Clément Cances
email: cances@ann.jussieu.fr

Nicolas Seguin
email: nicolas.seguin@upmc.fr

Error estimate for Godunov approximation of locally constrained conservation laws

Keywords: Locally constrained scalar conservation laws, monotone finite volume scheme, BV-estimate, error estimate. AMS 35L65, 65M15, 76M12, 90B20

We consider a model of traffic flow with unilateral constraint on the flux introduced by R. M. Colombo and P. Goatin (J. Differ. Equ. 234(2): 2007), for which the convergence of numerical approximation using monotone finite volume schemes has been performed by B. Andreianov et al. (Numer. Math. 115:609-645, 2010). We derive for this problem some new BV-estimate, and make use of it to provide an error estimate for the Godunov approximation of the problem of order h 1/3 that is improved into the optimal order h 1/2 under a reasonable assumption. Numerical experiments are then provided to illustrate the optimality of the result.

     ∂ t u + ∂ x f (u) = 0 for (x, t) ∈ R × R + , u(x, 0) = u 0 (x) for x ∈ R, f (u)(0, t) ≤ F (t) for t ∈ R + , (1)
where f is supposed to be Lipschitz continuous on [0, 1] and bell-shaped, i.e. there exists u ∈ (0, 1) such that

f (0) = f (1) = 0, f ≥ 0, f ′ (u)(u -u) > 0 for a.e. u ∈ [0, 1], (2)
and where the constraint F satisfies 0 ≤ F (t) ≤ f (u), for a.e. t ∈ R + .

This constraint F models toll gates or traffic lights. In the case where the flow is not constrained, i.e. F ≡ f (u), then it is well known that the good notion of solution for the problem [START_REF] Adimurthi | Optimal entropy solutions for conservation laws with discontinuous flux-functions[END_REF] is the notion of entropy solution [START_REF] Oleȋnik | Discontinuous solutions of non-linear differential equations[END_REF][START_REF]′ pert. Spaces BV and quasilinear equations[END_REF][START_REF] Kružkov | First order quasilinear equations with several independent variables[END_REF]. In the case where the constraint becomes active, i.e. F (t) < f (u) on a non negligible set of R + , then it is shown in [START_REF] Colombo | A well posed conservation law with a variable unilateral constraint[END_REF] that a non-classical shock with zero speed can appear at the interface {x = 0}, so that the constraint can be satisfied. It has then been pointed out by B. Andreianov, P. Goatin and N. Seguin [4] that the problem (1) can be immersed in the framework of scalar conservation laws with discontinuous flux functions, that has been widely studied during the last years. Among the numerous references available on this topic, let us mention that Adimurthi, S. Mishra and G.D. Veerappa Gowda [START_REF] Adimurthi | Optimal entropy solutions for conservation laws with discontinuous flux-functions[END_REF] exhibited that such scalar conservation laws with discontinuous flux function admit an infinite number of solutions.

More precisely, we look for solutions of

∂ t u + ∂ x f (u) = 0 for (x, t) ∈ R × R + , u(x, 0) = u 0 (x) for x ∈ R, (3)
that satisfy the usual entropy criterion away from the interface, i.e. such that

∂ t |u -κ| + ∂ x Φ κ (u) ≤ 0, in D ′ ([0, +∞) × R + ), (4)
where Φ κ (u) = sign(u -κ)(f (u) -f (κ)).

(

) 5
We also require the continuity of the flux at {x = 0}, yielding the Rankine-Hugoniot condition

f (u L ) = f (u R ), (6)
where u L and u R respectively denote the one-sided traces of u on {x = 0} from {x < 0} and {x > 0}, i.e.

u L (t) = lim

ǫ→0 + 1 ǫ 0 -ǫ u(x, t)dx, u R (t) = lim ǫ→0 + 1 ǫ ǫ 0 u(x, t)dx. (7)
Note that, following E. Yu. Panov [START_REF] Yu | Existence of strong traces for quasi-solutions of multidimensional conservation laws[END_REF], since the flux function f is non constant on any interval and since the solution u satisfies (4), then the traces u L,R defined by [START_REF] Bachmann | Analysis of a scalar conservation law with a flux function with discontinuous coefficients[END_REF] exist for almost all t > 0.

As it was first noticed in [START_REF] Adimurthi | Optimal entropy solutions for conservation laws with discontinuous flux-functions[END_REF], uniqueness of the solution of ( 3)-( 6) fails since some undercompressive shocks can be generated by the interface {x = 0}, yielding an infinite number of solutions. More precisely, define

U = (A, B) ∈ [0, 1] 2 | f (A) = f (B), A ≥ u and B ≤ u
the set of all the possible stationary undercompressive shocks. For all (A, B) ∈ U, the piecewise constant function

x → A1 1 x<0 (x) + B1 1 x>0 (x) [START_REF] Bouchut | Kružkov's estimates for scalar conservation laws revisited[END_REF] is a steady solution to the problem (3)- [START_REF] Attouch | Variational analysis in Sobolev and BV spaces, volume 6 of MPS/SIAM Series on Optimization[END_REF], being a usual entropy solution in the sense of [START_REF] Kružkov | First order quasilinear equations with several independent variables[END_REF][START_REF] Oleȋnik | Discontinuous solutions of non-linear differential equations[END_REF] if and only if A = B = u. It has been emphasized in [START_REF] Andreianov | A theory of L 1 -dissipative solvers for scalar conservation laws with discontinuous flux[END_REF] that for all (A, B) ∈ U, there exists a unique L 1 -contraction semi-group such that [START_REF] Bouchut | Kružkov's estimates for scalar conservation laws revisited[END_REF] is a solution to the problem. Reciprocally, given a time independent L 1 -contraction semi-group for the problem (3)- [START_REF] Attouch | Variational analysis in Sobolev and BV spaces, volume 6 of MPS/SIAM Series on Optimization[END_REF], there exists a unique (A, B) ∈ U such that [START_REF] Bouchut | Kružkov's estimates for scalar conservation laws revisited[END_REF] is a solution to the problem. Let u be the solution of (3)- [START_REF] Attouch | Variational analysis in Sobolev and BV spaces, volume 6 of MPS/SIAM Series on Optimization[END_REF] belonging to this latter L 1 -contraction semi-group, then its one-sided traces satisfy Φ B (u R (t)) -Φ A (u L (t)) ≤ 0, for a.e. t > 0.

Moreover, it has been pointed out in [START_REF] Andreianov | The Godunov scheme for scalar conservation laws with discontinuous bell-shaped flux functions[END_REF] that

f (u L ) = f (u R ) ≤ f (A) = f (B) for a.e. t > 0,
so that the choice of (A, B) ∈ U enforces a constraint on the flux at the interface. Symmetrically, let F ∈ L ∞ (R + ; [0, f (u)]), then there exists a unique couple (A, B)

∈ (L ∞ (R + ; [0, 1])) 2 such that A(t) ≥ u (resp. B(t) ≤ u), F (t) = f (A(t)) = f (B(t)). (9)
Obviously, (A(t), B(t)) ∈ U for a.e. t > 0.

Definition 1.1 Let F ∈ L ∞ (R + ; [0, f (u)]), and let A, B ∈ L ∞ (R + ) be defined almost everywhere by [START_REF] Bürger | An Engquist-Osher-type scheme for conservation laws with discontinuous flux adapted to flux connections[END_REF], then the subset G * (t) of [0, 1] 2 , defined by: for a.e. t > 0,

G * (t) = (c L , c R ) ∈ [0, 1] 2 | f (c L ) = f (c R ) and Φ B(t) (c R ) -Φ A(t) (c L ) ≤ 0 .
is said to be the L 1 -dissipative (dual) germ corresponding to F .

We refer to [START_REF] Andreianov | A theory of L 1 -dissipative solvers for scalar conservation laws with discontinuous flux[END_REF] for an extensive discussion about the notion of L 1 -dissipative germs and to [START_REF] Andreianov | The Godunov scheme for scalar conservation laws with discontinuous bell-shaped flux functions[END_REF] for a discussion on the correspondence between constraining the flux and choosing an undercompressive shock.

We focus now on the characterization of the relevant solution to [START_REF] Adimurthi | Optimal entropy solutions for conservation laws with discontinuous flux-functions[END_REF].

Definition 1.2 A function u ∈ L ∞ (R × R + ; [0, 1]
) is said to be a solution of the problem (1) if:

1. for all κ ∈ [0, 1], for all ψ ∈ D + (R × R + ) 1 such that ψ(0, •) = 0, +∞ 0 R |u(x, t) -κ|∂ t ψ(x, t)dxdt + R |u 0 (x) -κ|ψ(x, 0)dx + +∞ 0 R Φ κ (u)(x, t)∂ x ψ(x, t)dxdt ≥ 0; (10)
2. for almost every t ∈ R + , the one-sided traces (u L (t), u R (t)) belong to G * (t).

Remark 1.1 Three equivalent notions of solutions to the problem (1), among which the previous one, are proposed in [START_REF] Andreianov | Finite volume schemes for locally constrained conservation laws[END_REF]. Here, we choose to focus on only one of them, which is the only one that we will use in the sequel.

