Aekta Aggarwal

Paola Goatin

Crowd Dynamics through Non-Local Conservation Laws

Keywords: Nonlocal Conservation Laws, Lax-Friedrichs Scheme, Lane Formation AMS MSC: 35L65, 65M12

published or not. The documents may come

Introduction

We consider the following class of non-local systems of N conservation laws in 2 space dimensions:

∂ t U k + ∂ x f k (t, x, y, U k , η * U ) + ∂ y g k (t, x, y, U k , ϑ * U ) = 0, k ∈ {1, . . . , N }, (1.1)
where Here, (η * U )(t, x, y) and (ϑ * U )(t, x, y) are vectors in R m whose ℓ-component, for ℓ = 1, . . . , m, is

U (t,
(η * U ) ℓ (t, x, y) = R 2 N k=1 η ℓ,k (x -x ′ , y -y ′ ) U k (t, x ′ , y ′ ) dx ′ dy ′ ∈ R , (ϑ * U ) ℓ (t, x, y) = R 2 N k=1 ϑ ℓ,k (x -x ′ , y -y ′ ) U k (t, x ′ , y ′ ) dx ′ dy ′ ∈ R .
In (1.1), as usual, t is time, x, y the space variables, U the vector of the unknown densities, (f k , g k ) k=1,...,N the matrix valued flow and η and ϑ are smooth convolution kernels. The coupling among the equations is only due to the non-local terms η * U and ϑ * U .

The above class of systems can model many physical phenomena such as sedimentation [START_REF] Betancourt | On nonlocal conservation laws modelling sedimentation[END_REF], one-dimensional vehicular traffic [START_REF] Blandin | Well-posedness of a conservation law with non-local flux arising in traffic flow modeling[END_REF][START_REF] Colombo | Control of the continuity equation with a non local flow[END_REF], biological applications, e.g., in structured population dynamics [START_REF] Perthame | Transport equations in biology[END_REF], supply chains [START_REF] Colombo | Control of the continuity equation with a non local flow[END_REF], conveyor belts [START_REF] Göttlich | Modeling, simulation and validation of material flow on conveyor belts[END_REF], etc. The nonlocal nature of (1.1) is particularly suitable to describe the behavior of crowds, where each individual moves according to the average crowd density/density variation in a neighborhood of his position. The size of this neighborhood is related to the individual visual horizon. Other crowd dynamics models fitting into (1.1) were recently considered in the literature, see for example [START_REF] Colombo | A class of nonlocal models for pedestrian traffic[END_REF][START_REF] Colombo | Nonlocal crowd dynamics models for several populations[END_REF].

Here, we use the Lax-Friedrichs type algorithm that was recently proposed in [START_REF] Aggarwal | Nonlocal systems of conservation laws in several space dimensions[END_REF], generating a sequence of approximate solutions to (1.1) which, up to a subsequence, converges to a weak entropy solution of (1.1). The key point in the related convergence proof is to obtain a uniform bound on the total variation of the approximate solutions. However, a bound on the L ∞ norm and the L 1 Lipschitz continuous dependence of the solution on time are also obtained. These estimates allow to extend to a wider class of equations the existence results in [START_REF] Amorim | On the numerical integration of scalar nonlocal conservation laws[END_REF][START_REF] Betancourt | On nonlocal conservation laws modelling sedimentation[END_REF][START_REF] Colombo | A class of nonlocal models for pedestrian traffic[END_REF][START_REF] Colombo | Control of the continuity equation with a non local flow[END_REF][START_REF] Colombo | Nonlocal crowd dynamics models for several populations[END_REF]. In particular, the actual number of space dimensions is not relevant, no divergence free assumption is required, see [START_REF] Chainais-Hillairet | Finite volume schemes for a nonlinear hyperbolic equation. Convergence towards the entropy solution and error estimate[END_REF], and the convolution kernels η and ϑ are not required to have compact support.

We are interested in crowd dynamics models that capture the phenomenon of so-called lanes or paths or trails formation, which is a behavior commonly observed in the engineering literature, when two groups of people move in opposite directions. Some models were recently introduced in the mathematical literature in this regard, see [START_REF] Colombo | A class of nonlocal models for pedestrian traffic[END_REF][START_REF] Colombo | Nonlocal crowd dynamics models for several populations[END_REF][START_REF] Tory | An adaptive finite-volume method for a model of two-phase pedestrian flow[END_REF]. A new nonlocal model close to [8, eq. (3.5)] is proposed in Section 3, which is shown to fit the settings of Theorem 2.2, see also [START_REF] Aggarwal | Nonlocal systems of conservation laws in several space dimensions[END_REF]Theorem 2.3], and hence possesses a weak entropy solution. The numerical scheme developed in [START_REF] Aggarwal | Nonlocal systems of conservation laws in several space dimensions[END_REF] is then used to compute solutions of the proposed model, showing the formation of various patterns, in particular of lanes.

The next section revisits the main results of [START_REF] Aggarwal | Nonlocal systems of conservation laws in several space dimensions[END_REF], while Section 3 is devoted to sample numerical integrations of (1.1).

