Boris Andreianov
email: boris.andreianov@lmpt.univ-tours.fr

Carlotta Donadello
email: carlotta.donadello@univ-fcomte.fr

Ulrich Razafison
email: ulrich.razafison@univ-fcomte.fr

Massimiliano D Rosini
email: mrosini@hektor.umcs.lublin.pl

Qualitative behaviour and numerical approximation of solutions to conservation laws with non-local point constraints on the flux and modeling of crowd dynamics at the bottlenecks

Keywords: finite volume scheme, scalar conservation law, non-local point constraint, crowd dynamics, capacity drop, Braess' paradox, Faster Is Slower 2000 MSC: 35L65, 90B20, 65M12, 76M12

Qualitative behaviour and numerical approximation of solutions to conservation laws with non-local point constraints on the flux and modeling of crowd dynamics at the bottlenecks

Introduction

Andreianov, Donadello and Rosini developed in [START_REF] Andreianov | Crowd dynamics and conservation laws with nonlocal constraints and capacity drop[END_REF] a macroscopic model, called here ADR, aiming at describing the behaviour of pedestrians at bottlenecks. The model is given by the Cauchy problem for a scalar hyperbolic conservation law in one space dimension with non-local point constraint of the form

∂ t ρ + ∂ x f (ρ) = 0 (t, x) ∈ R + × R, (1a) ρ(0, x) = ρ(x) x ∈ R, (1b)
f (ρ(t, 0±)) ≤ p R - w(x) ρ(t, x) dx t ∈ R + , (1c)
where ρ(t, x) ∈ [0, R] is the (mean) density of pedestrians at x ∈ R at time t ∈ R + and ρ : R → [0, R] is the initial (mean) density, with R > 0 being the maximal density. Then, f : [0, R] → R + is the flow considered to be bell-shaped, which is an assumption commonly used in crowd dynamics (see [START_REF] Colombo | Pedestrian flows and non-classical shocks[END_REF][START_REF] Colombo | Existence of nonclassical solutions in a Pedestrian flow model[END_REF][START_REF] Rosini | Nonclassical interactions portrait in a macroscopic pedestrian flow model[END_REF] for examples of non bell-shaped flows used in crowd dynamics and [START_REF] Andreianov | On interface transmission conditions for conservation laws with discontinuous flux of general shape[END_REF][START_REF] Chalons | Numerical Approximation of a Macroscopic Model of Pedestrian Flows[END_REF][START_REF] Chalons | General constrained conservation laws. Application to pedestrian flow modeling[END_REF] for the numerical approximations). A typical example of such flow is the so-called Lighthill-Whitham-Richards (LWR) flux [START_REF] Lighthill | On Kinematic Waves. II. A Theory of Traffic Flow on Long Crowded Roads[END_REF][START_REF] Richards | Shock waves on the highway[END_REF][START_REF] Greenshields | A study of traffic capacity[END_REF] defined by

f (ρ) = ρ v max 1 - ρ ρ max ,
where v max and ρ max are the maximal velocity and the maximal density of pedestrians respectively. Throughout this paper the LWR flux will be used. Next p : R + → R + prescribes the maximal flow allowed through a bottleneck located at x = 0 as a function of the weighted average density in a left neighbourhood of the bottleneck and w : R -→ R + is the weight function used to average the density. Finally in (1c), ρ(t, 0-) denotes the left measure theoretic trace along the constraint, implicitly defined by

lim ε↓0 1 ε +∞ 0 0 -ε |ρ(t, x) -ρ(t, 0-)| φ(t, x) dx dt = 0 for all φ ∈ C ∞ c (R 2 ; R).
The right measure theoretic trace, ρ(t, 0+), is defined analogously.

In the last few decades, the study of the pedestrian behaviour through bottlenecks, namely at locations with reduced capacity, such as doors, stairs or narrowings, drawn a considerable attention. The papers [START_REF] Schadschneider | Evacuation Dynamics: Empirical Results, Modeling and Applications[END_REF][START_REF] Cepolina | Phased evacuation: An optimisation model which takes into account the capacity drop phenomenon in pedestrian flows[END_REF][START_REF] Hoogendoorn | Pedestrian behavior at bottlenecks[END_REF][START_REF] Kopylow | The study of people' motion parameters under forced egress situations[END_REF][START_REF] Kretz | Experimental study of pedestrian counterflow in a corridor[END_REF][START_REF] Seyfried | Capacity Estimation for Emergency Exits and Bottlenecks[END_REF][START_REF] Zhang | Empirical study of a unidirectional dense crowd during a real mass event[END_REF] present results of empirical experiments. However, for safety reasons, experiments reproducing extremal conditions such as evacuation and stampede are not available. In fact, the unique experimental study of a crowd disaster is proposed in [START_REF] Helbing | Dynamics of crowd disasters: An empirical study[END_REF]. The available data show that the capacity of the bottleneck (i.e. the maximum number of pedestrians that can flow through the bottleneck in a given time interval) can drop when highdensity conditions occur upstream of the bottleneck. This phenomenon is called capacity drop and can lead to extremely serious consequences in escape situations. In fact, the crowd pressure before an exit can reach very high values, the efficiency of the exit dramatically reduces and accidents become more probable due to the overcrowding and the increase of the evacuation time (i.e. the temporal gap between the times in which the first and the last pedestrian pass through the bottleneck). A linked phenomenon is the so-called Faster Is Slower (FIS) effect, first described in [START_REF] Helbing | Simulating dynamical features of escape panic[END_REF]. FIS effect refers to the jamming and clogging at the bottlenecks, that result in an increase of the evacuation time when the degree of hurry of a crowd is high. We recall that the capacity drop and the FIS effect are both experimentally reproduced in [START_REF] Cepolina | Phased evacuation: An optimisation model which takes into account the capacity drop phenomenon in pedestrian flows[END_REF][START_REF] Soria | Experimental evidence of the "Faster is Slower" effect in the evacuation of ants[END_REF]. A further related (partly counter-intuitive) phenomenon is the so-called Braess' paradox for pedestrian flows [START_REF] Hughes | The flow of human crowds[END_REF]. It is well known that placing a small obstacle before an exit door can mitigate the inter-pedestrian pressure and, under particular circumstances, it reduces the evacuation time by improving the outflow of people. Note that as it happens for any first order model, see for instance [32, Part III] and the references therein, ADR can not explain the capacity drop and collective behaviours at the bottlenecks. Therefore one of the difficulties we have to face is that the constraint p has to be deduced together with the fundamental diagram from the empirical observations. The aim of this paper is to validate ADR by performing simulations in order to show the ability of the model to reproduce the main effects described above and related to capacity drop that are 2 FIS and Braess' paradox. To this end we propose a numerical scheme for the model and prove its convergence. The scheme is obtained by adapting the local constrained finite volume method introduced in [START_REF] Andreianov | Finite volume schemes for locally constrained conservation laws[END_REF] to the non-local case considered in ADR, using a splitting strategy. The paper is organized as follows. In Section 1 we briefly recall the main theoretical results for ADR. In Section 2 we introduce the numerical scheme, prove its convergence and validate it with an explicit solution obtained in [START_REF] Andreianov | Crowd dynamics and conservation laws with nonlocal constraints and capacity drop[END_REF]. In Section 3 we perform simulations to show that ADR is able to reproduce the Braess' paradox and the FIS effect. In Subsection 3.3 we combine local and non-local constraints to model a slow zone placed before the exit. Conclusions and perspectives are outlined in Section 4.

