionary theory [START_REF] Commons | Institutional Economics[END_REF][START_REF] Veblen | Why Is Economics Not an Evolutionary Sciences?[END_REF]), but rising competition and cooperation are among the challenges that firms must face today, when innovating and selling on the world markets. Two opposite approaches have been recently proposed for firms in order to increase their market shares on the world markets. The "strategic approach of economics" insists on the increasing role of "competition" to win the actual "economic war" [START_REF] Baumard | Le vide stratégique[END_REF][START_REF] Harbulot | Techniques offensives et guerre économique[END_REF]. In this approach, building "knowledge networks" authorizes the seizure of "strategic knowledge." On the other hand, the "institutional approach of economics" stresses the endogenous characteristics of the "learning by sharing" process to reach an efficient level for both individuals and collective organizations [START_REF] Cohendet | Organisational Innovation, Communities of Practice and Epistemic Communities[END_REF][START_REF] Foray | L'économie de la connaissance[END_REF].

I argue that these opposite approaches must be mixed into dynamic networks, where institutions are as important as markets and demand policies are as useful as supply policies in increasing innovations in countries. In the first section of the paper, I analyze why it is so important to build "meso-institutions" for thinking and acting in the new economy. In the second part, I propose new long-and short-run strategies, based on dynamic networks, pertinent to both firms and countries. In the last section, I study two new functions that are useful for managing individuals and organizations within these networks.

Division of Cognitive Labor and Networks Organization Within Firms

The globalization of countries and their entry into a knowledge economy induce a contradictory movement of "competition" and "cooperation" between individuals, firms, and countries. Facing these quantitative and qualitative changes in the world economy, both previous regulation mechanisms -the "invisible hand" of private markets and the "visible hand" of the national institutions of nation states -have failed to allow the adaptation of firms to the new kind of competition. The interdependences between markets and institutions in the economic growth of countries have been studied for a long time in evolutionary theory. Among these analyses are those of Thorstein [START_REF] Veblen | Why Is Economics Not an Evolutionary Sciences?[END_REF][START_REF] Polanyi | La Grande Transformation: aux origines politiques et économiques de notre temps[END_REF] 1983), Albert [START_REF] Hirschman | Exit, Voice, and Loyalty: Responses to Decline in Firms, Organizations, and States[END_REF], and Frederic S. [START_REF] Lee | Competition, Going Enterprise, and Economic Activity[END_REF], to name a few. In thinking dynamic interactions between markets and institutions, Edgar [START_REF] Morin | La méthode : la nature de la nature[END_REF][START_REF] Morin | La méthode : la nature de la nature[END_REF]) and Arthur [START_REF] Koestler | Génie et folie des l'homme, le cheval dans la locomotive (le paradoxe humain)[END_REF] have already shown how an "intermediary institution" in systemic approaches could be more powerful than the "macro-institution" of the nation state. The meso-institutions, created by individuals within firms or countries, stabilize the behaviors of individuals (by producing "regularities by disorder") and create innovation (by producing "complexity by disorder") [START_REF] Atlan | Entre le cristal et la fumée: Essai sur l'organisation du vivant[END_REF]). John S. [START_REF] Brown | Organizational Learning and Communities of Practice: Towards a Unified View of Working, Learning and Innovation[END_REF] as well as Patrick [START_REF] Cohendet | Organisational Innovation, Communities of Practice and Epistemic Communities[END_REF] demonstrate how flexible meso-institutions (as the "communities of practice") must be today in order to help individuals adapt their strategies to the knowledge economy. In a global "inclusive economy,"1 it is important to conceptualize the contradictory relationships between markets and institutions, supply and demand, internal and external environment. From a systemic analysis perspective, firms and countries must be "open" to innovation and "constraint" by "path dependency" to produce and sell on the market. A traditional debate in economics proposes two opposite approaches to managing this evolution. On one side, the "market approach" -based on the methodological individualism -focuses on the "automatic go-back to the equilibrium." On the other side, the "institutional approach" -based on the holistic approach -centers on the "co-evolution of markets and institutions" inside the "path dependency" of a firm or country. 2To study more profoundly the consequences of global change on the behavior of firms, my analysis stresses the utility of building networks by individuals (Table 1). Today, this meso-institution seems more efficient than the micro-level of private firms or macro-institutions (which suffered from the decreasing power of nation states in international relationships). In the modern knowledge economy, innovation becomes a "collective" process and the relationships between individuals become as important as each inventive component [START_REF] Guilhon | Sous la direction.) De l'Intelligence Economique à l'économie de la connaissance[END_REF][START_REF] Muldoon | Diversity and the Division of Cognitive Labor[END_REF]. The DCL concept is so far away from the "international division of industrial labor" -one, where big international firms choose to implant different activities, such as research, raw materials, producing, and selling, in different countries. With the high economic growth rate of the 1990s and the 2000s, the place of new industrialized countries (NIC) in the world markets have increased both quantitatively (they now belong to G20 and G8) and qualitatively (they also enter into the knowledge economy). The most important "emerging countries" are competing against advanced countries in goods and services with the use of high technology. So, in order to define the new "sustainable competitive advantage specialization," all countries must analyze their specialization "on the whole value chain" [START_REF] Aghion | Competition and Innovations: An Inverted-U Relationship[END_REF][START_REF] Porter | Competitive Advantage of Nations[END_REF]). To build their long-run strategy, firms must exploit the interdependences that exist between supply, demand, cooperation, and competition. If these links are not sufficiently thought through, firms would fail to adapt to the world market changes. On the other hand, anticipating new innovation requires the creation of meso-networks to interlink cooperation and competition. In determining future demands for goods, countries compete on the world market. The competition among countries, not only concerns the "upstream business" (cost-intensive technologies pertaining to aerospace, planes, energy sectors, medicine), but also the "downstream business" (intensive in inexpensive technology like low-cost computer industries or new information and communication technologies (ICT).