Main Results

Throughout, we denote

R + = [0, +∞[, R N + = [0, +∞[ N ,
|ξ| for a the modulus of a scalar ξ ∈ R and, for ξ ∈ R 2 , ξ stands for the usual Euclidean norm. The following assumptions on the functions f k = f k (t, x, y, u, A), g k = g k (t, x, y, u, B) and η, ϑ in (1.1), for k = 1, . . . , N are required:

(H0) f k , g k ∈ C 2 (R + × R 2 × R × R m ; R); ∂ u f k , ∂ u g k ∈ L ∞ (R + × R 2 × R × R m ; R); for all t ∈ R + , (x, y) ∈ R 2 and A, B ∈ R m , f k (t,
x, y, 0, A) = g k (t, x, y, 0, B) = 0.

(H1) There exists a constant M > 0 such that max t,x,y,A

∂ x f k , ∂ 2 xx f k , ∂ 2 xy f k , ∇ A f k , ∂ x ∇ A f k , ∂ y ∇ A f k , ∇ 2 AA f k ≤ M |u|, max t,x,y,B ∂ y g k , ∂ 2 yy g k , ∂ 2 xy g k , ∇ B g k , ∂ x ∇ B g k , ∂ y ∇ B g k , ∇ 2 BB g k ≤ M |u|. (H2) ∂ u f k , ∂ u g k ∈ W 1,∞ (R + × R 2 × R × R m ; R). (H3) η, ϑ ∈ (C 2 ∩ W 2,∞ )(R 2 ; R m×N ).
We recall the definition of solution to the Cauchy problem for (1.1), see also [START_REF] Aggarwal | Nonlocal systems of conservation laws in several space dimensions[END_REF][START_REF] Betancourt | On nonlocal conservation laws modelling sedimentation[END_REF][START_REF] Blandin | Well-posedness of a conservation law with non-local flux arising in traffic flow modeling[END_REF][START_REF] Colombo | A class of nonlocal models for pedestrian traffic[END_REF][START_REF] Colombo | Control of the continuity equation with a non local flow[END_REF][START_REF] Colombo | Nonlocal crowd dynamics models for several populations[END_REF].

Definition 2.1. Let U o ∈ L ∞ (R 2 ; R N ). A map U : [0, T ] → L ∞ (R 2 ; R N ) is a solution to (1.1) with initial datum U o if, for k = 1, . . . , N , setting for all w ∈ R f k (t, x, y, w) = f k t, x, y, w, (η * U )(t, x, y) , gk (t, x, y, w) = g k t, x, y, w, (ϑ * U )(t, x, y) ,
the map U k is a Kružkov solution to the scalar Cauchy problem

∂ t U k + ∂ x f k (t, x, y, U k ) + ∂ y gk (t, x, y, U k ) = 0 U k (0, x, y) = U k o (x, y) . (2.1)
Above, by Kružkov solution we refer to [START_REF] Kružhkov | First order quasilinear equations with several independent variables[END_REF]Definition 1]. It is immediate to observe that (H0) ensures that U (t, x, y) ≡ 0 solves (1.1) in the sense of Definition 2.1. Fix a rectangular grid with mesh sizes ∆x and ∆y in R 2 and a time step ∆t. Also, define

t n = n ∆t, n ∈ Z ; x i = i ∆x, x i+1/2 = i + 1 2 ∆x, i ∈ Z ; y j = j ∆y, y j+1/2 = j + 1 2 ∆y, j ∈ Z ; λ x = ∆t/∆x , λ y = ∆t/∆y . Throughout, an initial datum U o ∈ (L ∞ ∩ BV)(R 2 ; R N
) is fixed and we denote

u k,0 ij = 1 ∆x ∆y x i+1/2 x i-1/2 y j+1/2 y j-1/2 U k o (x, y) dx dy for i, j ∈ Z , k ∈ {1, . . . , N }.
Iteratively, we define a piecewise constant approximate solution u ∆ (t, x, y)

≡ u 1 ∆ , . . . , u N ∆ (t, x, y) to (1.1) such that u k ∆ (t, x, y) = u k,n ij for      t ∈ [t n , t n+1 [ , x ∈ [x i-1/2 , x i+1/2 [ , y ∈ [y j-1/2 , y j+1/2 [ , where n ∈ N i, j ∈ Z k ∈ {1, . . . , N }
through the 5-points algorithm based on dimensional splitting, see [9, Section 3]:

u k,n+1/2 ij = u k,n ij -λ x F k,n i+1/2,j (u k,n ij , u k,n i+1,j ) - F k,n i-1/2,j (u k,n i-1,j , u k,n ij ) u k,n+1 ij = u k,n+1/2 ij -λ y G k,n i,j+1/2 (u k,n+1/2 ij , u k,n+1/2 i,j+1 ) -G k,n i,j-1/2 (u k,n+1/2 i,j-1 , u k,n+1/2 ij ) (2.2)
where for fixed α, β in 0, 2/3 , we define