Well-posedness for the ADR model

Existence, uniqueness and stability for the general Cauchy problem (1) are established in [START_REF] Andreianov | Crowd dynamics and conservation laws with nonlocal constraints and capacity drop[END_REF] under the following assumptions:

(F) f belongs to Lip ([0, R]; [0, +∞[) and is supposed to be bell-shaped, that is f (0) = 0 = f (R)
and there exists

σ ∈ ]0, R[ such that f ′ (ρ) (σ -ρ) > 0 for a.e. ρ ∈ [0, R]. (W) w belongs to L ∞ (R -; R + )
, is an increasing map, w L 1 (R -) = 1 and there exists i w > 0 such that w(x) = 0 for any x ≤ -i w . (P) p belongs to Lip [0, R] ; 0, f (σ) and is a non-increasing map.

The regularity w ∈ L ∞ (R -; R + ) is the minimal requirement needed in order to prove existence and uniqueness of [START_REF] Andreianov | New approches to describing admissibility of solutions of scalar conservation laws with discontinuous flux[END_REF]. In this paper, we shall consider continuous w.

The existence of solutions for the Riemann problem for (1) is proved in [START_REF] Andreianov | Riemann problems with non-local point constraints and capacity drop[END_REF] for piecewise constant p. However, such hypothesis on p is not sufficient to ensure uniqueness of solutions, unless the flux f and the efficiency p satisfy a simple geometric condition, see [START_REF] Andreianov | Riemann problems with non-local point constraints and capacity drop[END_REF] for details. In the present paper, we consider either continuous nonlinear p or a piecewise constant p that satisfies such geometric condition.

The definition of entropy solution for a Cauchy problem (1a), (1b) with a fixed a priori time dependent constraint condition

f (ρ(t, 0±)) ≤ q(t) t ∈ R + (2)
was introduced in [ [START_REF] Colombo | A well posed conservation law with a variable unilateral constraint[END_REF] is related to the theory of conservation laws with discontinuous flux functions. Indeed a limitation of the flux of the form (2) can be interpreted as the choice of an (A, B)-connection if the problem is reformulated as a discontinuous flux problem, see [START_REF] Andreianov | A theory of L 1 -dissipative solvers for scalar conservation laws with discontinuous flux[END_REF] for a full theoretical discussion of this point and [START_REF] Andreianov | New approches to describing admissibility of solutions of scalar conservation laws with discontinuous flux[END_REF][START_REF] Andreianov | On interface transmission conditions for conservation laws with discontinuous flux of general shape[END_REF][START_REF] Cancès | Error estimate for Godounov approximation of locally constrained conservation laws[END_REF] for related numerical investigations.

The following theorem on existence, uniqueness and stability of entropy solutions of the constrained Cauchy problem (1) is achieved under the hypotheses (F), (W) and (P). Theorem 1.1 (Theorem 3.1 in [START_REF] Andreianov | Crowd dynamics and conservation laws with nonlocal constraints and capacity drop[END_REF]). Let (F), (W), (P) hold. Then, for any initial datum ρ ∈ L ∞ (R; [0, R]), the Cauchy problem (1) admits a unique entropy solution ρ. Moreover, if ρ ′ = ρ ′ (t, x) is the entropy solution corresponding to the initial datum ρ′ ∈ L ∞ (R; [0, R]), then for all T > 0 and L > i w , the following inequality holds

ρ(T ) -ρ ′ (T ) L 1 ([-L,L]) ≤ e CT ρ -ρ′ L 1 ({|x|≤L+MT }) , (3)
where M = Lip( f

) and C = 2Lip(p) w L ∞ (R -) .
The total variation of the solution may in general increase due to the presence of the constraint. In [START_REF] Andreianov | Crowd dynamics and conservation laws with nonlocal constraints and capacity drop[END_REF] the authors provide an invariant domain D ⊂ L 1 (R; [0, R]) such that if ρ belongs to D, then one obtains a Lipschitz estimate with respect to time of the L 1 norm and an a priori estimate of the total variation of

Ψ(ρ) = sign(ρ -σ)[ f (σ) -f (ρ)] = ρ σ ḟ (r) dr.