Table 1. Consequences of the Two Main World Changes on External Competition and Internal Organization of Firms

Long-and Short-Run Strategies for Building Meso-Networks

How Firms Create Networks for Building Sustainable "Competitive Advantages"

In this section, I study the role of networks to create a long-run competitive advantages for small firms. With the globalization of the world economy, the optimal specialization strategy has changed, too. Today, most countries specialize in sectors that produce high value goods, so the competition for producing such goods has sharply increased since 1990. However, the "non-price specialization," proposed by Elhanan [START_REF] Helpman | Market Structure and Foreign Trade[END_REF] for advanced countries, is no longer sufficient to explain the countries' choices of specialization. "Emerging countries" represent a rising part of the actual world trade and they know how to compete with advanced countries. Since 2001, China has increased its patent deposits in the highly innovation intensive "industrial goods," while India is exporting high technology "computer services" and Brazil trades in biotechnological goods that are R&D intensive.

To help understand the actual strategies of firms, I expand on the "competitive advantage" theory of Mickael [START_REF] Porter | Competitive Advantage of Nations[END_REF]. The innovation process must include the Brazil exports "intelligent textile goods" that combine "cotton" with "spider's genes" to make textile elastic and resistant. specific role of "market innovation." In recombining some previously invented "radical innovations" into new products Apple Inc., for example, has succeeded in answering the new "needs" of consumers. So, I introduce into Porter's "diamond" the concept of "differentiation" for analyzing the different strategies of firms, which combine supply, demand, competition, and cooperation factors (Figure 1). My concept of "differentiation" allows all small firms to participate in "market innovations." The aim of innovations is taking into account the "real needs" of consumers in particular markets. Within the networks, firms build systemic interrelations between opposite factors to reach an output that will be "more than the sum of its parts" [START_REF] Koestler | Génie et folie des l'homme, le cheval dans la locomotive (le paradoxe humain)[END_REF]. Most small firms develop their competitive advantage by achieving some "market innovations." For example, since most of India's population still lives on limited income, Indian firms practice more "frugal innovations" (i.e., Tata Nano, iPpad, Smartphone, or 3D printer to build houses) than "radical innovation" (i.e., aerospace or pharmaceutical products) to build competitive advantages [START_REF] Prahalad | Serving the World's Poor, Profitably[END_REF]. In order to build "competitive advantages," small firm must invent new "cooperation relationships." Cooperation could be simple and simply based on a tacit sharing process. This kind of cooperation uses low-cost investments, such as, for example, the purchase of a cell phone for communicating and selling products. Finally, firms must invent new ways to practice "competition" in using different kinds of competitiveness for increasing their market shares, as shown in the next section.

Competition differentiation:

Price competitiveness Non-price competitiveness Information competitiveness

How Firms Create Networks to Increase Their Information Competitiveness

In the world market, firms must manage three kinds of competitiveness: "price competitiveness," "non-price competiveness," and "information competitiveness." This information competitiveness becomes more important because of the ICT revolution. Today, a "false information" can destroy the reputation of a firm even if their products are of good quality. In order to be "pro-active" in the world markets, all firms engaged in "business intelligence." 5 The methodology of the business intelligence is to transform "information" into "knowledge" and then to transform "knowledge" into "useful information." Information competiveness is thus quite different from price competiveness (to have low prices), as well as from non-price competitiveness (to develop their oligopolistic positions).