F k,n i+1/2,j (u, v) = f k,n i+1/2,j (u) + f k,n i+1/2,j (v) 2 - α(v -u) 2 λ x , f k,n i+1/2,j (u) = f k (t n , x i+1/2 , y j , u, A n i+1/2,j ) , G k,n i,j+1/2 (u, v) = g k,n+1/2 i,j+1/2 (u) + g k,n+1/2 i,j+1/2 (v) 2 - β(v -u) 2 λ y , g k,n+1/2 i,j+1/2 (u) = g k (t n+1/2 , x i , y j+1/2 , u, B n+1/2 i,j+1/2 ) .
The convolution terms are computed through quadrature formulae, so that they are given by

A n i+ 1 2 ,j =∆x∆y l,p∈Z N k=1 η 1,k i+1/2-l,j-p u k,n l+1/2,p , . . . , l,p∈Z N k=1 η m,k i+1/2-l,j-p u k,n l+1/2,p B n+1/2 i,j+ 1 2 =∆x∆y l,p∈Z N k=1 ϑ 1,k i-l,j+1/2-p u k,n+1/2 l,p+1/2 , . . . , l,p∈Z N k=1 ϑ m,k i-l,j+1/2-p u k,n+1/2 l,p+1/2 (2.3)
where, for instance, u k,n l+1/2,p is a convex combination of u k,n l,p and u k,n l+1,p , moreover

η ℓ,k i+1/2,j = η ℓ,k (x i+1/2 , y j ) and ϑ ℓ,k i,j+1/2 = ϑ ℓ,k (x i , y j+1/2 ) . (2.4)
Throughout, ∆t must satisfy the CFL condition

λ x ≤ min{6α, 4 -6α, 1} 1 + 6 max k ∂ u f k L ∞ , λ y ≤ min{6β, 4 -6β, 1} 1 + 6 max k ∂ u g k L ∞ , (2.5)
and we assume that the mesh sizes are sufficiently small, in the sense that ∆x, ∆y ≤ 1/(3M ), where M is as in (H1). The main result in [START_REF] Aggarwal | Nonlocal systems of conservation laws in several space dimensions[END_REF] is as follows:

Theorem 2.2. Let assumptions (H0), (H1), (H2), (H3) and conditions (2.5) hold. Fix an initial datum

U o ∈ (L 1 ∩ L ∞ ∩ BV)(R 2 ; R N + ).
Then, the algorithm (2.2) defines a sequence of approximate solutions which converges, up to a subsequence, to a solution U ∈ C 0 R + ; L 1 (R 2 ; R N + ) of (1.1) in the sense of Definition 2.1. Moreover, U is locally Lipschitz continuous with respect to the L 1 norm and for all k ∈ {1, . . . , N } and for all t ∈ R + , the following bounds hold:

U (t) L ∞ (R 2 ;R N ) ≤ e C t(1+ Uo L 1 ) U o L ∞ (R 2 ;R N ) , U k (t) L 1 (R 2 ;R) = U k o L 1 (R 2 ;R) , TV(U k (t)) ≤ e K1 t TV(U k o ) + K 2 e K1 t -1 , U (t + τ ) -U (t) L 1 (R 2 ;R N ) ≤ C(t) τ,
where C, K

∈ (L 1 ∩ L ∞ ∩ BV)(R 2 ; R N + ).
Then, the approximate solution u ∆ defined by the algorithm

(2.2) satisfies (a) (Positivity) u k ∆ (t, x, y) ≥ 0 for all k ∈ {1, . . . , N }, t ∈ R + and (x, y) ∈ R 2 . (b) (L 1 bound) u k ∆ (t) L 1 = u k ∆ (0) L 1 for all k ∈ {1, . . . , N } and t ∈ R + . (c) (L ∞ bound) For all t ∈ R + , u ∆ (t) L ∞ ≤ e C t(1+ Uo L 1 ) U o L ∞ , (2.6)
where C depends only on η, ϑ, f 1 , . . . , f N , g 1 , . . . , g N .

(d) (BV bound) For all n, for all t ∈ t n , t n+1 and for all k = 1, . . . ,

N ij u k,n i+1,j -u k,n ij ∆y + u k,n i,j+1 -u k,n ij ∆x ≤ e K1 t ij u k,0 i+1,j -u k,0 ij ∆y + u k,0 i,j+1 -u k,0 ij ∆x + K 2 e K1 t -1 .
where K 1 and K 2 are constants depending on U o L 1 and on the functions η, ϑ, f k and g k .

(e) (Discrete entropy condition) For all i, j ∈ Z, for all k = 1, . . . , N and for all κ ∈ R,