Numerical method for approximation of ADR

In this section we describe the numerical scheme based on finite volume method that we use to solve [START_REF] Andreianov | New approches to describing admissibility of solutions of scalar conservation laws with discontinuous flux[END_REF]. Then we prove the convergence of our scheme and validate it by comparison with an explicit solution of [START_REF] Andreianov | New approches to describing admissibility of solutions of scalar conservation laws with discontinuous flux[END_REF]. In what follows, we assume that (F), (W) and (P) hold.

Non-local constrained finite volume method

Let ∆x and ∆t be the constant space and time steps respectively. We define the points x j+1/2 = j∆x, the cells K j = [x j-1/2 , x j+1/2 [ and the cell centers x j = ( j -1/2)∆x for j ∈ Z. We define the time discretization t n = n∆t. We introduce the index j c such that x j c +1/2 is the location of the constraint (a door or an obstacle). For n ∈ N and j ∈ Z, we denote by ρ n j the approximation of the average of ρ(t n , • ) on the cell K j , namely

ρ 0 j = 1 ∆x x j+1/2 x j-1/2 ρ(x) dx and ρ n j ≃ 1 ∆x x j+1/2 x j-1/2 ρ(t n , x) dx if n > 0.
We recall that for the classical conservation law (1a)-(1b), a standard finite volume method can be written into the form

ρ n+1 j = ρ n j - ∆t ∆x F n j+1/2 -F n j-1/2 , (4)
where F n j+1/2 = F ρ n j , ρ n j+1 is a monotone, consistent numerical flux, that is, F satisfies the following assumptions:

• F is Lipschitz continuous from [0, R] 2 to R with Lipschitz constant Lip(F), • F(a, a) = f (a) for any a ∈ [0, R], • (a, b) ∈ [0, R] 2 → F(a, b) ∈ R
is non-decreasing with respect to a and non-increasing with respect to b.

We also recall that in [START_REF] Andreianov | Finite volume schemes for locally constrained conservation laws[END_REF] the numerical flux for the time dependent constraint (2) is modified as follow in order to take into account the constraint condition

F n j+1/2 =          F ρ n j , ρ n j+1 if j j c , min F ρ n j , ρ n j+1 , q n if j = j c , (5)
where q n is an approximation of q(t n ). In the present paper, when dealing with a Cauchy problem subject to a non-local constraint of the form (1c) we will use the approximation

q n = p         ∆x j≤ j c w(x j ) ρ n j         . (6)
Roughly speaking

• we apply the numerical scheme (4) for the problem (1a)-(1b),

• we apply the numerical scheme ( 4)-( 5) for the problem (1a)-( 1b)-( 2),

• we apply the numerical scheme ( 4)-( 5)-( 6) for the problem (1).

Convergence of the scheme

Let us introduce the finite volume approximate solution ρ ∆ defined by

ρ ∆ (t, x) = ρ n j for x ∈ K j and t ∈ [t n , t n+1 [, (7)
where the sequence (ρ n j ) j∈Z, n∈N is obtained by the numerical scheme (4)- [START_REF] Andreianov | Riemann problems with non-local point constraints and capacity drop[END_REF]. Analogously, we also define the approximate constraint function

q ∆ (t) = q n for t ∈ [t n , t n+1 [. (8)
First, we prove a discrete stability estimate valid for any domain Q = [0, T ]×R with T > 0, for the scheme (4)-( 5) applied to problem (1a)-(1b)-( 2). This estimate can be seen as the equivalent, in this framework, of the stability result established in [4, Proposition 2.10].

Proposition 2.1. Let ρ be in L ∞ (R; [0, R]) and q ∆ , q∆ be piecewise constant functions of the form [START_REF] Bale | A wave propagation method for conservation laws and balance laws with spatially varying flux functions[END_REF]. If ρ ∆ and ρ∆ are the approximate solutions of (1a)-(1b)-( 2) corresponding, respectively, to q ∆ and q∆ and constructed by applying the scheme (4)-( 5), then we have

ρ ∆ -ρ∆ L 1 (Q) ≤ 2T q ∆ -q∆ L 1 ([0,T ]) .
Proof. For notational simplicity, let N = ⌊T/∆t⌋. Let us also introduce ( ρn j ) j∈Z, n∈N defined by,

ρn+1 j = ρ n j - ∆t ∆x F n j+1/2 -F n j-1/2 , for any j ∈ Z, n ∈ N,
where F n j+1/2 is defined by

F n j+1/2 =          F ρ n j , ρ n j+1 if j j c , min F ρ n j , ρ n j+1 , qn if j = j c .
Then using the definitions of (ρ n j ) j∈Z, n∈N and ( ρn j ) j∈Z, n∈N , we have for any n = 1, . . . , N,

ρ n j = ρn j if j { j c , j c + 1}
and

ρ n j c -ρn j c = - ∆t ∆x min F ρ n-1 j c , ρ n-1 j c +1 , q n-1 -min F ρ n-1 j c , ρ n-1 j c +1 , qn-1 , ρ n j c +1 -ρn j c +1 = ∆t ∆x min F ρ n-1 j c , ρ n-1 j c +1 , q n-1 -min F ρ n-1 j c , ρ n-1 j c +1 , qn-1 ,
which implies that

ρ n j c -ρn j c ≤ ∆t ∆x q n-1 -qn-1 , ρ n j c +1 -ρn j c +1 ≤ ∆t ∆x q n-1 -qn-1 .
Therefore we deduce that, for any n = 1, . . . , N, j∈Z