Let us describe now the L1 -dissipative germ G * involved in the problem [START_REF] Adimurthi | Optimal entropy solutions for conservation laws with discontinuous flux-functions[END_REF]. Given a constraint F ∈ L ∞ (R + , [0, f (u)]) and A, B the functions defined by ( 9), then following [START_REF] Andreianov | Finite volume schemes for locally constrained conservation laws[END_REF], the L 1 -dissipative germ G * corresponding to the problem (1), represented in Figure 1, can be split into three parts

G * (t) = G 1 (t) ∪ G 2 (t) ∪ G 3 (t),

where

• G 1 (t) = (A(t), B(t)) corresponds to the unique undercompressive shock allowed at time t > 0;

• G 2 (t) = {(c, c) ∈ [0, 1] 2 | f (c) ≤ F (t)
} corresponds to a continuous solution across the interface and a flux satisfying the constraint; We now state the existence and uniqueness result for the solution u of the problem (1) whose proof is detailed in [START_REF] Colombo | A well posed conservation law with a variable unilateral constraint[END_REF][START_REF] Andreianov | Finite volume schemes for locally constrained conservation laws[END_REF]. The time continuity of the solution u prescribed below is a consequence of the result stated in [START_REF] Cancès | On the time continuity of entropy solutions[END_REF].

• G 3 (t) = {(c L , c R ) ∈ [0, 1] 2 | c L ≤ u, c R ≥ u and f (c L ) = f (c R ) ≤ F (t)} correspond to the compressive stationary shocks satisfying the constraint. u F (t) 0 B(t) A(t) u f (u) 1 uR uL 1 A(t) B(t) B(t) 0 1 A(t) (A(t), B(t))
Theorem 1 ([14, 4]) Let u 0 ∈ L ∞ (R; [0, 1]), and let F ∈ L ∞ (R + ; [0, f (u)]
), then there exists a unique u solution to the problem (1) in the sense of the Definition 1.2, which furthermore can be assumed to belong to

C(R + , L 1 loc (R)). Moreover, if v is another solution corresponding to the initial data v 0 ∈ L ∞ (R; [0, 1]) such that (u 0 -v 0 ) ∈ L 1 (R), then one has, for all t ∈ R + , u(•, t) -v(•, t) L 1 (R) ≤ u 0 -v 0 L 1 (R) .
2 Godunov approximation of the problem and main result

The Godunov approximation

In this section, we introduce the Godunov approximation of the problem [START_REF] Adimurthi | Optimal entropy solutions for conservation laws with discontinuous flux-functions[END_REF]. For the sake of simplicity, we choose to deal with uniform discretizations of R × R + . Nevertheless, note that all the following results can be adapted to the case of non-uniform discretizations.

Let h > 0 the space step, we denote, for all i ∈ Z,

x i = ih, x i+1/2 = (i + 1/2)h and u 0 i+1/2 = 1 h xi+1 xi u 0 (x)dx.
Let k > 0 the time step, we denote by t n = nk (n ∈ N), and by

F n = 1 k t n+1 t n F (t)dt, A n = max{s ∈ [0, 1] | f (s) = F n }, B n = min{s ∈ [0, 1] | f (s) = F n }. (11)
Note that

F n ≤ f (u) (n ∈ N).
We define the constrained Godunov scheme by

u n+1 i+1/2 -u n i+1/2 k h + G n i+1 (u n i+1/2 , u n i+3/2 ) -G n i (u n i-1/2 , u n i+1/2 ) = 0, ( 12
)
where G n i is the Godunov numerical flux through the edge x i , given by

• the classical Godunov numerical flux G if i = 0, i.e. G n i (a, b) = G(a, b) = min (f (min{a, u}), f (max{u, b})) =    min s∈[a,b] f (s) if a ≤ b, max s∈[b,a] f (s) if b ≤ a; (13)
• the constrained Godunov numerical flux G n 0 if i = 0, i.e. G n 0 (a, b) = min (F n , G(a, b)) , (14)
as proposed in [START_REF] Andreianov | Finite volume schemes for locally constrained conservation laws[END_REF].

Proposition 2.1 The constrained Godunov numerical flux [START_REF] Colombo | A well posed conservation law with a variable unilateral constraint[END_REF] coincides with the classical Godunov numerical flux for the constrained problem, i.e. f (U(0; a, b)) where U(x/t; a, b) is the self-similar solution of the constrained Riemann problem (described in [START_REF] Colombo | A well posed conservation law with a variable unilateral constraint[END_REF]).

Proof: Consider u 0 (x) = a1 1 x<0 + b1 1 x>0 . First, solving the Riemann problem without constraint, i.e.

∂ t v + ∂ x f (v) = 0 in R × R + , v |t=0 = a1 1 x<0 + b1 1 x>0 in R
provides that either the solution v is continuous at x = 0 and t > 0, with v(0, t) = c that does not depend on t, or we have an compressive shock f (a) = f (b) and a < b. In any case, the solution is self-similar, yielding that t → f (v)(0, t) is constant on (0, +∞), and is exactly given by the formula [START_REF] Cockburn | An error estimate for finite volume methods for multidimensional conservation laws[END_REF]. Assume first that f (v)(0, t) ≤ F n , then, clearly, v satisfies [START_REF] Cancès | Asymptotic behavior of two-phase flows in heterogeneous porous media for capillarity depending only on space. I. Convergence to the optimal entropy solution[END_REF]. Moreover, (v(0 -, t), v(0 + , t)) either belongs to G 2 (t) or G 3 (t) (that do not depend on time, since we consider here the constraint F ≡ F n ). Therefore, v is the unique solution of the problem (1) for the constant constraint F ≡ F n .

Assume now that f (v(0, t)) > F n , then one deduces from the formula [START_REF] Cockburn | An error estimate for finite volume methods for multidimensional conservation laws[END_REF] and from the fact that f is bell-shaped (2) that a ≥ B n and b ≤ A n . In this case, define u as the solution of two distinct initial boundary value problems

   ∂ t u + ∂ x f (u) = 0 in R -× R + u |t=0 = a in R - u |x=0 = A n in R + ,    ∂ t u + ∂ x f (u) = 0 in R + × R + u |t=0 = b in R + u |x=0 = B n in R + .
Since a ≥ B n (respectively b ≤ A n ), the wave connecting a to A n (resp. B n to b) has a non-positive (resp. non-negative) speed, so that the boundary condition is fulfilled in a strong sense in each case. Clearly, u satisfies [START_REF] Cancès | Asymptotic behavior of two-phase flows in heterogeneous porous media for capillarity depending only on space. I. Convergence to the optimal entropy solution[END_REF], and its traces on the interface belong to G 3 (t), thus u is the unique solution to the problem (1) for the constant constraint F ≡ F n , and the flux at the interface is exactly given by F n .

All along this paper, we assume that the following CFL condition is fulfilled:

2L f k h ≤ 1, (15)
and also that the time step is bounded, let say, without loss of generality,

k ≤ 1. ( 16
)
Definition 2.1 We denote by u h the so-called approximate solution, defined almost everywhere by

u h (x, t) = u n i+1/2 if (x, t) ∈ (x i , x i+1 ) × [t n , t n+1 ).
We now state the L ∞ stability of the scheme. We refer to [START_REF] Andreianov | Finite volume schemes for locally constrained conservation laws[END_REF]Proposition 4.2] for the proof of Proposition 2.2. Proposition 2.2 Under the CFL condition (15), then

0 ≤ u h ≤ 1 a.e. in R × R + .

Approximate traces on the interface

In this section, we seek to introduce, for all n ∈ N, two artificial approximate traces u n L , u n R such that (u n L , u n R ) belongs to the approximate germ G n , defined by

G n = G n 1 ∪ G n 2 with G n 1 = (A n , B n ), G n 2 = {(c, c) ∈ [0, 1] 2 | f (c) < F n }
, and then to derive some properties on them.

Proposition 2.3 For all n ∈ N, there exists (u n L , u n R ) ∈ G n such that G n 0 (u n -1/2 , u n 1/2 ) = G(u n -1/2 , u n L ) = G(u n R , u n 1/2 ) = f (u n L ) = f (u n R ). ( 17
)
In the case where

G(u n -1/2 , u n 1/2 ) < F n , one has either u n L = u n R = u n -1/2 or u n L = u n R = u n 1/2 . Moreover, the case u n L = A n and u n R = B n only occurs when u n 1/2 ≤ A n and u n -1/2 ≥ B n .
Proof: We can prove the above Proposition by a case by case study. For reader's convenience, we drop the index n. Let u -1/2 and u 1/2 belong to [0, 1], then we define by u

⋆ -1/2 ∈ [u, 1] and u ⋆ 1/2 ∈ [0, u] by f (u ⋆ -1/2 ) = f (u -1/2 ), f (u ⋆ 1/2 ) = f (u 1/2 ). 1. Assume that 0 ≤ u -1/2 < B. (a) If u 1/2 ≤ u ⋆ -1/2 , then u L = u R = u -1/2 satisfies (17). (b) If u 1/2 > u ⋆ -1/2 , then u L = u R = u 1/2 satisfies (17). 2. Assume that B ≤ u -1/2 ≤ 1. (a) If 0 ≤ u 1/2 ≤ A, then u L = A and u R = B satisfies (17). (b) If A < u 1/2 ≤ 1, then u L = u R = u 1/2 satisfies (17
u n+1 i+1/2 -u n i+1/2 k h + G(u n i+1/2 , u n i+3/2 ) -G(u n i-1/2 , u n i+1/2 ) = 0, for i / ∈ {-1, 0}, u n+1 -1/2 -u n -1/2 k h + G(u n -1/2 , u n L ) -G(u n -3/2 , u n -1/2 ) = 0, u n+1 1/2 -u n 1/2 k h + G(u n 1/2 , u n 3/2 ) -G(u n R , u n 1/2 ) = 0. ( 18
)
In the sequel, we denote by u L,h and u R,h the functions defined by

u L,h (t) := u n L , u R,h (t) = u n R for t ∈ [t n , t n+1 ). ( 19
)

Convergence of the scheme and error estimate

We state here the following convergence result, which is the main result of [START_REF] Andreianov | Finite volume schemes for locally constrained conservation laws[END_REF].