u k,n+1 ij -κ -u k,n ij -κ + λ x Φ k,n,κ i+1/2,j (u k,n ij , u k,n i+1,j ) -Φ k,n,κ i-1/2,j (u k,n i-1,j , u k,n ij ) + λ x sgn(u k,n+1/2 ij -κ) f k,n i+1/2,j (κ) -f k,n i-1/2,j (κ) + λ y Γ k,n,κ i,j+1/2 (u k,n+1/2 ij , u k,n+1/2 i,j+1 ) -Γ k,n,κ i,j-1/2 (u k,n+1/2 i,j-1 , u k,n+1/2 ij ) + λ y sgn(u k,n+1 ij -κ) g k,n+1/2 i,j+1/2 (κ) -g k,n+1/2 i,
Φ k,n,κ i+1/2,j (u 1 , u 2 ) = f k,n i+1/2,j (u 1 ∨ κ, u 2 ∨ κ) -f k,n i+1/2,j (u 1 ∧ κ, u 2 ∧ κ) , Γ k,n,κ i,j+1/2 (u 1 , u 2 ) = g k,n+1/2 i,j+1/2 (u 1 ∨ κ, u 2 ∨ κ) -g k,n+1/2 i,j+1/2 (u 1 ∧ κ, u 2 ∧ κ) . . (f ) (L 1 Lipschitz Continuity in time) For all n = 1, . . . , n, n ∈ N fixed, u ∆ (t n+1 ) -u ∆ (t n ) L 1 ≤ C ∆t .
where C is a constant depending on U o L 1 , TV(U o ), t n , λ x , λ y α, β and on the functions η, ϑ, f k and g k , for k ∈ {1, . . . , N }.

For the proofs of Theorem 2.2 and Lemma 2.3, refer to [START_REF] Aggarwal | Nonlocal systems of conservation laws in several space dimensions[END_REF].

Numerical Integrations

We refer to the class of nonlocal crowd dynamics models introduced in [START_REF] Colombo | Nonlocal crowd dynamics models for several populations[END_REF], for two populations trying to avoid each other and described by their densities U 1 , U 2 . In particular, we consider y) defines the preferred path for i-th population where each pedestrian moves in the preferred direction v i (x, y) but scales his speed according to (1

                   ∂ t U 1 + div   c 1 U 1 (1 -U 1 ) 1 -ε 1 U 1 * µ 1+ U 1 * µ 2 v 1 (x, y) -ε 2 ∇U 2 * µ 1+ ∇U 2 * µ 2   = 0, ∂ t U 2 + div   c 2 U 2 (1 -U 2 ) 1 -ε 1 U 2 * µ 1+ U 2 * µ 2 v 2 (x, y) -ε 2 ∇U 1 * µ 1+ ∇U 1 * µ 2   = 0. (3.1) For i, j = 1, 2, j = i, c i U i (1 -U i ), c i > 0 is the crowding factor and v i (x, y) = [v i1 , v i2 ](x,
-ε i (U i * µ)/ 1 + U i * µ
2 ), i.e., according to the density of its own population, which he sees in his horizon. The i-th population also deviates from its preferred trajectory due to the interaction with the individuals of the j-th population according to -ε j (∇U j * µ)/ 1 + ∇U j * µ 2 .

Lemma 3.1. System (3.1) fits into (1.1) setting, N = 2, m = 6 and

f (t, x, y, U, A) =      c 1 U 1 (1 -U 1 ) 1 -ε 1 A1 √ 1+A1 2 v 11 (x, y) -ε 2 A5 √ 1+A5 2 +A6 2 c 2 U 2 (1 -U 2 ) 1 -ε 1 A4 √ 1+A4 2 v 21 (x, y) -ε 2 A2 √ 1+A2 2 +A3 2      , g(t, x, y, U, B) =      c 1 U 1 (1 -U 1 ) 1 -ε 1 B1 √ 1+B1 2 v 12 (x, y) -ε 2 B6 √ 1+B5 2 +B6 2 c 2 U 2 (1 -U 2 ) 1 -ε 1 B4 √ 1+B4 2 v 22 (x, y) -ε 2 B3 √ 1+B2 2 +B3 2      , and
η = ϑ = µ ∂ x µ ∂ y µ 0 0 0 0 0 0 µ ∂ x µ ∂ y µ T , with A = η * U , B = ϑ * U , A, B ∈ R 6 . Moreover, if v i ∈ (C 2 ∩ W 2,∞ )(R 2 ; R 2 ) and µ ∈ (C 3 ∩ W 3,∞ )(R 2 ; R), for any compactly supported initial datum (U 1 o , U 2 o ) ∈ (L 1 ∩ L ∞ ∩ BV)(R 2 ; [0, 1]
2 ) problem (3.1) admits a solution defined for all t ∈ R + and satisfying the estimates in Theorem 2.2.

The proof is immediate and follows the same lines of [START_REF] Aggarwal | Nonlocal systems of conservation laws in several space dimensions[END_REF]Lemma 3.1]. Two different situations are considered and numerical integrations obtained with the algorithm described in Section 2 are exhibited. By Definition 2.1, [12, Theorem 3], the choices (3.5) and (3.9) of the initial data, the maximal density of solutions can not exceed 1, which is consistent with the results of the integrations.