ρ n j -ρn j ≤ 2 ∆t ∆x q n-1 -qn-1 . (9)
Besides, observe that the modification of the numerical flux at the interface x j c +1/2 introduced in ( 5) does not affect the monotonicity of the scheme ( 4)-( 5) (see [START_REF] Andreianov | Finite volume schemes for locally constrained conservation laws[END_REF]Proposition 4.2]). Therefore, for any n = 1, . . . , N, we have

j∈Z ρn j -ρn j ≤ j∈Z ρ n-1 j -ρn-1 j . (10)
Hence thanks to ( 9) and ( 10), we can write

j∈Z |ρ 1 j -ρ1 j | ≤ j∈Z |ρ 1 j -ρ1 j | + j∈Z | ρ1 j -ρ1 j | ≤ 2 ∆t ∆x q 0 -q0 + j∈Z |ρ 0 j -ρ0 j | = 2 ∆t ∆x q 0 -q0 .
Then an induction argument shows that for any n = 1, . . . , N, j∈Z

ρ n j -ρn j ≤ 2 ∆t ∆x n-1 k=0 |q k -qk | ≤ 2 ∆x q ∆ -q∆ L 1 ([0,t n ]) .
In conclusion, we find that

ρ ∆ -ρ∆ L 1 (Q) = ∆t ∆x N n=1 j∈Z |ρ n j -ρn j | ≤ 2 q ∆ -q∆ L 1 ([0,T ]) N n=1 ∆t ≤ 2T q ∆ -q∆ L 1 ([0,T ])
and this ends the proof.

Let us now notice that as in [4, Proposition 4.2], under the CFL condition

Lip(F) ∆t ∆x ≤ 1 2 , (11)
we have the L ∞ stability of the scheme (4)-( 5)-( 6) that is

0 ≤ ρ ∆ (t, x) ≤ R for a.e. (t, x) ∈ Q. ( 12
)
This stability result allows to prove the statement below.

Proposition 2.2. Let q ∆ be defined by ( 6)- [START_REF] Bale | A wave propagation method for conservation laws and balance laws with spatially varying flux functions[END_REF]. Assume moreover that w belongs to Lip(R -; R + ).

Then under the CFL condition [START_REF] Chalons | Numerical Approximation of a Macroscopic Model of Pedestrian Flows[END_REF], for any T > 0, there exists C > 0 only depending on T , f , F, p, w and R such that:

|q ∆ | BV([0,T ]) ≤ C. ( 13
)
Proof. Let N = ⌊T/∆t⌋ and j w be an integer such that supp(w) ⊂ ∪ j w ≤ j≤ j c K j . Then for any n = 0, . . . , N -1, we have

q n+1 -q n = p         ∆x j w ≤ j≤ j c w(x j )ρ n+1 j         -p         ∆x j w ≤ j≤ j c w(x j )ρ n j         ≤ ∆x Lip(p) j w ≤ j≤ j c w(x j )(ρ n+1 j -ρ n j ) = ∆t Lip(p) j w ≤ j≤ j c w(x j ) F n j+1/2 -F n j-1/2 .
Now, using a summation by part, we have

j w ≤ j≤ j c w(x j ) F n j+1/2 -F n j-1/2 = w(x j c )F n j c +1/2 -w(x j w )F n j w -1/2 - j w ≤ j≤ j c -1 w(x j+1 ) -w(x j ) F n j+1/2 .
Then, using the fact that w belongs to Lip(R -; R + ), it follows that

j w ≤ j≤ j c w(x j ) F n j+1/2 -F n j-1/2 ≤ w L ∞ (R -;R + ) |F n j c +1/2 | + |F n j w -1/2 | + ∆x Lip(w) j w ≤ j≤ j c -1 |F n j+1/2 |.
This yields

|q n+1 -q n | ≤ ∆t Lip(p) w L ∞ (R -;R) |F n j c +1/2 | + |F n j w -1/2 | + ∆t∆x Lip(p) Lip(w) j w ≤ j≤ j c -1 |F n j+1/2 |.
Now, from (5), for any j ∈ Z we have the estimate

F n j+1/2 ≤ F(ρ n j , ρ n j+1 ) ≤ F(ρ n j , ρ n j+1 ) -F(ρ n j , ρ n j ) + f (ρ n j ) ≤ Lip(F) ρ n j+1 -ρ n j + Lip( f ) ρ n j ≤ R Lip(F) + Lip( f ) .
Hence we deduce that

|q ∆ | BV([0,T ]) = N-1 n=0 q n+1 -q n ≤ C, where C = T Lip(p) R(Lip(F) + Lip( f ) 2 w L ∞ (R -;R) + ∆x( j c -j w )Lip(w) .
We are now in a position to prove a convergence result for the scheme ( 4)-( 5)-( 6).

Theorem 2.1. Under the CFL condition [START_REF] Chalons | Numerical Approximation of a Macroscopic Model of Pedestrian Flows[END_REF] and if w belongs to Lip(R -; R + ), the constrainted finite volume scheme (4)-( 5)-( 6) converges in L 1 (Q) to the unique entropy solution to [START_REF] Andreianov | New approches to describing admissibility of solutions of scalar conservation laws with discontinuous flux[END_REF].