Theorem 2 ([4]) Let F ∈ L ∞ (R + ; [0, f (u)]), and let u 0 ∈ L ∞ (R; [0, 1]), then, under the CFL condition 2L f k h ≤ 1 -ξ, with ξ ∈ (0, 1), (20)
then the approximate solution u h defined in Definition 2.1 converges in L 1 loc (R × R + ) towards the unique solution u to the problem in the sense of Definition 1.2 as h tends to 0. Remark 2.2 Note that the CFL condition (20) is slightly more restrictive than (15), so that some numerical diffusion stabilizes the scheme. However, we will deduce from this work that, if the data are sufficiently regular (roughly speaking in BV loc ), then the scheme still converges under the weaker CFL condition [START_REF] Eymard | Error estimates for the approximate solutions of a nonlinear hyperbolic equation given by finite volume schemes[END_REF].

Assumption 1

The functions A and B defined in (9) belong to BV loc (R + ).

In order to improve the error estimate, we shall make the following assumption.

Assumption 2 There exists C BV depending only on T such that, for all discretization parameters h, k fulfilling [START_REF] Eymard | Error estimates for the approximate solutions of a nonlinear hyperbolic equation given by finite volume schemes[END_REF], and for all τ ∈ (0, T ),

T -τ 0 (|u L,h (t + τ ) -u L,h (t)| + |u R,h (t + τ ) -u R,h (t)|) dt ≤ C BV .
In the sequel, we denote by ω R the subset of R × R + given by

ω R = {(x, t) ∈ R × R + | |x| ≤ R -L f t}. ( 21
)
Because the solution u propagates with finite speed lower or equal to L f , the restriction to ω R of the solution u depends only on the restriction of

u 0 to [-R, R].
Let us now state the main result of this paper. It relies on two error estimates between the approximate solution u h and the solution u. As usual, it is derived in the BV setting, but due to the (relatively slight) loss of control of the approximate traces (u n L , u n R ) n , the optimal order, that is h 1/2 , is obtained under assumption 2.

Theorem 3 Let u 0 ∈ BV loc (R), let u be the unique solution of the problem (1), and let u h be the approximate solution given by its Godunov approximation. Then, under the CFL condition (15) and Assumption 1, for all R > 0, there exists C depending only u 0 , f , A, B and R such that

ωR |u h (x, t) -u(x, t)|dxdt ≤ Ch 1/3 . (22)
Moreover, if Assumption 2 holds, then for all R > 0, there exists C depending only u 0 , f , A, B, C BV and R such that

ωR |u h (x, t) -u(x, t)|dxdt ≤ Ch 1/2 . ( 23
)
Remark 2.3 In the particular case of a Riemann problem, then u L,h and u R,h are constant w.r.t. time, then Assumption 2 holds for C BV = 0, and the error estimate (23) holds.

The proof of Theorem 3 is based on the doubling variable technique introduced by S. N. Kružkov [START_REF] Kružkov | First order quasilinear equations with several independent variables[END_REF] for proving the uniqueness of the solution of the entropy solution of a multidimensional scalar conservation law, and then used by N. N. Kuznetsov [20] in order to obtain some error estimate for the approximation of scalar conservation laws by monotone finite differences methods.

In order to obtain the convergence rates 1/3 and 1/2 stated in ( 22) and ( 23), we need to show that (i) the exact solution u belongs to BV loc (R × R + ),

(ii) the approximate solution u h is uniformly bounded w.r.t. the discretization in BV loc (R × R + ).

Outline of the paper

We derive in Section 3 a uniform (w.r.t. the space step h) estimate for the local total variation of the approximate solution u h . As a direct consequence, this will yield an estimate on the local total variation of the exact solution u. In section 4, we provide the discrete and continuous entropy inequalities that the approximate solution verifies. Section 5 is devoted to the proof of Theorem 3, following N. N. Kuznetsov [START_REF] Kuznetsov | Accuracy of some approximate methods for computing the weak solutions of a firstorder quasi-linear equation[END_REF]. We illustrate our error estimates by numerical tests in Section 6 and discuss in Section 7 several perspectives which could follow this work.

BV estimates

We first derive BV estimates on the approximate solution. This requires a careful study of the approximate traces, based on an extended definition of the numerical total variation which incorporates the discrete traces. Secondly, we deduce BV estimates on the exact solution.

3.1 BV estimate on the approximate constraint

Lemma 3.1 Under Assumption 1, the functions A h , B h defined respectively by A h (t) = A n , B h (t) = B n if t ∈ [t n , t n+1 ) belong to BV loc (R + )
, and, for all T > 0,

T V [0,T ] (A h -B h ) = T V [0,T ] (A h ) + T V [0,T ] (B h ) ≤ T V [0,T ] (A) + T V [0,T ] (B) + 1. ( 24
)
Proof: Assume first that the functions A, B are smooth, so

F = f (A) = f (B) is Lipschitz continuous.
Then, for all n ∈ N, there exists tn ∈ (t n , t n+1 ) such that F n = F ( tn ). Hence,

A n = A( tn ), B n = B( tn ).
Thus, denoting by N the index of the time step where t N ≤ T < t N +1 , one gets that

T V [0,T ] (A h ) = N -1 n=0 |A n+1 -A n | = N -1 n=0 |A( tn+1 ) -A( tn )| = N -2 n=0 |A( tn+1 ) -A( tn )| + |A( tN ) -A( tN-1 )|. Since |A( tN ) -A( tN-1 )| ≤ 1 -u,
and

N -2 n=0 |A( tn+1 ) -A( tn )| ≤ T V [0,T ] (A), one obtains that T V [0,T ] (A h ) ≤ T V [0,T ] (A) + (1 -u).
Similarly, we can state that

T V [0,T ] (B h ) ≤ T V [0,T ] (B) + u,
so that the estimate (24) holds for smooth functions A, B. Assume now that A, B only belong to BV loc (R + ), then there exists some sequences (A ν ) ν∈N and (B ν ) ν∈N of smooth functions (obtained for example by convolution with smoothing kernels) such that

A ν → A, B ν → B a.e. in R + as ν → ∞, T V [0,T ] (A ν ) ≤ T V [0,T ] (A), T V [0,T ] (B ν ) ≤ T V [0,T ] (B),
and

A ν,h → A h , B ν,h → B h a.e. in R + as ν → ∞.
Then we can pass to the limit and extend [START_REF] Yu | Existence of strong traces for quasi-solutions of multidimensional conservation laws[END_REF] to functions A, B merely in BV loc (R + ). It only remains to check that due to their definition [START_REF] Cancès | On the time continuity of entropy solutions[END_REF] and to the bell-shaped behavior of the function f , the functions n → A n and n → B n have variations of opposite signs, i.e.

A n+1 ≥ A n ⇔ F n+1 ≤ F n ⇔ B n+1 ≤ B n , yielding T V (A h -B h ) = T V (A h ) + T V (B h ).

Space BV estimate on the approximate solution

In the sequel, we introduce a modified total variation, that takes the approximate traces into account:

T V(u h (•, t n )) = i =0 u n i+1/2 -u n i-1/2 + u n -1/2 -u n L + u n 1/2 -u n R ,
while the classical total variation of u h (•, t n ) is given by

T V (u h (•, t n )) = i∈Z u n i+1/2 -u n i-1/2 .
We first state the following technical lemma.

Lemma 3.2 Let (c, d) ∈ [0, 1] 2 such that c ≥ d. Then, for all (a, b) ∈ [d, 1] × [0, c], one has ||c -a| + |d -b| -|a -b|| ≤ c -d. Proof: Denote by Ψ(a, b) = |c -a| + |d -b| -|a -b|.
The proof is performed using a case by case study, summarized in the following table. Note that only 4 cases are used in the proof of Lemma 3.2, but the other cases will be used later. In this table, we denote by a⊤b = max(a, b) and a⊥b = min(a, b).

a ∈ [0, d] a ∈ [d, c] a ∈ [c, 1] b ∈ [0, d] Ψ(a, b) = c + d -2(a⊤b) Ψ(a, b) = c + d -2a Ψ(a, b) = d -c b ∈ [d, c] Ψ(a, b) = c -d Ψ(a, b) = c -d -2(a -b) + Ψ(a, b) = 2b -(d + c) b ∈ [c, 1] Ψ(a, b) = c -d Ψ(a, b) = c -d Ψ(a, b) = 2(a⊥b) -(c + d) (25)
Lemma 3.3 For all t ≥ 0, one has

|T V(u h (•, t)) -T V (u h (•, t))| ≤ 1. Proof: Let t ∈ [t n , t n+1 ). Assume that (u n L , u n R ) ∈ G n 2 , then T V(u h (•, t)) = T V (u h (•, t))
. We now focus on the case where (

u n L , u n R ) = (A n , B n ). As seen in Proposition 2.3, this implies that u n -1/2 ≥ B n and u n 1/2 ≤ A n . Since T V(u h (•, t)) -T V (u h (•, t)) = |u n -1/2 -A n | + |u n 1/2 -B n | -|u n -1/2 -u n 1/2 |, then using the lemma 3.2 with a = u n -1/2 , b = u n 1/2 , c = A n , d = B n provides the result.
Denote by Λ ⊂ N the set of the time step indices where the constraint is saturated, i.e.

p ∈ Λ ⇔ u p -1/2 ≥ B p and u p 1/2 ≤ A p (⇔ G(u p -1/2 , u p 1/2 ) ≥ F p ), by Λ = {p ∈ N | p / ∈ Λ, p + 1 ∈ Λ}, Λ = {p ∈ Λ | p + 1 / ∈ Λ}, • Λ = Λ \ Λ,
and by

Υ = N \ (Λ ∪ Λ) = {p ∈ N | p / ∈ Λ, (p + 1) / ∈ Λ}, so that we have N = Υ ∪ • Λ ∪ Λ ∪ Λ. Lemma 3.4 Let p ∈ Υ, then, under the CFL condition (15), T V(u h (•, t p+1 )) ≤ T V(u h (•, t p )). Proof: Since p / ∈ Λ, then u h (•, t p+1
) is the solution computed by the classical Godunov scheme without constraint. Hence, it follows from classical computations (see e.g. [START_REF] Godlewski | Hyperbolic systems of conservation laws, volume 3/4 of Mathématiques & Applications[END_REF] or Lemma 5.7 in [START_REF] Eymard | Finite volume methods[END_REF]) that

i∈Z |u p+1 i+1/2 -u p+1 i-1/2 | ≤ i∈Z |u p i+1/2 -u p i-1/2 |. (26)
Now, since p ∈ Υ then, thanks to Proposition 2.3, for q ∈ {p, p + 1}, either u q L = u q R = u q -1/2 or u q L = u q R = u q 1/2 . As a consequence,

T V(u h (•, t q )) = i∈Z |u q i+1/2 -u q i-1/2 |. (27)
Lemma 3.4 is a direct consequence of ( 26) and [START_REF] Vila | Convergence and error estimates in finite volume schemes for general multidimensional scalar conservation laws. I. Explicit monotone schemes[END_REF].