To increase their market shares in the world market, firms can use the three kinds of proactive meso-networks that support information competitiveness. Here I present, as a long-run strategy, the case of small firms that can use the power of these networks. On one side of the competitiveness triangle (Figure 2), the "sharing networks" are useful for firms to initiate some cooperation between all partners. On the other side, the "positive lobbying networks" are used by firms to explain to their partners how their products are better than their competitors'. At the summit of the triangle are the "institutional networks." They are useful because the co-building institutions are able to change as quickly as economic rules in the world markets. These intermediary institutions (industrial patents organizations, non-governmental organizations, regional councils, etc.) help small firms to patent industrial innovations. Intermediary organizations can also inform small firms about a new change in international norms or laws. Even if small firms are not powerful enough to change the existing law, they can adapt their products to new norms before other competitors. As all firms today must be "proactive" in the world market, the mesonetworks are particularly efficient in increasing the power of small firms.

How to Manage Meso-Networks Inside and Outside Companies

Changes in the world market have involved new strategies based on dynamic networks. The co-building of these networks is based on private markets and institutional regulations. Today, national institutions are not powerful enough to regulate the new economy, where everything moves quickly. Building mesoinstitutions constitutes an opportunity to challenge competition and build cooperation relationships. I apply Geoffrey M. [START_REF] Hodgson | What Are Institutions?[END_REF] definition of A research conducted by Ronald A. [START_REF] Hites | Global Assessment of Organic Contaminants in Farm Salmon[END_REF], published in Nature, concluded in 2004 that Norwegian salmon was more cancerous than salmon from North America. This information, largely diffused by media, subsequently caused a 40 percent decrease in the sales of Norwegian salmon on the French market, even after this information was corrected by further scientific studies.

While business intelligence is not new [START_REF] Ansoff | Managing strategic surprise by response to weak signal[END_REF][START_REF] Wilensky | Organizational Intelligence: Knowledge and Policy in Government and Industry[END_REF], business intelligence practices have sharply increased since 1990, with the end of the Cold War.

Figure 2. Using Meso-Networks to Raise Short-Run Information Competitiveness of Small Firms

"institution" (in Ülgen 2014) to the reality of the knowledge economy. It holds that "the institution can be defined as systems of established and prevalent social rules that structure social interactions in order to make them compatible with the survival of a given society" (Hodgson in Ülgen 2014, 577). With the DCL, social institutions become "intermediary institutions" (as community of practice, clusters, and cooperatives) that help agents adapt to new environments. These new institutional networks are important in stabilizing the behavior of firms. They also help firms create dynamic "strengths-weaknesses opportunity threat" (SWOT) by transforming threats into opportunities. To build dynamic SWOTs, firms must create some "organized proximity," defined by Alain Rallet and André Torre (2005, 49) as "the capacity to encourage individuals to interact." Organized proximity "helps" individuals work together, but also "constraints" them [START_REF] Rutherford | Toward a History of American Institutional Economics[END_REF]). Here, I analyze how "organized proximity" permits the co-building of intermediary networks. For creating efficient "organized proximity" within a network, two kinds of leaders must be identified: the "economic leader" and "go-between leader." Whereas the former conducts a long-run strategy, the latter diffuses "useful information" to the right agents at the right moment (Table 2).

"Economic Leaders" Conduct the Long-Run Strategy

In organized networks, the systemic interrelations between individuals and organizations are not sufficient to generate innovations. Different works in sociology and management [START_REF] Avolio | The Full Range of Leadership Development: Basic and AdvaNced Manuals[END_REF]Drown and Duguid 2000) explain how recent changes in the world economy transformed regulation from a "hierarchic Table 2. Consequence of the Two World Changes on the Long-Run Specialization and the Short-Run Competitiveness of Firms leader" into a "transactional leader," proposing rewards for individuals in order to increase their efficiency. Since 1990, the "transactional leader" had also changed into a "transformational leader" [START_REF] Karaszewski | The Role of Leadership to Stimulate Pro-Developmental Positive Organisation Potential[END_REF]Muller and Pénin 2006, 7). This "transformational leader" co-builds a common strategy wherein all issues are discussed by all partners in the network [START_REF] Kotter | Leading Change: Why Transformation Effort Fails[END_REF]). Here, I analyze what the minimum factors are for defining an efficient "economic leader." From the studies on different experiments of cluster organizations in different countries [START_REF] Amisse | The Concept of Cluster and Its Two Logics[END_REF][START_REF] Baulant | De l'Europe des nations à l'Europe des régions: l'intérêt de l'approche des clusters pour relancer la coopération européenne[END_REF], I find that the "economic leader" must have two main characteristics. First, the "economic leader" must be a "visionary," capable of seeing into a thirty-year horizon. Second, the "economic leader" must be able to stabilize the cooperation relationships. Thus, the "economic leader" must make sure that all participants of his/her network would really "obtain an advantage" by cooperating out of his/her own motivation. In such a configuration, the people who are involved in networks are looking to cooperate for co-building a new strategy, and, therefore, would not engage in a "free-riding behavior." So, the "economic leader" does not have to monitor all the actions of his/her partners because he/she trusts them. As individuals feel both linked with other indviduals and have autonomy within the network, they are very motivated to co-build the collective strategy.