Proof. Let (ρ ∆ , q ∆ ) be constructed by the scheme (4)-( 5)-( 6). Proposition 2.2 and Helly's lemma give the existence of a subsequence, still denoted q ∆ and a constraint function

q ∈ L ∞ ([0, T ]) such that q ∆ converges to q strongly in L 1 ([0, T ]) as ∆t → 0. Let ρ ∈ L ∞ (R + × R; [0, R]
) be the unique entropy solution to (1a)-(1b)-( 2) associated to q. It remains to prove that the subsequence ρ ∆ converges to ρ strongly in L 1 (Q) as ∆t, ∆x → 0. The uniqueness of the entropy solution to (1a)-(1b)-( 2) will then imply that the full sequence ρ ∆ converges to ρ and, as a consequence, the full sequence q ∆ converges to q = p R - w(x) ρ(t, x) dx . Let q∆ be a piecewise constant approximation of q such that q∆ converges to q strongly in L 1 ([0, T ]). Furthermore, we also introduce ρ∆ constructed by the scheme (4)-( 5) and associated to q∆ . Now we have

ρ -ρ ∆ L 1 (Q) ≤ ρ -ρ∆ L 1 (Q) + ρ ∆ -ρ∆ L 1 (Q) .
But, thanks to [4, Theorem 4.9], under the CFL condition [START_REF] Chalons | Numerical Approximation of a Macroscopic Model of Pedestrian Flows[END_REF], ρ -ρ∆ L 1 (Q) tends to 0 as ∆t, ∆x → 0. Furthermore, thanks to Proposition 2.1, we have

ρ ∆ -ρ∆ L 1 (Q) ≤ 2 T q ∆ -q∆ L 1 ([0,T ])
which shows that also ρ ∆ -ρ∆ L 1 (Q) tends to 0 as ∆t, ∆x → 0. We propose here to validate the numerical scheme (4)-( 5)-( 6) by using the Godounov numerical flux (see e.g. [START_REF] Godlewski | Numerical approximation of hyperbolic systems of conservation laws[END_REF][START_REF] Leveque | Finite volume methods for hyperbolic problems[END_REF]) which will be used in the remaining of this paper:

Validation of the numerical scheme

F(a, b) =          min [a,b] f if a ≤ b, max [b,a] f if a > b.
We consider the explicit solution to (1) constructed in [START_REF] Andreianov | Crowd dynamics and conservation laws with nonlocal constraints and capacity drop[END_REF]Section 6] by applying the wavefront tracking algorithm. The set up for the simulation is as follows. Consider the domain of computation [-6, 1], take a normalized flux f (ρ) = ρ(1ρ) (namely the maximal velocity and the maximal density are assumed to be equal to one) and a linear weight function w

(x) = 2(1 + x) χ [-1,0] (x). Assume a uniform distribution of maximal density in [x A , x B ] ⊂ ]-6, 0[ at time t = 0, namely ρ = χ [x A ,x B ] .
The efficiency of the exit, p, see Figure 1, is of the form

p(ξ) =          p 0 if 0 ≤ ξ < ξ 1 , p 1 if ξ 1 ≤ ξ < ξ 2 , p 2 if ξ 2 ≤ ξ ≤ 1.
The explicit solution ρ corresponding to the values p 0 = 0.21, p 1 = 0.168, p 2 = 0.021, ξ 1 ∼ 0.566, x (d) ρ ∆ (10, x) is represented in Figure 2. The above choices for the flux f and the efficiency p ensure that the solution to each Riemann problem is unique, see [START_REF] Andreianov | Riemann problems with non-local point constraints and capacity drop[END_REF]. We refer to [START_REF] Andreianov | Crowd dynamics and conservation laws with nonlocal constraints and capacity drop[END_REF]Section 6] for the details about the construction of the solution ρ and its physical interpretation. A qualitative comparison between the numerically computed solution x → ρ ∆ (t, x) and the explicitly computed solution x → ρ(t, x) at different fixed times t is in Figure 3. We observe good agreements between x → ρ(t, x) and x → ρ ∆ (t, x). The parameters for the numerically computed solution are ∆x = 3.5 × 10 -4 and ∆t = 0.4∆x = 1.4 × 10 -4 . A convergence analysis is also performed for this test. We introduce the relative L 1 -error for the density ρ, at a given time t n , defined by

x A = -5.75, x B = -2, ξ 2 ∼ 0.731,
x 1 0 -1 -2 -3 -4 -5 -6 0.0 0.2 0.4 0.6 0.8 1.0 (e) ρ(0, x) x 1 0 -1 -2 -3 -4 -5 -6 0.0 0.2 0.4 0.6 0.8 1.0 (f) ρ(1, x) x 1 0 -1 -2 -3 -4 -5 -6 0.0 0.2 0.4 0.6 0.8 1.0 (g) ρ(7.325, x) x 1 0 -1 -2 -3 -4 -5 -6 0.0 0.2 0.4 0.6 0.8 1.0 (h) ρ(10, x) 0 0.2 0.4 0.6 0.8 1 -6 -5 -4 -3 -2 -1 0 1 x (i) ρ ∆ (11.939, x) 0 0.2 0.4 0.6 0.8 1 -6 -5 -4 -3 -2 -1 0 1 x (j) ρ ∆ (85.2, x) 0 0.2 0.4 0.6 0.8 1 -6 -5 -4 -3 -2 -1 0 1 x (k) ρ ∆ (85.5526, x) 0 0.2 0.4 0.6 0.8 1 -6 -5 -4 -3 -2 -1 0 1 x (l) ρ ∆ (87.4981, x) x 1 0 -1 -2 -3 -4 -5 - 6
E n L 1 =         j ρ(t n , x j ) -ρ n j                 j ρ(t n , x j )         .
In Table 1, we computed the relative L 1 -errors for different numbers of space cells at the fixed time t = 10. The time space is fixed to ∆t = 1.4 × 10 -4 . We deduce that the order of convergence 10 is approximatively 0.93. As in [START_REF] Andreianov | Finite volume schemes for locally constrained conservation laws[END_REF], we observe that the modification (5) of the numerical flux does not affect the accuracy of the scheme.

Number of cells

Numerical simulations

This section is devoted to the phenomenological description of some collective effects in crowd dynamics related to the capacity drop, namely the Faster Is Slower (FIS) effect and the Braess' paradox.