Lemma 3.5 Let p ∈ Λ, then, under the CFL condition (15),

T V(u h (•, t p+1 )) ≤ T V(u h (•, t p )) + (A p+1 -B p+1 ).
Proof: Since p / ∈ Λ, then, as previously,

i∈Z |u p+1 i+1/2 -u p+1 i-1/2 | ≤ i∈Z |u p i+1/2 -u p i-1/2 | = T V(u h (•, t p )). Since (u p+1 L , u p+1 R ) = (A p+1 , B p+1 ), then T V(u h (•, t p+1 )) ≤ T V(u h (•, t p )) + R p+1 , (28)
where

R p+1 = |u p+1 -1/2 -A p+1 | + |u p+1 1/2 -B p+1 | -|u p+1 1/2 -u p+1 -1/2 |. (29)
Since p + 1 ∈ Λ, then one has, thanks to Proposition 2.3,

u p+1 -1/2 ≥ B p+1 and u p+1 1/2 ≤ A p+1 .
Hence we can apply Lemma 3.2 to claim that

R p+1 ≤ A p+1 -B p+1 .
We investigate now the cases where the constraint at the time step t p is saturated, i.e. if p ∈

• Λ ∪ Λ. In these cases, (u p L , u p R ) = (A p , B p ), and adapting once again the computations of [START_REF] Eymard | Finite volume methods[END_REF]Lemma 5.7] on the formulation [START_REF] Kröner | A posteriori error estimates for upwind finite volume schemes for nonlinear conservation laws in multidimensions[END_REF], we get that, under the CFL condition [START_REF] Eymard | Error estimates for the approximate solutions of a nonlinear hyperbolic equation given by finite volume schemes[END_REF],

i>0 u p+1 i+1/2 -u p+1 i-1/2 + |u p+1 1/2 -B p | ≤ i>0 u p i+1/2 -u p i-1/2 + |u p 1/2 -B p |. i<0 u p+1 i+1/2 -u p+1 i-1/2 + |u p+1 -1/2 -A p | ≤ i<0 u p i+1/2 -u p i-1/2 + |u p -1/2 -A p |.
As a direct consequence of the inequalities stated above, one has

T V(u h (•, t p+1 )) ≤ T V(u h (•, t p )) + R p+1 , (30)
where

R p+1 = |u p+1 -1/2 -u p+1 L | -|u p+1 -1/2 -A p | + |u p+1 1/2 -u p+1 R | -|u p+1 1/2 -B p |. ( 31
) Lemma 3.6 Let p ∈ • Λ, then, under the CFL condition (15), T V(u h (•, t p+1 )) ≤ T V(u h (•, t p )) + |(A p+1 -B p+1 ) -(A p -B p )|. Proof: Since p + 1 ∈ Λ, one has (u p+1 L , u p+1 R ) = (A p+1 , B p+1 ).
Replacing in (31) leads to, thanks to the triangle inequality,

R p+1 ≤ |A p+1 -A p | + |B p+1 -B p | Since A p+1 ≥ A p ⇔ B p+1 ≤ B p , one obtains that R p+1 ≤ |A p+1 -A p | + |B p+1 -B p | = |(A p+1 -B p+1 ) -(A p -B p )|. ( 32
)
We conclude by using (32) in (30).

Lemma 3.7 Let p ∈ Λ, then, under the CFL condition (15),

T V(u h (•, t p+1 )) ≤ T V(u h (•, t p )) + (B p+1 -A p+1 ) + |A p+1 -A p | + |B p+1 -B p |.
Proof: Since, thanks to Proposition 2.3,

u p+1 L = u p+1 R ∈ {u p+1 -1/2 , u p+1 1/2 }, then the expression (31) turns to R p+1 ≤ R p+1 1 + |A p+1 -A p | + |B p+1 -B p |, with R p+1 1 = |u p+1 1/2 -u p+1 -1/2 | -|u p+1 -1/2 -A p+1 | -|u p+1 1/2 -B p+1 | Since p + 1 /
∈ Λ, we known from the case by case study carried out in Proposition 2.3 that u p+1 -1/2 ≤ B p+1 or u p+1

1/2 ≥ A p+1 . We deduce from the first column and the last line of (25

) that R p+1 1 ≤ B p+1 -A p+1 .
Lemma 3.8 Assume that u 0 ∈ BV(R), then, under the CFL condition (15) and Assumption 1, there exists C depending only on A, B and T (but neither on h nor on k) such that

T V (u h (•, T )) ≤ T V (u 0 ) + C. ( 33
) Proof: Let n ∈ N be such that T ∈ [t n , t n+1 ), then T V(u h (•, T )) = T V(u h (•, 0)) + n-1 p=0 T V(u h (•, t p+1 )) -T V(u h (•, t p )) .
From Lemmata 3.4, 3.5, 3.6 and 3.7, we deduce that

T V(u h (•, t p+1 )) -T V(u h (•, t p )) ≤              0, if p ∈ Υ, A p+1 -B p+1 , if p ∈ Λ, |(A p+1 -B p+1 ) -(A p -B p )|, if p ∈ • Λ, B p+1 -A p+1 + |(A p+1 -B p+1 ) -(A p -B p )| if p ∈ Λ.
Therefore,

T V(u h (•, T )) ≤ T V(u h (•, 0)) + p≤n-1 p∈Λ |(A p+1 -B p+1 ) -(A p -B p )| + p≤n-1 p∈Λ (A p+1 -B p+1 ) + p≤n-1 p∈Λ (B p+1 -A p+1 ). ( 34
)
Since for all p, q ∈ Λ with p < q, there exists r ∈ Λ such that p < r < q, and since

|A k -B k | ≤ 1 for all k, it follows that p≤n-1 p∈Λ (A p -B p ) -(A p+1 -B p+1 ) ≥ (A 0 -B 0 )1 1 Λ (0) + p≤n-1 p∈Λ (A p+1 -B p+1 ) + p≤n-1 p∈Λ (B p+1 -A p+1 ) -1
where the last term -1 is a lower bound to A n -B n . This inequality can also be written as p≤n-1 p∈Λ

(A p+1 -B p+1 ) + p≤n-1 p∈Λ (B p+1 -A p+1 ) ≤ 1 + p≤n-1 p∈Λ |(A p+1 -B p+1 ) -(A p -B p )| (35)
which, taking (35) into account in (34), yields

T V(u h (•, T )) ≤ T V(u h (•, 0)) + 1 + 2 p≤n-1 p∈Λ |(A p+1 -B p+1 ) -(A p -B p )| ≤ T V(u h (•, 0)) + 1 + 2T V [0,T ] (A h -B h ).
We conclude by using Lemmata 3.1 and 3.3.

Proposition 3.9 Let u 0 ∈ BV(R), then, under the CFL condition (15) and Assumption 1, there exists C depending only on u 0 , T, A, B such that, for all ξ > 0,

T 0 R |u h (x + ξ, t) -u h (x, t)|dxdt ≤ Cξ.
Proof: It follows from Lemma 3.8 that the function u h (•, t) has a bounded variation for all t ∈ [0, T ], thus there exists C depending only on u 0 , A, B, T such that

R |u h (x + ξ, t) -u h (x, t)|dx ≤ Cξ.
We conclude by integrating w.r.t. to t ∈ [0, T ].

Time BV estimate on the approximate solution

Lemma 3.10 Let u 0 ∈ BV(R), then, one has

i∈Z |u n+1 i+1/2 -u n i+1/2 | ≤ 2L f k h T V(u h (•, t n )). ( 36
)
Proof: The scheme ( 12) can be rewritten

u n+1 i+1/2 -u n i+1/2 = k h G(u n i-1/2 , u n i+1/2 ) -f (u n i+1/2 ) -G(u n i+1/2 , u n i+3/2 ) -f (u n i+1/2 ) if i / ∈ {-1, 0}, u n+1 -1/2 -u n -1/2 = k h G(u n -3/2 , u n -1/2 ) -f (u n -1/2 ) -G(u n -1/2 , u n L ) -f (u n -1/2 ) , u n+1 1/2 -u n 1/2 = k h G(u n R , u n 1/2 ) -f (u n 1/2 ) -G(u n 1/2 , u n 3/2 ) -f (u n 1/2 ) .
Using the fact that G is L f -Lipschitz continuous w.r.t. each of its variables, we obtain that

|u n+1 i+1/2 -u n i+1/2 | ≤ L f k h |u n i-1/2 -u n i+1/2 | + |u n i+1/2 -u n i+3/2 | if i / ∈ {-1, 0}, ( 37
)
|u n+1 -1/2 -u n -1/2 | ≤ L f k h |u n L -u n -1/2 | + |u n -1/2 -u n -3/2 | , (38)
|u n+1 1/2 -u n 1/2 | ≤ L f k h |u n 1/2 -u n R | + |u n 3/2 -u n 1/2 | . (39)
Summing (37) for i ∈ Z \ {-1, 0} with (38) and ( 39) yields (36).