Two Crowds Moving Radially in Opposite Directions

In the domain [-4, 4] × [-4, 4], two groups of people, one concentrated in a circle of radius √ 4.3 at the origin and other concentrated in an annulus between concentric circles (centered at origin) of radii √ 4.5 and √ 8.8, move radially outward and inward respectively. Thus, in (3.1) we choose: As in [START_REF] Colombo | Nonlocal crowd dynamics models for several populations[END_REF], the preferred path of each pedestrian is given by the sum g+δ. The vector g is tangent to the geodesic towards the pedestrian's target or 0 when pedestrians would not move. The vector δ describes the discomfort of pedestrians when walking close to a wall. It is perpendicular to the walls and pointing inward. Its maximum modulus is δ max = 0.8 at walls and decreases to zero at a distance δ r = 0.6 from walls. As indicated earlier, both crowds scale their speed due to their own density and deviate from their preferred trajectory due the presence of the other group present in their horizon. In (3.1) we choose: (3.9)

c 1 = c 2 = 4, ε 1 = 0.3, ε 2 = 0.7 (3.2) v 1 (x, y) = -v 2 (x, y) = [x, y] 1 + x 2 + y 2 , ( 3
c 1 = 2, c 2 = 1, ε 1 = 0.3, ε 2 = 0.7 (3.6)
The numerical integration of (3.1)-(3.9) is shown in Figure 3. The first group moves to the right. The second crowd, though prefers not to move, moves and forms lanes to pave the path for the first crowd to move. After the first crowd has completely vacated the room, the density of the second crowd keeps moving for a while.

Figure 4 shows the L 1 norm of the solution (U 1 , U 2 ) as a function of time t. It can be seen that the mass of the first crowd decreases to zero, while the mass of second crowd, remains constant.

  x, y) = (U 1 , . . . , U N )(t, x, y) ∈ R N , U k (t, x, y) ∈ R , k = 1, . . . y) ∈ R m×N , η ℓ,k (x, y) ∈ R , ℓ = 1, . . . , m , k = 1, . . . , N, ϑ(x, y) = , y) ∈ R m×N , ϑ ℓ,k (x, y) ∈ R , ℓ=1, . . . , m , k = 1, . . . , N.

. 3 ) 1 R 2 μFigure 1 Figure 1 :

 31211 Figure1shows the L 1 norm of the solution (U 1 , U 2 ) as a function of time t. It can be seen that the

3. 2 A

 2 Moving Crowd Interacting with a Standing Crowd Consider two groups of people in the domain [-1.7, 4.2] × [-1.7, 1.7] with a single exit at x = 4.2. At time 0, one group represented by U 1 starts moving towards the exit while the other group, U 2 , wants to stay and moves only as a reaction to the exiting population. The space available to the pedestrians is the rectangle [-1.4, 4.2] × [-1.4, 1.4] due to the presence of the walls of 0.3 width along {-1.7} × [-1.7, 1.7], and [-1.7, 4.2] × {-1.7} and [-1.7, 4.2] × {1.7}.

Figure 2 : 1 R 2 μ

 212 Figure 2: Solution to (3.1)-(3.5) with space mesh with ∆x = ∆y = .0125, λ x = λ y = .05, ∆t = 6.25E -4 and α = β = .3333. A symmetric(radial) formation of lanes is noticed for both the crowds. Each crowd forms lanes to pave paths for the other crowd to move in/out of the domain. The lanes formed by the two crowds are complimentary to each other. The formation of the lanes is noticed till the first crowd U 1 completely exits the domain.

Figure 3 :

 3 Figure 3: Solution to (3.1)-(3.9) computed with ∆x = ∆y = .01, λ x = λ y = .05, ∆t = .005 and α = β = .3333. Note that though the second crowd U 2 does not prefer to move, the model makes U 2 to adjust and develop lanes to allow the first crowd U 1 to move along its preferred path to right. The lane formation ceases to exist after the first crowd U 1 has completely vacated the domain.

Figure 4 :

 4 Figure 4: Time evolution of the L 1 norm of (U 1 , U 2 ) as a function of t.

  [START_REF] Aggarwal | Nonlocal systems of conservation laws in several space dimensions[END_REF] , K 2 and C(t) are constants defined in Lemma 2.3 below.The proof is based on Lemma 2.3 stated below. In view of [1, Remark 2.1], (H0) and (H3), Theorem 2.2 ensures the existence of a solution and the validity of the bounds above on any finite time interval, whenever the initial datum is compactly supported.

Lemma 2.3. Let assumptions (H0), (H1), (H2), (H3) and condition (2.5) hold. Fix U o

Acknowledgements. This work was partially supported by the European Research Council under the European Union's Seventh Framework Program (FP/2007-2013) ERC Grant Agreement n. 257661. The authors would like to thank the third co-author R. M. Colombo (INDAM Unit, University of Brescia) who contributed significantly to this work.


Paola Goatin
email: paola.goatin@inria.fr

Sheila Scialanga
email: sheila.scialanga@gmail.com

WELL-POSEDNESS AND FINITE VOLUME APPROXIMATIONS OF THE LWR TRAFFIC FLOW MODEL WITH NON-LOCAL VELOCITY

Keywords: 2010 Mathematics Subject Classification. Primary: 35L65; Secondary: 65M12, 90B20 Scalar conservation laws, non-local flux, macroscopic traffic flow models, finite volume schemes

We consider an extension of the traffic flow model proposed by Lighthill, Whitham and Richards, in which the mean velocity depends on a weighted mean of the downstream traffic density. We prove well-posedness and a regularity result for entropy weak solutions of the corresponding Cauchy problem, and use a finite volume central scheme to compute approximate solutions. We perform numerical tests to illustrate the theoretical results and to investigate the limit as the convolution kernel tends to a Dirac delta function.