Faster is Slower effect

The FIS effect was first described in [START_REF] Helbing | Simulating dynamical features of escape panic[END_REF][START_REF] Parisi | Microscopic dynamics of pedestrian evacuation[END_REF] in the context of the room evacuation problem. The authors studied the evolution of the evacuation time as a function of the maximal velocity reached by the pedestrians, and they have shown that there exists an optimal velocity for which the evacuation time attains a minimum. Therefore, any acceleration beyond the optimal velocity worses the evacuation time. Following the studies above, the curve representing the evacuation time as a function of the average velocity takes a characteristic shape [30, Figure 1].

The first numerical tests we performed aim to verify if such shape is obtained starting from the ADR model. To this end, we consider the corridor modeled by the segment [-6,1], with an exit at x = 0. We consider the flux f (ρ) = ρ v max (1ρ), where v max is the maximal velocity of the pedestrians and the maximal density is equal to one. We use the same weight function as for the validation of the scheme, w(x) = 2(1 + x)χ [-1,0] (x) and, the same initial density, ρ = χ [-5.75,-2] . The efficiency of the exit p is now given by the following continuous function

p(ξ) =                    p 0 if 0 ≤ ξ < ξ 1 , (p 0 -p 1 )ξ + p 1 ξ 1 -p 0 ξ 2 ξ 1 -ξ 2 if ξ 1 ≤ ξ < ξ 2 , p 1 if ξ 2 ≤ ξ ≤ 1, (14)
where p 0 = 0.24, p 1 = 0.05, ξ 1 = 0.5, ξ 2 = 0.9.

The space and time steps are fixed to ∆x = 5 × 10 -3 and ∆t = 5 × 10 -4 . In Figure 4 are plotted the flux f corresponding to the maximal velocity v max = 1 and the above efficiency of the exit. Figure 5 represents the evacuation time as a function of the maximal velocity v max , as v max varies in the interval [0.1, 5]. As we can observe, the general shape described in [30, Figure 1] is recovered. The numerical minimal evacuation time is 19.007 and is obtained for v max = 1.

In addition, we present in Figure 6 the density at the exit as a function of time for different values of the maximal velocity v max around the optimal one. We notice that the maximal density at the exit increases with the velocity. This expresses the jamming at the exit that leads to the FIS effect.

Then we performed some series of tests to see how the general shape obtained in Figure 5 changes with respect to variations of the parameters of the model. In Figure 8 (a), we show this variation when we consider different initial densities, namely, ρ, ρ1 and ρ2 with ρ1 (x) = 0.8χ [-5.75,-2] and ρ2 (x) = 0.6χ [-5.75,-2] . The general shape of the curves is conserved. We observe that the evacuation time increases with the initial amount of pedestrians while the optimal velocity decreases as the initial amount of pedestrians increases. The minimal evacuation time and the corresponding optimal maximal velocity are 12.259 and 1.07 for ρ2 and 15.691 and 1.03 for ρ1 .

Next we explore the case where the efficiency of the exit varies. We consider the function p defined in [START_REF] Colombo | A well posed conservation law with a variable unilateral constraint[END_REF] and the modification p β such that p β (ξ) = p(βξ). In Figure 7, we plotted the functions p, p β for β = 0.8 and β = 0.9. Then, in Figure 8 obtain 18.586 and 18.827 for β = 0.8, 0.9 respectively. As expected, the minimal evacuation time increases with lower efficiency of the exit. The corresponding velocities are approximatively 1.06 and 1.02 respectively.

Finally, we change the location of the initial density. In addition to the corridor [-6, 1], we consider two other corridors modeled by the segments [-12, 1] and [-20, 1]. In these two corridors we take as initial densities ρ3 (x) = χ [-11.75,-8] and ρ4 (x) = χ [-19.75,-16] respectively. We have reported the obtained evacuation time curves in Figure 8 (c). As expected, the minimal evacuation time increases with the distance between the exit and the initial density location.

Braess' paradox

The presence of obstacles, such as columns upstream from the exit, may prevent the crowd density from reaching dangerous values and may actually help to minimize the evacuation time, since in a moderate density regime the full capacity of the exit can be exploited. From a microscopic point of view, the decrease of the evacuation time may seem unexpected, as some of the pedestrians are forced to choose a longer path to reach the exit.

The ADR model is able to reproduce the Braess' paradox for pedestrians, as we show in the following simulations. We consider, as in the previous subsection, the corridor modeled by 13 In these following simulations we place an obstacle at x = d, with -2 < d < 0. The obstacle reduces the capacity of the corridor and can be seen as a door, which we assume larger than the one at x = 0. Following these ideas we define an efficiency function p κ (ξ) = κ p(ξ), where κ = 1.15 and a weight function

w d (x) = 2(x -d + 1)χ [d-1,d] (
x) associated to the obstacle.

In Figure 9 we have reported the evolution of the evacuation time when the position of the obstacle varies in the interval [-1.9, -0.01] with a step of 0.01. We get the characteristic shape already obtained in [START_REF] Colombo | A macroscopic model for pedestrian flows in panic situations[END_REF]Figure 13] for the model introduced in [START_REF] Colombo | Pedestrian flows and non-classical shocks[END_REF]. We observe that for -1.8 ≤ d ≤ -1.72, the evacuation time is much lower than in the absence of the obstacle. The optimal position of the obstacle is obtained for d = -1.72 and the corresponding evacuation time is 24.246. We compare in Figure 10 five snapshots of the solution without obstacle and the solutions 14 with an obstacle placed at d = -1.72 and d = -1.85. This latter location corresponds to a case where the evacuation time is greater than the one without an obstacle. In these snapshots, we see that the obstacle placed at d = -1.85 becomes congested very soon. This is due to the fact that the obstacle is too close to the location of the initial density. When the obstacle is placed at d = -1.72, it mitigates the congestion at the exit.