Proposition 3.11 Let u 0 ∈ BV(R), then, under the CFL condition (15), ( 16) and Assumption 1, for all T > 0, there exists C depending only on A, B, T, u 0 and L f such that, for all τ ∈ (0, T ),

T -τ 0 R |u h (x, t + τ ) -u h (x, t)|dxdt ≤ Cτ. (40)
Proof: One has (with a slight abuse of notation, since ∂ t u h is a bounded Radon measure on R × [0, T ] which is not absolutely continuous w.r.t. to the Lebesgue measure)

R T 0 |∂ t u h (x, t)| dxdt = i∈Z ⌊T /k⌋ n=0 |u n+1 i+1/2 -u n i+1/2 |h.
Then it follows from Lemmata 3.10 and 3.8 that

R T 0 |∂ t u h (x, t)| dxdt ≤ C ⌊T /k⌋ n=0 k.
Using [START_REF] Eymard | Finite volume methods[END_REF], we obtain that

R T 0 |∂ t u h (x, t)| dxdt ≤ C. (41)
The inequality (40) is a classical consequence of the previous estimate (see e.g. [START_REF] Attouch | Variational analysis in Sobolev and BV spaces, volume 6 of MPS/SIAM Series on Optimization[END_REF]).

BV estimates on the exact solution

Letting now h tend to 0, since we know, thanks to Theorem 2, that u h tends to the unique solution u (at least under the more restrictive CFL condition (20)) we obtain the following regularity result on the exact solution u.

Proposition 3.12 Let u be the exact solution to the problem corresponding to u 0 ∈ BV(R). Then, under Assumption 1, then, for all T > 0, u ∈ BV(R × [0, T ]).

Remark 3.1 Because of the finite speed propagation property, the solution u to the problem depends, on the set ω R defined in [START_REF] Lighthill | On kinematic waves. II. A theory of traffic flow on long crowded roads[END_REF], only on the restriction of the initial data u 0 to [-R, R]. So, if u 0 ∈ BV loc (R), extending u 0 by a constant outside of [-R, R] will not affect the solution u on ω R . Thus the Proposition 3.12 can be generalized in the following way. If u 0 ∈ BV loc (R), then, under Assumption 1, u belongs to BV loc (R × R + ).

4 Entropy formulations for the approximate solution

Discrete entropy inequalities

Using the approximate traces u n L , u n R introduced in Section 2.2, the scheme (18) can be rewritten under the form

H u n+1 i+1/2 , u n i+1/2 , u n i-1/2 , u n i+3/2 = 0, ∀i / ∈ {-1, 0}, H u n+1 -1/2 , u n -1/2 , u n -3/2 , u n L = 0, H u n+1 1/2 , u n 1/2 , u n R , u n 3/2 =
0, where, under the CFL condition [START_REF] Eymard | Error estimates for the approximate solutions of a nonlinear hyperbolic equation given by finite volume schemes[END_REF], the function H is non-decreasing w.r.t. its first argument, and nonincreasing w.r.t. its three last arguments. As a consequence, if (a, b, c, d) ∈ [0, 1] 4 satisfies H(a, b, c, d) = 0, then, thanks to the fact that, for all κ ∈ [0, 1], H(κ, κ, κ, κ) = 0, it follows from classical computations (see e.g. [START_REF] Eymard | Finite volume methods[END_REF]) that

H(a⊤κ, b⊤κ, c⊤κ, d⊤κ) -H(a⊥κ, b⊥κ, c⊥κ, d⊥κ) ≤ 0, ( 42
)
where a⊤κ = max(a, κ) and a⊥κ = min(a, κ). In the sequel, we denote by

Φ κ (a, b) = G(a⊤κ, b⊤κ) -G(a⊥κ, b⊥κ).
Note that for all a ∈ [0, 1], for all κ ∈ [0, 1], Φ κ (a, a) = Φ κ (a), where the notation Φ κ (a) has been introduced in ( 5). The following proposition follows from (42).

Proposition 4.1 For all κ ∈ [0, 1], one has

|u n+1 i+1/2 -κ| -|u n i+1/2 -κ| k h + Φ κ (u n i+1/2 , u n i+3/2 ) -Φ κ (u n i-1/2 , u n i+1/2 ) ≤ 0, ∀i ∈ Z \ {-1, 0}, (43) |u n+1 -1/2 -κ| -|u n -1/2 -κ| k h + Φ κ (u n -1/2 , u n L ) -Φ κ (u n -3/2 , u n -1/2 ) ≤ 0, ( 44
)
|u n+1 1/2 -κ| -|u n 1/2 -κ| k h + Φ κ (u n 1/2 , u n 3/2 ) -Φ κ (u n R , u n 1/2 ) ≤ 0. ( 45
) Lemma 4.2 For all κ ∈ [0, 1], Φ κ (u n -1/2 , u n L ) -Φ κ (u n L ) ≥ 0, ( 46
) Φ κ (u n R ) -Φ κ (u n R , u n 1/2 ) ≥ 0. ( 47
)
Proof: We only prove (46), since the proof of (47) is similar. Here again, for readers convenience, we drop the index n. We denote by

I(a, b) the interval [a, b] if a ≤ b and [b, a] otherwise. Firstly, if κ / ∈ I(u -1/2 , u L ), then, using that G(u -1/2 , u L ) = f (u L ), one has Φ κ (u -1/2 , u L ) = Φ κ (u L ).
Consider now the case where

κ ∈ I(u -1/2 , u L ). Since G(u -1/2 , u L ) = f (u L ), the function a → G(a, u L ) is constant on I(u -1/2 , u L ). Assume that u -1/2 ≥ u L , then Φ κ (u -1/2 , u L ) = G(u -1/2 , κ) -G(κ, u L ) = G(u -1/2 , κ) -f (u L ).
Since G is non-increasing w.r.t. its second argument, G(u -1/2 , κ) ≥ f (κ), hence one has

Φ κ (u -1/2 , u L ) ≥ f (κ) -f (u L ) = Φ κ (u L ). Similarly, if u -1/2 ≤ u L , one obtains Φ κ (u -1/2 , u L ) ≥ f (u L ) -f (κ) = Φ κ (u L ).
We now state the straightforward corollary, obtained by subtracting ( 46) to ( 44) and ( 47) to (45).

Corollary 4.3 For all κ ∈ [0, 1], |u n+1 -1/2 -κ| -|u n -1/2 -κ| k h + Φ κ (u n L ) -Φ κ (u n -3/2 , u n -1/2 ) ≤ 0, ( 48
)
|u n+1 1/2 -κ| -|u n 1/2 -κ| k h + Φ κ (u n 1/2 , u n 3/2 ) -Φ κ (u n R ) ≤ 0. (49)

Continuous entropy inequalities

For a C 1 (R × R + ; R) function ϕ, we denote by

∇ϕ = ∂ t ϕ ∞ + ∂ x ϕ ∞ .
Recall that when ϕ is compactly supported, i.e. if ϕ ∈ C 1 c ((-R, R) × [0, T )), then there exists C depending only on R, T such that ϕ ∞ ≤ C ∇ϕ . R+ R-

|u h -κ|∂ t ϕdxdt + R- |u 0 -κ|ϕ(•, 0)dx + R+ R- Φ κ (u h )∂ x ϕdxdt - R+ Φ κ (u L,h )ϕ(0, •)dt ≥ -C ∇ϕ h. ( 50
) R+ R+ |u h -κ|∂ t ϕdxdt + R+ |u 0 -κ|ϕ(•, 0)dx + R+ R+ Φ κ (u h )∂ x ϕdxdt + R+ Φ κ (u R,h )ϕ(0, •)dt ≥ -C ∇ϕ h. ( 51
)
Proof: We only prove (50) since the proof of (51) is similar. Let ϕ ∈ D + ((-R, R) × [0, T )), we denote by

ϕ n i = ϕ(x i , t n ), ϕ n i+1/2 = ϕ(x i+1/2 , t n ), ∀i ∈ Z, ∀n ∈ N.
Multiplying equations (43) by kϕ n+1 i+1/2 and (48) by kϕ n+1 -1/2 , then summing on i < -1, one obtains after reorganization of the sums,

T 1 + T 2 + T 3 + T 4 ≥ 0, (52)
with

T 1 = n∈N i<0 |u n i+1/2 -κ|(ϕ n+1 i+1/2 -ϕ n i+1/2 )h + i<0 |u 0 i+1/2 -κ|ϕ 0 i+1/2 h, T 2 = n∈N k i<0 Φ κ (u n i-1/2 , u n i+1/2 )(ϕ n+1 i+1/2 -ϕ n+1 i-1/2 )h, T 3 = - n∈N kΦ κ (u n L )ϕ n+1 0 , T 4 = - n∈N kΦ κ (u n L )(ϕ n+1 i-1/2 -ϕ n+1 0 ).
Firstly, it is easy to check that

|T 4 | ≤ C ∇ϕ h, (53)
and that

T 3 + R+ Φ κ (u L,h )ϕ(0, •)dt ≤ C ∇ϕ h, (54)
It follows from Propositions 3.9 and 3.11 (we use here classical computations that we can deduce for example from [START_REF] Chainais-Hillairet | Finite volume schemes for a nonlinear hyperbolic equation. Convergence towards the entropy solution and error estimate[END_REF]) and the CFL condition (15) that

T 1 - R+ R- |u h -κ|∂ t ϕdxdt - R- |u 0 -κ|ϕ(•, 0)dx ≤ Ch ∇ϕ , (55)
T 2 - R+ R- Φ κ (u h )∂ x ϕdxdt ≤ Ch ∇ϕ . ( 56
)
Then (50) follows from ( 52)-( 56).