1. Introduction. Macroscopic traffic flow models provide nowadays a validated and powerful approach to simulate and manage traffic on road networks (we refer the interested reader to [START_REF] Treiber | Traffic Flow Dynamics[END_REF] for an overview of modelling approaches and practical applications and to [START_REF] Garavello | Traffic Flow on Networks[END_REF] for a detailed review of the mathematical theory of macroscopic models on networks). These models are based on hyperbolic equations derived from fluid dynamics, and describe the spatio-temporal evolution of macroscopic quantities like vehicle density and mean velocity. One of the seminal macroscopic models was introduced in the mid 1950s by Lighthill and Whitham [START_REF] Lighthill | On kinematic waves. II. A theory of traffic flow on long crowded roads[END_REF] and Richards [START_REF] Richards | Shock waves on the highway[END_REF], who proposed to complete the one-dimensional mass conservation equation ∂ t ρ(t, x) + ∂ x f (t, x) = 0 with a closure relation between speed and density, leading to the fundamental diagram f (t, x) = f (ρ(t, x)) = ρ(t, x)v(ρ(t, x)), which can be derived from empirical speed-density or flow-density data. The classical LWR model therefore reads

∂ t ρ(t, x) + ∂ x (ρ(t, x)v(ρ(t, x))) = 0, x ∈ R, t > 0,
where ρ(t, x) ≥ 0 is the traffic density and the non-increasing function v : R + → R + is the mean velocity. Thanks to its simplicity and its established analytical and numerical properties, this model is widely used in modern traffic engineering, see e.g. [START_REF] De Wit | Grenoble Traffic Lab: An experimental platform for advanced traffic monitoring and forecasting[END_REF][START_REF] Herrera | Incorporation of lagrangian measurements in freeway traffic state estimation[END_REF] for some recent applications. Nevertheless, like every model describing complex systems, it suffers form some limitations that limit its applicability.

In particular, like all classical macroscopic models, it allows for speed discontinuities, which translate in infinite acceleration values. This restrains the use of these equations in combination with consumption and atmospheric and noise pollution models, which require an accurate estimation of the acceleration terms. Aiming to overcome this difficulty, a prototype of non-local traffic flow models has been recently introduced in [START_REF] Blandin | Well-posedness of a conservation law with non-local flux arising in traffic flow modeling[END_REF], where the speed is assumed to depend on a weighted mean of the downstream traffic conditions. Due to the action of a convolution product, the speed becomes a Lipschitz function of the space and time variables, guaranteeing bounded acceleration.

In this paper, we provide a generalization and some extensions of the results presented in [START_REF] Blandin | Well-posedness of a conservation law with non-local flux arising in traffic flow modeling[END_REF]: instead of focussing on the linear decreasing function v(ρ) = 1 -ρ, we consider general non-increasing velocities, we provide a regularity result and study a higher order finite volume central scheme for a better approximation of the solutions. More precisely, we consider the following mass conservation equation for traffic flow with non-local mean velocity:

∂ t ρ(t, x) + ∂ x ρ(t, x)v x+η x ρ(t, y)w η (y -x) dy = 0 , (1)
defined for t ∈ R + and x, η ∈ R, η > 0. The convolution kernel w η ∈ C 1 ([0, η]; R + ) is a non-increasing function such that η 0 w η (x)dx = 1 (for example, w η (x) ≡ 1/η or w η (x) = 2(1 -x/η)/η).
In (1), we consider a speed function v : [0,

ρ max ] → R + such that -A ≤ v ≤ 0, with A ∈ R + , v(0) = v max , v(ρ max ) = v min ≥ 0 ,
where ρ max denotes the maximal car density. We denote the downstream convolution product as

ρ * d w η (t, x) := x+η x ρ(t, y)w η (y -x) dy .
This term is intended to reproduce the fact that drivers adapt their velocity to the downstream traffic, assigning a greater importance to closer vehicles. Setting V (t, x) = v(ρ * d w η (t, x)), we rewrite (1) as

∂ t ρ(t, x) + ∂ x (ρ(t, x)V (t, x)) = 0 , with initial datum ρ(0, x) = ρ 0 (x) ∈ [0, ρ max ]. (2)
We will consider solutions ρ = ρ(t, x) satisfying the following definition (see [START_REF] Betancourt | On nonlocal conservation laws modelling sedimentation[END_REF][START_REF] Blandin | Well-posedness of a conservation law with non-local flux arising in traffic flow modeling[END_REF][START_REF] Colombo | Control of the continuity equation with a non local flow[END_REF][START_REF] Kružkov | First order quasilinear equations with several independent variables[END_REF]):

Definition 1.1. A function ρ ∈ (L 1 ∩ L ∞ ∩ BV)(R + × R; R) is an entropy weak solution if +∞ 0 +∞ -∞ (|ρ -κ|ϕ t +|ρ -κ| V ϕ x -sgn(ρ -κ) κ V x ϕ) (t, x)dxdt + +∞ -∞ |ρ 0 (x) -κ|ϕ(0, x) dx ≥ 0 (3)
for all ϕ ∈ C 1 c (R 2 ; R + ) and κ ∈ R.