So far we have fixed the coefficient κ to 1.15. In order to highlight the influence of κ, we performed simulations where we let now κ vary and for each value of κ, we used the position of the obstacle d as a parameter to minimize the associate evacuation time. Figure 11 shows the evolutions of the evacuation time and the corresponding optimal position of the obstacle when κ varies in [1, 1.19]. The interval is chosen so that the obstacle, interpreted as a door, is at least larger than the exit and has a capacity that is inferior to the one of the corridor. We observe the existence of an optimal value of κ, κ = 1.12 for which the evacuation time is T = 23.187 when the position of the obstacle is d = -1.03. From the extreme values of κ, we also see that, taking an obstacle with the same size as the exit or almost the same size as the corridor leads to optimal evacuation times that are close to the one without obstacle; we have T = 29.53 for κ = 1 and T = 29.3 for κ = 1.19. Finally we notice that the optimal distance between the obstacle and the exit is an increasing function of κ.

Zone of low velocity

In this section, we perform a series of simulations where the obstacle introduced in Subsection 3.2 is now replaced by a zone where the velocity of pedestrians is lower than elsewhere in the domain. The effect we want to observe here is similar to the one we see in Braess' paradox. Namely we prevent a high concentration of pedestrians in front of the exit by constraining their flow in an upstream portion of the corridor. In this case however the constraint is local, as the maximal value allowed for the flow only depends on the position in the corridor.

We consider again the corridor modeled by the segment [-6, 1] with an exit at x = 0. The efficiency of the exit and the initial density are the same as in the previous subsection. Assume that the slow zone is of size one and is centred at x = d, where -1.

(x) =                          1 if x ≤ d -0.5, -2(x -d) if d -0.5 ≤ x ≤ d, 2(x -d) if d ≤ x ≤ d + 0.5, 1 if x ≥ d + 0.5, (15)
and the following velocity Since the flux function considered here is Lipschitz continuous with respect to x, the xdependence of the flux can be taken into account within the classical Kruzhkov theory which is easily combined with taking into account the constraint. From the numerical point of view, this generalization is also straightforward: we adapt the scheme presented in Subsection 2.1, by assuming a distinct flux function f j+1/2 (ρ) = f (x j+1/2 , ρ) at each interface x j+1/2 of the space grid. The reader can refer for instance to [START_REF] Bale | A wave propagation method for conservation laws and balance laws with spatially varying flux functions[END_REF] and references therein for more details on finite volume methods with spatially varying flux functions. Figure 12 (a) shows the evolution of the evacuation time as a function of the parameter λ varying in the interval [0.1, 1] when the center of the slow zone is fixed at d = -1.5. We observe that the optimal minimal velocity in the slow zone is for λ = 0.88 and the corresponding evacuation time is 20.945. Recalling that without the slow zone the evacuation time is 29.496, we see that the introduction of the slow zone allows to reduce the evacuation time. In Figure 12 (b), we show the evolution of the evacuation time when varying the center of the slow zone d in the interval [-1.9, 0] and when the minimal and the maximal velocities are fixed and correspond to λ = 0.88 and v max = 1. We observe here that, unlike in the Braess' paradox test cases, the evacuation time does not depend on the location of the slow zone, except when this latter is close enough to the exit. Indeed, when the slow zone gets too close to the exit, the evacuation time grows. This is due to the fact that pedestrians do not have time to speed up before reaching the exit.

v(x, ρ) = [λ + (1 -λ) k(x)] v max (1 -ρ),
Fix now d = -1.5 and λ = 0.88 and assume that v max varies in the interval [0. [START_REF] Andreianov | New approches to describing admissibility of solutions of scalar conservation laws with discontinuous flux[END_REF][START_REF] Andreianov | Riemann problems with non-local point constraints and capacity drop[END_REF]. The evolution of the evacuation time as a function of v max is reported in Figure 12 (c). We observe that we get the characteristic shape already obtained for the FIS effect in Subsection 3.1.

Finally we present in Figure 13 five snapshots of the solution computed with a zone of low velocity centered at d = -1.72, λ = 0.88 and v max = 1. In order to have a good resolution of this third solution, the space and time steps were fixed to ∆x = 3.5 × 10 -4 and ∆t = 7 × 10 -5 . We note that in the case where a zone of low velocity is placed in the domain, the capacity drop at the exit is mitigated with respect to the cases with and without an obstacle, as the density of pedestrians never attains very high values in the region next to the exit. Moreover, the minimal evacuation time we obtain by introducing a zone of low velocity takes lower values than the ones obtained in Subsection 3.2, even when we take the optimal values of κ and of d. This is due to the fact that in all Braess' paradox simulations we can observe effects related to the capacity drop upstream the exit and upstream the obstacle. By construction, these phenomena cannot be observed upstream the zone of low velocity.

Conclusions

Qualitative features that are characteristic of pedestrians' macroscopic behaviour at bottlenecks (Faster is Slower, Braess' paradox) are reproduced in the setting of the simple scalar model with non-local point constraint introduced in [START_REF] Andreianov | Crowd dynamics and conservation laws with nonlocal constraints and capacity drop[END_REF]. These effects are shown to be persistent for large intervals of values of parameters. The validation is done by means of a simple and robust time-explicit splitting finite volume scheme which is proved to be convergent, with experimental rate close to one.

The results presented in this paper allow to consider more complex models. Indeed, as ADR is a first order model, it is not able to capture more complicated effects related to crowd dynamics. Typically, ADR fails to reproduce the amplification of small perturbations. This leads to consider second order model such as the model proposed by Aw, Rascle and Zhang [START_REF] Aw | Resurrection of "second order" models of traffic flow[END_REF][START_REF] Zhang | A non-equilibrium traffic model devoid of gas-like behavior[END_REF] in the framework of vehicular traffic.

Another extension of this work is to consider the ADR model with constraints that are nonlocal in time. Such constraints allow to tackle optimal management problems in the spirit of [START_REF] Colombo | On the continuum modeling of crowds. In Hyperbolic problems: theory, numerics and applications[END_REF][START_REF] Colombo | On the modelling and management of traffic[END_REF].