As a direct consequence of Lemma 4.4, following the idea of R. Eymard et al. [START_REF] Eymard | Error estimates for the approximate solutions of a nonlinear hyperbolic equation given by finite volume schemes[END_REF], exploited by F. Bouchut and B. Perthame [START_REF] Bouchut | Kružkov's estimates for scalar conservation laws revisited[END_REF], we can state the following proposition.

L,h , µ R,h belonging to (C c (R × R + )) ′ such that there exists C depending only on R, L f µ L,h (ω R ) ≤ Ch, µ R,h (ω R ) ≤ Ch, ( 57
)
and such that, for all ϕ

∈ D + (R × R + ), one has R+ R- |u h -κ|∂ t ϕdxdt + R- |u 0 -κ|ϕ(•, 0)dx + R+ R- Φ κ (u h )∂ x ϕdxdt - R+ Φ κ (u L,h )ϕ(0, •)dt ≥ -µ L,h , |∂ t ϕ| + |∂ x ϕ| , (58)
R+ R+ |u h -κ|∂ t ϕdxdt + R+ |u 0 -κ|ϕ(•, 0)dx + R+ R+ Φ κ (u h )∂ x ϕdxdt + R+ Φ κ (u R,h )ϕ(0, •)dt ≥ -µ R,h , |∂ t ϕ| + |∂ x ϕ| . (59)
In section 5, we will use the doubling variable technique introduced by S. N. Krǔzkov [START_REF] Kružkov | First order quasilinear equations with several independent variables[END_REF] and adapted to this frame by N. N. Kuznetsov [START_REF] Kuznetsov | Accuracy of some approximate methods for computing the weak solutions of a firstorder quasi-linear equation[END_REF]. For this reason, we will assume that the exact solution u depends on the variable (y, s) instead of (x, t). Since u admits strong traces u L , u R ∈ L ∞ (R + ; [0, 1]) on {y = 0} × R + (either thanks to Proposition 3.12 or to [START_REF] Yu | Existence of strong traces for quasi-solutions of multidimensional conservation laws[END_REF]), then u satisfies the following entropy inequalities :

∀ψ ∈ D + (R × R + ), R+ R- |u -κ|∂ s ψdyds + R- |u 0 -κ|ψ(•, 0)dy + R+ R- Φ κ (u)∂ y ψdyds - R+ Φ κ (u L )ψ(0, •)ds ≥ 0, ( 60
)
R+ R+ |u -κ|∂ s ψdyds + R+ |u 0 -κ|ψ(•, 0)dy + R+ R+ Φ κ (u)∂ y ψdyds + R+ Φ κ (u R )ψ(0, •)ds ≥ 0. ( 61
)
5 Proof of Theorem 3

As mentioned before, the proof of the error estimates is based on the doubling variable technique, introduced by S. N. Kružkov [START_REF] Kružkov | First order quasilinear equations with several independent variables[END_REF] for proving the uniqueness of the entropy solution to a multidimensional scalar conservation law and later adapted by N. N. Kuznetsov [START_REF] Kuznetsov | Accuracy of some approximate methods for computing the weak solutions of a firstorder quasi-linear equation[END_REF] to derive error estimates on the solutions provided by monotone finite difference schemes. First of all, we need to introduce approximation of the unit.

Approximation of the unit

Because of the presence of the interface {x = 0}, we need to introduce a family of non-even smoothing kernels (ρ ǫ ) ǫ>0 . It is built as follows. Let ρ ∈ D + (R) such that supp(ρ) ⊂ [0, 1], R ρ(a)da = 1 and such that (x -1/2)ρ ′ (x) ≤ 0. Let ǫ ∈ (0, 1], we denote by ρ ǫ (x) = 1 ǫ ρ x ǫ , so that supp(ρ ǫ ) ⊂ [0, ǫ], and

R |ρ ′ ǫ (a)|da = 2 ǫ ρ(1/2). (62)
5.2 The case where F h ≡ F

In this section, we first assume that for almost all t ∈ [t n , t n+1 ), F (t) = F n , and thus (u

L,h (t), u R,h (t)) ∈ G * (t). Let ϕ ∈ D + ((-R, R) × [0, T )), then we define the functions ξ L , ξ R by ξ L (x, t, y, s) = ϕ(x, t)ρ ǫ (x -y)ρ η (s -t), ξ R (x, t, y, s) = ϕ(x, t)ρ ǫ (y -x)ρ η (s -t),
for some ǫ, η > 0 to be fixed later, and where ρ ǫ (or ρ η ) is the approximation of the unit introduced in Section 5.1. The functions ξ L and ξ R are built so that

ξ L (x, t, y, 0) = ξ R (x, t, y, 0) = 0, ∀(x, y, t) ∈ R 2 × R + , (63)
ξ L (x, t, 0, s) = 0, ∀(x, t, s) ∈ R -× (R + ) 2 , ξ R (x, t, 0, s) = 0, ∀(x, t, s) ∈ (R + ) 3 . (64)
Let us choose κ = u(y, s) in ( 58) and integrate on R -× R + w.r.t. (y, s), and κ = u h (x, t) in (60) and integrate on R -× R + w.r.t. (x, t), and then sum both contributions. This provides

D L 1 + D L 2 + D L 3 + D L 4 ≥ D L 5 , (65)
where

D L 1 = R+ R-R+ R- |u h (x, t) -u(y, s)|∂ t ϕ(x, t)ρ ǫ (x -y)ρ η (s -t)dxdtdyds, D L 2 = R+ R-R- |u 0 (x) -u(y, s)|ϕ(x, 0)ρ ǫ (x -y)ρ η (s)dxdyds, D L 3 = R+ R-R+ R- Φ u(y,s) (u h (x, t))∂ x ϕ(x, t)ρ ǫ (x -y)ρ η (s -t)dxdtdyds, D L 4 = - R+ R-R+ Φ u(y,s) (u L,h (t))ϕ(0, t)ρ ǫ (-y)ρ η (s -t)dtdyds, D L 5 = - R+ R- µ L,h , (|∂ t ϕ| + |∂ x ϕ|) ρ ǫ (• -y)ρ η (s -•) dyds - R+ R- µ L,h , ϕ |ρ ′ ǫ (• -y)|ρ η (s -•) + ρ ǫ (• -y)|ρ ′ η (s -•)| dyds
Among the above terms, only D L 4 is original in the sense that its treatment has not already been performed in the already mentioned works [START_REF] Kuznetsov | Accuracy of some approximate methods for computing the weak solutions of a firstorder quasi-linear equation[END_REF][START_REF] Godlewski | Hyperbolic systems of conservation laws, volume 3/4 of Mathématiques & Applications[END_REF][START_REF] Eymard | Finite volume methods[END_REF]. Let us first recall the classical results concerning the other terms.

Concerning the term D L 5 , it follows from Fubini-Tonelli theorem and from estimate (57) that

R+ R- µ L,h , (|∂ t ϕ| + |∂ x ϕ|) ρ ǫ (• -y)ρ η (s -•) dyds ≤ Ch ∇ϕ .
On the other hand, thanks to (62), one has

R+ R- µ L,h , ϕ |ρ ′ ǫ (•, -y)|ρ η (s -•) + ρ ǫ (• -y)|ρ ′ η (s -•)| dyds ≤ C h ǫ + h η ϕ ∞ , hence D L 5 ≥ -C h + h ǫ + h η ∇ϕ . (66)
Let us now consider the term D 1 , for which one has

D L 1 ≤ D L 1,1 + D L 1,2 , (67)
where

D L 1,1 = R+ R- |u h (x, t) -u(x, t)|∂ t ϕ(x, t)dxdt, D L 1,2 = R+ R-R+ R- |u(x, t) -u(y, s)||∂ t ϕ(x, t)|ρ ǫ (x -y)ρ η (s -t)dxdtdyds.
Using the fact that u belongs to BV(suppϕ) (cf. Proposition 3.12), one thus obtains that

D L 1,2 ≤ C(ǫ + η) ∇ϕ , (68)
where C only depends on suppϕ, u 0 , f , A, B.