Conservation laws with non-local in space terms arose recently in several application domains, such as granular flows [START_REF] Amadori | An integro-differential conservation law arising in a model of granular flow[END_REF], sedimenatation [START_REF] Betancourt | On nonlocal conservation laws modelling sedimentation[END_REF], pedestrian flows [START_REF] Colombo | A class of nonlocal models for pedestrian traffic[END_REF], conveyor belts [START_REF] Göttlich | Modeling, simulation and validation of material flow on conveyor belts[END_REF] and aggregation phenomena [START_REF] James | Numerical methods for one-dimensional aggregation equations[END_REF]. Also, other traffic flow models with non-local terms have been proposed and studied in [START_REF] Herty | Coupling of non-local driving behaviour with fundamental diagrams[END_REF][START_REF] Li | Shock formation in a traffic flow model with Arrhenius look-ahead dynamics[END_REF][START_REF] Sopasakis | Stochastic modeling and simulation of traffic flow: asymmetric single exclusion process with Arrhenius look-ahead dynamics[END_REF]. Some general analytical results on non-local conservation laws, proving existence and eventually uniqueness of solutions, can be found in [START_REF] Amorim | On the numerical integration of scalar nonlocal conservation laws[END_REF] for scalar equations in one space dimension, in [START_REF] Colombo | Control of the continuity equation with a non local flow[END_REF] for equations in several space dimension and in [START_REF] Aggarwal | Nonlocal systems of conservation laws in several space dimensions[END_REF][START_REF] Colombo | Nonlocal crowd dynamics models for several populations[END_REF][START_REF] Crippa | Existence and uniqueness of measure solutions for a system of continuity equations with non-local flow[END_REF] for multi-dimensional systems of conservation laws. Unlike all the above mentioned results, in the case of equation ( 1) we deal with a very specific (non-increasing, discontinuous) convolution kernel. Despite its lack of regularity at the origin, the monotonicity of the kernel allows to obtain refined L ∞ and BV estimates on Lax-Friedrichs approximate solutions, which as a by-product allow to prove the existence of weak entropy solutions by a standard compactness argument. In particular, we prove that a maximum principle holds, which guarantees the positivity of solutions and a uniform upper bound, given by the maximum of the initial density. We stress that these properties, which are of utmost importance in traffic flow applications, depend heavily on the non-increasing monotonicity of the kernel, as numerical counterexamples show, see [START_REF] Blandin | Well-posedness of a conservation law with non-local flux arising in traffic flow modeling[END_REF]Section 5].

The paper is organized as follows: Section 2 is devoted to the analytical results, summarized in Theorem 2.1 and some intermediate relevant statements. Section 3 describes a generalisation of the Nessyahu-Tadmor finite volume central scheme and Section 4 collects the results of the numerical tests. The final Appendix contains some interesting details of the proof of the BV-estimates.

2. Well-posedness and regularity. The main results are reported in the following theorem: Theorem 2.1. Let ρ 0 ∈ BV(R; [0, ρ max ]) and w η ∈ C 1 ([0, η]; R + ) be a non-increasing function such that η 0 w η (x)dx = 1. Then the Cauchy problem

∂ t ρ + ∂ x (ρv(ρ * d w η )) = 0, x ∈ R, t > 0, ρ(0, x) = ρ 0 (x), x ∈ R,
admits a unique weak entropy solution in the sense of Definition 1.1. Moreover it holds min

R {ρ 0 } ≤ ρ(t, x) ≤ max R {ρ 0 }, for a.e. x ∈ R, t > 0. (4)
Uniqueness of solutions is proved in [START_REF] Blandin | Well-posedness of a conservation law with non-local flux arising in traffic flow modeling[END_REF]. The proof of existence is based on the convergence of a sequence of approximate solutions as in [START_REF] Blandin | Well-posedness of a conservation law with non-local flux arising in traffic flow modeling[END_REF], see also [START_REF] Amorim | On the numerical integration of scalar nonlocal conservation laws[END_REF][START_REF] Betancourt | On nonlocal conservation laws modelling sedimentation[END_REF][START_REF] Colombo | A class of nonlocal models for pedestrian traffic[END_REF][START_REF] Colombo | Control of the continuity equation with a non local flow[END_REF][START_REF] Colombo | Nonlocal crowd dynamics models for several populations[END_REF] for related results. Here we briefly describe the finite volume scheme used to construct approximate solutions and we state its key proprieties: maximum principle, bounded total variation and discrete entropy inequalities, which allow to prove the convergence to a weak entropy solution.

Let us consider a space step ∆x such that η = N ∆x, for some N ∈ N, and a time step ∆t subject to a CFL condition which will be specified later. For j ∈ Z and n ∈ N, let x j+1/2 = j∆x be the cells interfaces, x j = (j -1/2)∆x the cells centers and t n = n∆t the time mesh. To construct a finite volume approximate solution

ρ ∆x (t, x) = ρ n j for (t, x) ∈ C n j = [t n , t n+1 [×]x j-1/2 , x j+1/2
[, we take a piece-wise constant approximation of the initial datum ρ 0 ,

ρ 0 j = 1 ∆x x j+1/2 x j-1/2 ρ 0 (x) dx ,
we denote w k η := w η (k∆x) for k = 0, . . . , N -1 and

V n j := v ∆x N -1 k=0 w k η ρ n j+k . (5)
Note that the discretization chosen for w η implies ∆x

N -1 k=0 w k η ≤ 1 + w η (0)∆x. ( 6
)
so it slightly underestimates traffic mean velocity and may introduce unphysical negative velocities. Other discretizations are possible, see for example [4, eq. (3.