Finally, this work can also be extended to two-dimensional models where experimental validations may be possible.

Figure 1 :

 1 Figure 1: The functions [ρ → f (ρ)] and [ξ → p(ξ)] as in Section 2.3.

  (a) The solution in the (t, x, ρ)-coordinates. (b) The solution in the (x, t)-coordinates. (c) The solution in the (t, x, ρ)-coordinates for 0 ≤ t ≤ 15. (d) The solution in the (t, x, ρ)-coordinates for 85 ≤ t ≤ 87.5.

Figure 2 :

 2 Figure 2: Representation of the solution constructed in [3, Section 6] and described in Subsection 2.3.

  ρ(87.4981, x)

Figure 3 :

 3 Figure 3: With reference to Subsection 2.3: The numerically computed solution x → ρ ∆ (t, x) and the explicitly computed solution x → ρ(t, x) at different fixed times t.

Figure 4 :

 4 Figure 4: The normalized flux ρ → f (ρ) and the constraint ξ → p(ξ) defined in (14).

Figure 5 :

 5 Figure 5: With reference to Subsection 3.1: Evacuation time as a function of the velocity v max .

  Figure 5 represents the evacuation time as a function of the maximal velocity v max , as v max varies in the interval [0.[START_REF] Andreianov | New approches to describing admissibility of solutions of scalar conservation laws with discontinuous flux[END_REF][START_REF] Andreianov | Riemann problems with non-local point constraints and capacity drop[END_REF]. As we can observe, the general shape described in [30, Figure1] is recovered. The numerical minimal evacuation time is 19.007 and is obtained for v max = 1.In addition, we present in Figure6the density at the exit as a function of time for different values of the maximal velocity v max around the optimal one. We notice that the maximal density at the exit increases with the velocity. This expresses the jamming at the exit that leads to the FIS effect.Then we performed some series of tests to see how the general shape obtained in Figure5changes with respect to variations of the parameters of the model. In Figure8 (a), we show this variation when we consider different initial densities, namely, ρ, ρ1 and ρ2 with ρ1 (x) = 0.8χ [-5.75,-2] and ρ2 (x) = 0.6χ [-5.75,-2] . The general shape of the curves is conserved. We observe that the evacuation time increases with the initial amount of pedestrians while the optimal velocity decreases as the initial amount of pedestrians increases. The minimal evacuation time and the corresponding optimal maximal velocity are 12.259 and 1.07 for ρ2 and 15.691 and 1.03 for ρ1 .Next we explore the case where the efficiency of the exit varies. We consider the function p defined in[START_REF] Colombo | A well posed conservation law with a variable unilateral constraint[END_REF] and the modification p β such that p β (ξ) = p(βξ). In Figure7, we plotted the functions p, p β for β = 0.8 and β = 0.9. Then, in Figure8(b) are plotted the evacuation time curves corresponding to these three efficiencies of the exit. As minimum evacuation times, we 12

Figure 6 :Figure 7 :

 67 Figure 6: With reference to Subsection 3.1: Densities at the exit as a function of time for different velocities.

  Evacuation time as a function of v max for different amounts of initial densities. Evacuation time as a function of v max for different efficiencies of the exit.

  Evacuation time as a function of v max for different locations of the initial density.

Figure 8 :

 8 Figure 8: With reference to Subsection 3.1: Evacuation time as a function of v max for different parameters of the model.

Figure 9 :

 9 Figure 9: With reference to Subsection 3.2: Evacuation time as a function of the position of the obstacle.

  9 ≤ d ≤ 0. Define the following function Without obstacle Obstacle at d = -1.85 Obstacle at d = -1

Figure 11 :

 11 Figure 11: With reference to Subection 3.2: Optimal evacuation time and corresponding optimal position of the obstacle as a function of the strength κ

  Evacuation time as a function of d.

  Evacuation time as a function of v max .

Figure 12 :

 12 Figure 12: With reference to Subection 3.3: Evacuation time as a function of different parameters of the model.

  k

  where λ ∈ [0, 1] and v max ≥ 1 is the maximal velocity. With such velocity, the maximal velocity of pedestrians decreases in the interval [d -0.5, d], reaching its minimal value λ v max at x = d. Then the velocity increases in the interval [d, d + 0.5] reaching the maximum value v max , that corresponds to the maximal velocity away from the slow zone. Finally we consider the flux f (x, ρ) = ρ v(x, ρ) and the space and time steps are fixed to ∆x = 5 × 10 -3 and ∆t = 5 × 10 -4 . Let us underline that by definition f is Lipschitz.

  t = 20.945

Figure 13 :

 13 Figure 13: With reference to Subection 3.3: zone of low velocity simulations: density profiles at different times.

Table 1 :

 1 Relative L 1 -error at time t = 10.

		L 1 -error	rate of convergence
	625	1.1491 × 10 -2	-
	1250	4.641 × 10 -3	1.3
	2500	3.5968 × 10 -3	0.83
	5000	1.5106 × 10 -3	0.91
	10000	8.1705 × 10 -4	0.92
	20000	4.243 × 10 -4	0.93

Acknowledgement. All the authors are supported by French ANR JCJC grant CoToCoLa and Polonium 2014 (French-Polish cooperation program) No.331460NC. The second author is also supported by the Université de Franche-Comté, soutien aux EC 2014. The last author is also partially supported by Interdyscyplinarne Centrum Modelowania Matematycznego i Komputerowego, Uniwersytet Warszawski. The first author is grateful to IRMAR, Université de Rennes, for the hospitality during the preparation of this paper. The last author thanks Gran Sasso Science Institute, L'Aquila, and LMB Laboratoire de Mathématiques CNRS UMR6623, Université de Franche-Comté, for the hospitality during the preparation of this paper.