In order to estimate D L 2 , we mimic the method proposed in [START_REF] Eymard | Error estimates for the approximate solutions of a nonlinear hyperbolic equation given by finite volume schemes[END_REF][START_REF] Chainais-Hillairet | Finite volume schemes for a nonlinear hyperbolic equation. Convergence towards the entropy solution and error estimate[END_REF][START_REF] Ohlberger | Error estimate for the approximation of nonlinear conservation laws on bounded domains by the finite volume method[END_REF]. Therefore, we choose

(y, s) → ψ(x, t, y, s) = ϕ(x, 0)ρ ǫ (x -y) ∞ s ρ η (τ )dτ
as test function in (60) for κ = u 0 (x) and integrate w.r.t. x ∈ R -. This provides that

D L 2 ≤ D L 2,1 + D L 2,2 + D L 2,3 , (69)
where

D L 2,1 = R-R- |u 0 (y) -u 0 (x)|ϕ(x, 0)ρ ǫ (x -y)dxdy, D L 2,2 = - R-R+ R- Φ u0(y) (u(y, s))ϕ(x, 0)ρ ′ ǫ (x -y) ∞ s ρ η (τ )dτ dydsdx. D L 2,3 = - R-R+ R- Φ u0(x) (u(y, s)) -Φ u0(y) (u(y, s)) ϕ(x, 0)ρ ′ ǫ (x -y) ∞ s ρ η (τ )dτ dydsdx.
Since u 0 ∈ BV(R -), one has

D L 2,1 ≤ Cǫ ϕ ∞ . ( 70
) Recall that b → Φ b (a) is L f -Lipschitz continuous for all a ∈ [0, 1]; besides, supp s → ∞ s ρ η (τ )dτ ⊂ [0, η] and 0 ≤ ∞ s ρ η (τ )dτ ≤ 1, which gives |D L 2,3 | ≤ L f η ϕ ∞ R-R- |u 0 (x) -u 0 (y)||ρ ′ ǫ (x -y)|dxdy. Now, noting that a → ǫ|ρ ′ ǫ (a)| 2ρ(1/2)
is an approximation of the unit leads to

|D L 2,3 | ≤ Cη ϕ ∞ . (71)
Integrating D L 2,2 by parts w.r.t. the variable x provides

D L 2,2 = R-R+ R- Φ u0(y) (u(y, s))∂ x ϕ(x, 0)ρ ǫ (x -y) ∞ s ρ η (τ )dτ dydsdx + R+ R- Φ u0(y) (u(y, s))ϕ(0, 0)ρ ǫ (-y) ∞ s ρ η (τ )dτ dyds.
Using again that supp s → ∞ s ρ η (τ )dτ ⊂ [0, η] and 0 ≤ ∞ s ρ η (τ )dτ ≤ 1, one obtains that

|D L 2,2 | ≤ Cη( ϕ ∞ + ∇ϕ ). ( 72
)
Concerning D L 3 , one has D L 3 ≤ D L 3,1 + D L 3,2 , (73)
with

D L 3,1 = R+ R- Φ u(x,t) (u h (x, t))∂ x ϕ(x, t)dxdt, D L 3,2 = L f R+ R-R+ R- |u(x, t) -u(y, s)||∂ x ϕ(x, t)|ρ ǫ (x -y)ρ η (s -t)dxdtdyds.
Therefore, using again that u ∈ BV loc (R × R + ), we deduce that

D L 3,2 ≤ C(ǫ + η) ∇ϕ . ( 74
)
The treatment of D L 4 is quite similar to the treatment of D L 2 . Indeed, choosing

(y, s) → ϕ(0, t)ρ η (s -t) ∞ -y
ρ ǫ (a)da as test function in (60) and integrating for t ∈ R + yield

D L 4 ≤ D L 4,1 + D L 4,2 , (75)
where

D L 4,1 = - R+ R+ Φ uL(s) (u L,h (t))ϕ(0, t)ρ η (s -t)dtds, D L 4,2 = R+ R+ R- |u(y, s) -u L,h (t)|ϕ(0, t)ρ ′ η (s -t) ∞ -y ρ ǫ (a)dadydsdt.
We deduce from the triangle inequality that

D L 4,2 ≤ D L 4,3 + D L 4,4 , (76)
where

D L 4,3 = R+ R+ R- |u(y, s) -u L,h (s)|ϕ(0, t)ρ ′ η (s -t) ∞ -y ρ ǫ (a)dadydsdt, D L 4,4 = ǫ R+ R+ |u L,h (t) -u L,h (s)||ρ ′ η (s -t)|dsdt.
Integrating D L 4,3 by parts w.r.t. the variable t leads to

D L 4,3 = R+ R- |u(y, s) -u L,h (s)| ∞ -y ρ ǫ (a)da ϕ(0, 0)ρ η (s) + ∞ 0 ∂ t ϕ(0, t)ρ η (s -t)dt dyds.
Since 0 ≤ u(y, s), u L,h (t) ≤ 1, since, y → ∞ -y ρ ǫ (a)da is compactly supported in [-ǫ, 0] and bounded by 1, we obtain that

D L 4,3 ≤ Cǫ ( ϕ ∞ + ∇ϕ ) . (77)
We provide now two estimates for D L 4,4 according to the regularity of t → u L,h (t).

• First, we do not assume that Assumption 2 holds, then, using the fact that 0 ≤ u L,h ≤ 1 a.e. in R + , one has

D L 4,4 ≤ C ǫ η ∇ϕ . (78)
• Assume now that Assumption 2 holds, then one obtains that

D L 4,4 ≤ 2ǫρ(1/2)C BV ∇ϕ . ( 79
)
Thus it follows from (65)-( 79) that

R+ R- |u h (x, t) -u(x, t)|∂ t ϕ(x, t)dxdt + R+ R- Φ u(x,t) (u h (x, t))∂ x ϕ(x, t)dxdt - R+ R+ Φ uL(s) (u L,h (t))ϕ(0, t)ρ η (s -t)dtds ≥ -C ∇ϕ Θ(h, ǫ, η), (80)
where

Θ(h, ǫ, η) =        h + ǫ + η + h ǫ + h η if
+ R+ R- Φ u(x,t) (u h (x, t))∂ x ϕ(x, t)dxdt + R η,h (ϕ) ≥ -C ∇ϕ Θ(h, ǫ, η), (83)
where

R η,h (ϕ) = R+ R+ Φ uR(s) (u R,h (t)) -Φ uL(s) (u L,h (t)) ϕ(0, t)ρ η (s -t)dtds.
(c L , c R ) ∈ G * (t), for all (κ L , κ R ) ∈ [0, 1] 2 , Φ κR (c R ) -Φ κL (c L ) ≤ L f dist 1 ((κ L , κ R ), G * (t)) , where dist 1 ((κ L , κ R ), G * (t)) = min (aL,aR)∈G * (t) (|κ L -a L | -|κ R -a R |) . Proof: Let (a L , a R ) ∈ G * (t), then, thanks to the definition of G * (t), one has Φ aR (c R ) -Φ aL (c L ) ≤ 0. Now, since κ → Φ κ (s) is L f -Lipschitz continuous for all s ∈ [0, 1], we obtain that Φ κR (c R ) -Φ κL (c L ) ≤ L f (|κ L -a L | -|κ R -a R |) .
Since G * (t) is closed in [0, 1] 2 , the above relation thus still holds for the minimum (a L , a R ) ∈ G * (t).

Lemma 5.2 There exists C depending only on f, T, A, B such that

R η,h (ψ) ≤ Cη ψ ∞ , ∀ψ ∈ C c ([0, T ); R).
Proof: Using the fact that Φ κ (u) = Φ u (κ), it follows from Lemma 5.1 that

Φ uR(s) (u R,h (t)) -Φ uL(s) (u L,h (t)) ≤ L f max dist 1 ((u L (s), u R (s)), G * (t)) , dist 1 ((u L,h (t), u R,h (t)), G * (s)) . Now, its appears clearly that if (u L (s), u R (s)) ∈ G * (t) or (u L,h (t), u R,h (t)) ∈ G * (s), then Φ uR(s) (u R,h (t)) -Φ uL(s) (u L,h (t)) ≤ 0.
Assume now that (u L (s), u R (s)) / ∈ G * (t) and (u L,h (t), u R,h (t)) / ∈ G * (s). This implies that either

F (t) > F (s) and (u L (s), u R (s)) = (A(s), B(s)), (84)
or

F (t) < F (s) and (u L,h (t), u R,h (t)) = (A(t), B(t)). (85)
In the first case (84), one has

dist 1 ((u L (s), u R (s)); G * (t)) ≤ |A(t) -A(s)| + |B(t) -B(s)|,
while in the second case (85), one has

dist 1 ((u L,h (t), u R,h (t)); G * (s)) ≤ |A(t) -A(s)| + |B(t) -B(s)|. Hence, Φ uR(s) (u R,h (t)) -Φ uL(s) (u L,h (t)) ≤ L f (|A(t) -A(s)| + |B(t) -B(s)|) . (86) Now, for ψ ∈ C c ([0, T ); R), recalling that supp(ρ η ) ⊂ [0, η], one has R η,h (ψ) ≤ L f ψ ∞ sup τ ∈[0,η] T 0 |A(t + τ ) -A(t)|dt + sup τ ∈[0,η] T 0 |B(t + τ ) -B(t)|dt , thus we obtain R η,h (ψ) ≤ L f T V [0,T +1] (A -B)η ψ ∞ . Using Lemma 5.2 in (83) provides that R+ R |u h (x, t) -u(x, t)|∂ t ϕ(x, t)dxdt + R+ R Φ u(x,t) (u h (x, t))∂ x ϕdxdt ≤ C ∇ϕ Θ(h, ǫ, η). (87)
In order to conclude the proof of Theorem 3, it only remains to choose a convenient ϕ, that is

ϕ(x, t) =    ζ(|x| -L f t) T -t T if (x, t) ∈ R × [0, T ], 0 if t ≥ T. where ζ(r) = max (0, min(1, R + 1 -r)) , ∀r ∈ R + ,
and to notice that choosing ǫ = η = h 1/2 (under Assumption 2) or ǫ = h 2/3 and η = h 1/3 (general case) provides

min ǫ,η Θ(h, ǫ, η) ≤ Ch 1/2 if Assumption 2 holds,
Ch 1/3 otherwise.