"Go-Between Leaders" Manage the Short-Run Strategy

The work of the "go-between leader" (the "gate keeper" of Crozier 1977) is also very important to the successful building of meso-networks. This leader constructs a kind of "alchemy" among employees, whereby the whole would become greater than the sum of its parts in the network. Getting different people to work together within the network is the "go-between leader's" role, and it is complementary to the "economic leader's" one. The "go-between leader" knows how to make use of the network's "weak ties" [START_REF] Granovetter | The Strength of Weak Ties[END_REF]. When some partners of his/her network refuse to cooperate, the "go-between leader" must negotiate compromises. In practice, the "go-between leader" must understand how people with different goals and cultures can cooperate. He/she has to formulate some "intermediary goals" that are useful in order to reach the final goal presented by the "economic leader." The "economic leader" builds organized proximity in the network by using the individuals' capacity to build "new norms" together. The "go-between leader" also builds organized proximity in the network by using the capability of individuals to share the "same frame of mind" [START_REF] Rallet | Proximity and Localization[END_REF]. To keep the network active, the "gobetween leader" is proactive in helping people communicate and cooperate (see the "nudge" analysis of [START_REF] Thaler | Nudge, la méthode douce pour inspirer la bonne décision[END_REF] 2010). This function within the network is paramount, even if society seems to give the "go-between leader" less consideration than to the "economic leader." 7

Concluding Notes

Rising globalization and the knowledge economy have led to an uncertain world. In such a world, the firms must use proactive networks to sustain their innovation strategies, as well as their competitiveness practices. To manage these long-and shortrun strategies inside the firms, I discussed two kinds of leaders: "economic leaders" and "go-between leaders." Whereas the "economic leader" conducts the innovation strategy, the "go-between leader" manages the producing and selling strategy. The networks organization of firms induces new regulation policies that are far removed from the "top down hierarchy." Effectively, networks must be flexible enough to be both open to innovations (by monitoring, for example) and closed to internal objectives and cultural habits in order to induce learning outcomes. Managing innovative and selling networks is a hard task. As these networks have powerful leverage in building "virtuous circles" of economic growth, firms have to concentrate their efforts on controlling some "key points" of their network and allowing a degree of liberty to each individual within the network. My analysis stresses the key role of the differentiation process within the meso-networks in innovating and competing. This study reconciles the two opposite approaches in economics: the strategic vision of competition and the participative vision of innovation. Further research should explore these new ways of "thinking the complexity" and "acting efficiently" by taking into account not only economic and quantitative efficiency, but also social and qualitative efficiency. The main objective would be to define a new regime of

In the knowledge economy, the leaders are "hill-climbers on an unknown landscape." The "maverick leader" (i.e., the "economic leader") "discover[s] new land" and the "follower leader" (i.e., the "go -between leader") explores this new land in detail (Muldoon 2013, 125).

The concept of "bounded rationality," created by Herbert Simon and Dominique [START_REF] Foray | L'économie de la connaissance[END_REF], shows how Michel [START_REF] Crozier | L'Acteur et le Système[END_REF] played a "go-between leader" role in diffusing this concept in France.

Figure

  Figure 1. Using Meso-Networks to Co-Build Long-Run Competitive Advantages for Small Firms

The "social provisioning process," as defined by[START_REF] Lee | Competition, Going Enterprise, and Economic Activity[END_REF], stresses the following main qualitative factors in analysing the "inclusive economy": incorporation of caring and wellbeing of agencies; global analysis (economic, political, and social processes); the key role of ethical values; and analysis of the variety of agents who are involved in the production and consumption process.

[START_REF] Kirdina | Institutions and the Importance of Social Control in a Nation's Development[END_REF] analyzes the interactions of "revolution" with "social evolution" in different countries. The "path-dependency evolution" depends on the interaction of "individual" and "communitarian" values in different spheres (economic, political, and ideological).

 [START_REF] Stiglitz | Commission sur la mesure des performances économiques et du progrès social[END_REF]of the "Gross Product Happiness" index, which seems more adapted to the world knowledge economy than the "Gross Domestic Product" index.