The general case

Denote by ũ the unique solution to the problem corresponding to the constraint F h . Then it has been proven previously that

ωR |u h -ũ|dxdt ≤ Ch α with α ∈ {1/2, 1/3}.
In order to achieve the proof of Theorem 3, it only remains to show that

ωR |u -ũ|dxdt ≤ Ch 1/2 .
In fact, one has a better estimate, thanks to the following Proposition, proved in Appendix of [START_REF] Andreianov | Finite volume schemes for locally constrained conservation laws[END_REF].

Proposition 5.3 ([4]) Let F, F ∈ L ∞ (R + ; [0, f (u)]
), and let u, ȗ be the solutions corresponding respectively to the constraint F, F and to a similar initial data u 0 . Then,

T 0 R |u -ȗ|dxdt ≤ 2 T 0 |F -F |dt.
Since A is supposed to belong to BV(0, T ), then F = f (A) also belongs to BV(0, T ). As a consequence, there exists C depending only on A, f, T such that

F -F h L 1 (0,T ) ≤ Ch.
We deduce from the above estimate and from Proposition 5.3 the following corollary that achieves the proof of Theorem 3.

Numerical illustration

We now present some numerical simulations in order to illustrate the error estimate [START_REF] Ohlberger | Error estimate for the approximation of nonlinear conservation laws on bounded domains by the finite volume method[END_REF]. Two conservation laws are investigated: the first one is based on the flux (called the hat flux in the following)

f (u) = 1/2 -|u -1/2|
which has the particularity of having linear two parts and the second one is based on the flux (called the GNL flux -as genuinely non linear -in the following)

f (u) = u(1 -u)
which is strictly concave. While the present work is devoted to the analysis of the Godunov scheme, we also present the results obtained with the Rusanov scheme:

G(a, b) = f (a) + f (b) 2 - max(|f ′ (a)|, |f ′ (b)|) 2 (b -a)
and the constraint is still handled using the trick [START_REF] Colombo | A well posed conservation law with a variable unilateral constraint[END_REF]. The initial data for the test case is u(x, 0) = 0.4 if -1/2 ≤ x < 0, 0.5 if 0 ≤ x ≤ 1/2, and the final time is 0.3. For each flux, the constraint is set to F = 0.2 and is activated (see Figure 2, left). For the hat flux, the solution is composed of a left-going shock wave, a nonclassical stationary shock and a right-going linear wave. For the GNL flux, the solution is composed of a left-going shock wave, a nonclassical stationary shock and a right-going shock wave. The rates of convergence are displayed in Figure 2, right. They are the same for both numerical schemes, which let us think that our result should be extended for any monotone numerical scheme. For the hat flux, the measured rate is 1/2 (and therefore it attests the optimality of our result) while the measured rate is 1 for the GNL flux. Note that in the latter case, it means that the constraint does not alter the classical rate of convergence.

Concluding remarks 7.1 A posteriori error estimate

As noticed by D. Kröner and M. Ohlberger [START_REF] Kröner | A posteriori error estimates for upwind finite volume schemes for nonlinear conservation laws in multidimensions[END_REF], the doubling variable approach used for obtaining error estimates provides a posteriori estimators, i.e. that for all compact subset K of R × R + , there exists η K depending only f, K, u 0 , A, B (but not on the exact solution u) such that K |u h (x, t) -u(x, t)|dxdt ≤ η K (u h ). Since the right-hand side in the above estimate is fully computable, this permits the localization of the error, and an adaptive mesh refinement strategy (we refer to [START_REF] Kröner | A posteriori error estimates for upwind finite volume schemes for nonlinear conservation laws in multidimensions[END_REF] for more details on both the derivation of the a posteriori estimator and the mesh refinement algorithm). As a consequence, our estimates can be used to develop a posteriori estimators for constrained conservation laws.

Comments on the optimality of the result

The order h 1/2 is optimal in the sense that it can be recovered in some particular cases. Indeed, choosing f (u) = 1/2 -|u -1/2|, F ≡ 1/2 (this means that A ≡ B ≡ 1/2, so that the constraint is always inactive), and u 0 in BV(R) such that 0 ≤ u 0 ≤ 1/2. Then the problem turns to be the standard linear equation ∂ t u + ∂ x u = 0, and the Godunov scheme becomes the upwind scheme. It is well known that in this case, the error behaves as h 1/2 , as illustrated in Figure 2.

In the case where f is uniformly concave, the numerical experiments provide an error of order h.

The case of discontinuous flux function

Consider the case of a scalar conservation law with discontinuous flux function, i.e.

∂ t u + ∂ x f (x, u) = 0, where f (x, u) = f L (u) if x < 0 and f (x, u) = f R (u) if x > 0, with f L , f R bell-shaped reaching their maximum respectively in u L ,u R . As pointed out by Adimurthi and Veerappa Gowda [START_REF] Adimurthi | Conservation law with discontinuous flux[END_REF], an infinite number of L 1 -contractive semi-groups can be built for such an equation, and a criterion has to be taken into account in order to select one. We refer to the recent contributions of R. Bürger et al. [START_REF] Bürger | An Engquist-Osher-type scheme for conservation laws with discontinuous flux adapted to flux connections[END_REF], B. Andreianov et al. [START_REF] Andreianov | A theory of L 1 -dissipative solvers for scalar conservation laws with discontinuous flux[END_REF] and references therein for an overview of this topic, and in particular to the resolution of the Riemann problem arising at the interface, thanks to which we can define the Godunov scheme, and its approximate solution u h . In the case where f L = f R , no BV estimate is available on u (neither on u h ), but we can prove that the Temple function (x, t) → Φ u (u(x, t), x) :=    sign(u(x, t) -u L )(f L (u(x, t)) -f L (u L )) if x < 0, sign(u(x, t) -u R )(f R (u(x, t)) -f R (u R )) if x > 0 belong to BV loc (R × R + ) (see e.g. [START_REF] Seguin | Analysis and approximation of a scalar conservation law with a flux function with discontinuous coefficients[END_REF][START_REF] Bachmann | Analysis of a scalar conservation law with a flux function with discontinuous coefficients[END_REF][START_REF] Cancès | Asymptotic behavior of two-phase flows in heterogeneous porous media for capillarity depending only on space. I. Convergence to the optimal entropy solution[END_REF]). By the use of numerical diffusion introduced by the scheme (see [START_REF] Cockburn | An error estimate for finite volume methods for multidimensional conservation laws[END_REF][START_REF] Vila | Convergence and error estimates in finite volume schemes for general multidimensional scalar conservation laws. I. Explicit monotone schemes[END_REF][START_REF] Eymard | Error estimates for the approximate solutions of a nonlinear hyperbolic equation given by finite volume schemes[END_REF][START_REF] Chainais-Hillairet | Finite volume schemes for a nonlinear hyperbolic equation. Convergence towards the entropy solution and error estimate[END_REF][START_REF] Ohlberger | Error estimate for the approximation of nonlinear conservation laws on bounded domains by the finite volume method[END_REF]), it is still possible to derive an error estimate. Indeed, all the tools introduced in the paper, excepted in Section 3, can be adapted to the case of discontinuous flux functions. Nevertheless, the theoretical convergence speed will depend on the continuity modulus of the function (Φ u ) -1 , and will be furthermore degraded by the fact that no strong BV-estimate is available on the exact solution itself.

Figure 1 :

 1 Figure 1: Left: graphical representation of the values A(t) and B(t) corresponding to the constraint F (t). Right: graphical representation of the corresponding L 1 -dissipative germ G * (t).

Lemma 4 . 4

 44 Let T > 0, and let ϕ ∈ D + ((-R, R) × [0, T )), then there exists C depending only on u 0 , f , A, B, R and T such that, for all κ ∈ [0, 1],

Proposition 4 . 5

 45 There exist positive local Radon measures µ

Lemma 5 . 1

 51 Let t > 0 and let G * (t) be the L 1 -dissipative germ introduced in Definition 1.1, then, for all

Corollary 5 . 4

 54 Under Assumption 1, there exists C depending only on A, f, T such that ωR |u -ũ|dxdt ≤ Ch.

  the hat flux Exact solution for the GNL flux

Figure 2 :

 2 Figure 2: Left: Exact solutions for each flux (u vs x). Right: Rates of convergence for each flux and each numerical scheme (L 1 error vs h in Log-scale)

  ).

	Remark 2.1 u n 1/2 = u n,⋆ 1. Note that u n L , u n R are not in general unique. Indeed, assume that u n -1/2 < B n , and that -1/2 , then (u n L , u n R ) can be chosen equal to either (u n -1/2 , u n -1/2 ) or (u n 1/2 , u n 1/2 ). Of course, whatever this choice is, the flux f (u n L ) = f (u n R ) through the interface {x = 0} is unique.
	2. The introduction of these approximate traces enables us to rewrite the constrained Godunov scheme
	(12)-(14) as two classical Godunov schemes on {x < 0} and {x > 0} with respective Dirichlet boundary
	conditions u n L and u n R :

  Similar calculations carried out for (x, t, y, s) ∈ (R + )4 with the test function ξ R yield

								Assumption 2 holds,
		h + ǫ + η +	h ǫ	+	h η	+	η ǫ	otherwise.	(81)
	|u h (x, t) -u(x, t)|∂ t ϕ(x, t)dxdt +		Φ u(x,t) (u h (x, t))∂ x ϕ(x, t)dxdt
	R+ R+						R+ R+
	+	Φ uR(s) (u R,h (t))ϕ(0, t)ρ η (s -t)dtds ≥ -C ∇ϕ Θ(h, ǫ, η).	(82)
	R+ R+
	Adding (80) and (82) provides
	|u h (x, t) -u(x, t)|∂ t ϕ(x, t)dxdt
	R+ R

The set D + (R × R + ) denotes the space of C ∞ (R × R + ; R + ) functions with compact support.