2)], but our choice is simpler to implement and does not sensibly affect the results. We consider the following modified Lax-Friedrichs flux adapted to (1):

F n j+1/2 := 1 2 ρ n j V n j + 1 2 ρ n j+1 V n j+1 + α 2 (ρ n j -ρ n j+1 ) ,
where α ≥ 1 is the viscosity coefficient. This gives the N + 2 points finite volume scheme

ρ n+1 j = ρ n j + λα 2 ρ n j-1 -2ρ n j + ρ n j+1 + λ 2 ρ n j-1 V n j-1 -ρ n j+1 V n j+1 (7)
with λ = ∆t/∆x. Easy calculations, detailed in [START_REF] Goatin | The Lighthill-Whitham-Richards traffic flow model with nonlocal velocity: analytical study and numerical results[END_REF], show that the numerical scheme is not monotone w. r. to the variables ρ j+k , k = 2, . . . , N -2. Therefore, classical convergence arguments do not apply in this context. To prove the convergence of the numerical approximations, we will rely on the following results, see [START_REF] Blandin | Well-posedness of a conservation law with non-local flux arising in traffic flow modeling[END_REF][START_REF] Goatin | The Lighthill-Whitham-Richards traffic flow model with nonlocal velocity: analytical study and numerical results[END_REF] for details of the proofs.

Proposition 1. (Maximum principle and L ∞ -estimates) For any initial datum ρ 0 j , j ∈ Z, let

ρ m = min j∈Z {ρ 0 j } ∈ [0, ρ max ] and ρ M = max j∈Z {ρ 0 j } ∈ [0, ρ max ].
Then the finite volume approximation ρ n j , j ∈ Z and n ∈ N, constructed using scheme [START_REF] Colombo | A class of nonlocal models for pedestrian traffic[END_REF], satisfies the bounds

ρ m ≤ ρ n j ≤ ρ M for all j ∈ Z and n ∈ N, under the CFL condition ∆t ≤ 2 2α + A∆x w η (0) ∆x . (8)
This is an important property for traffic flow applications. In particular, we expect solutions to (1) not to exceed the maximal density bound ρ max . It would be interesting to see if different convolution choices are able to preserve this upper bound property while allowing density to increase with respect to the initial values.

Proposition 2. (BV-estimates) Let ρ 0 ∈ BV(R; [0, ρ max ]), and let ρ ∆x be given by [START_REF] Colombo | A class of nonlocal models for pedestrian traffic[END_REF]. If α ≥ v max + A w η (0)∆x (and α ≥ 2A w η (0)∆x) and the CFL condition ∆t ≤ ∆x/(α + 2A w η (0)∆x)) holds, then for every T > 0 the following discrete space BV estimate holds

TV(ρ ∆x (T, •)) ≤ C(w η , v, ρ 0 , T ) := e wη(0) 5A+7 v ∞ T 2 TV(ρ 0 ). (9)
In particular, solutions to (1) are not total variation decreasing, unlike the solutions of the corresponding classical (non non-local) conservation law. Moreover, it is worth noticing that the bound provided by [START_REF] Colombo | Nonlocal crowd dynamics models for several populations[END_REF] depends on the L ∞ -norm of the derivatives of the velocity function v. Also, unlike the case of linear velocity studied in [START_REF] Blandin | Well-posedness of a conservation law with non-local flux arising in traffic flow modeling[END_REF], the scheme is not monotonicity preserving for general functions v. We refer the reader to the Appendix for a sketch of the proof.

Following [3, Proposition 2.8] and [5, Section 3.3], we derive a discrete entropy inequality for the approximate solution generated by [START_REF] Colombo | A class of nonlocal models for pedestrian traffic[END_REF]. Let us denote

G j+1/2 (u, v) := 1 2 u V n j + 1 2 v V n j+1 + α 2 (u -v) , F κ j+1/2 (u, v) := G j+1/2 (u ∧ κ, v ∧ κ) -G j+1/2 (u ∨ κ, v ∨ κ), with a ∧ b = max(a, b) and a ∨ b = min(a, b).
Proposition 3. (Discrete entropy inequalities) Let ρ n j , j ∈ Z, n ∈ N, be given by [START_REF] Colombo | A class of nonlocal models for pedestrian traffic[END_REF]. Then, if α ≥ 1 and the CFL condition (8) holds, for all j ∈ Z, n ∈ N we have

ρ n+1 j -κ -ρ n j -κ + λ F κ j+1/2 (ρ n j , ρ n j+1 ) -F κ j-1/2 (ρ n j-1 , ρ n j ) + λ 2 sgn(ρ n+1 j -κ) κ V n j+1 -V n j-1 ≤ 0 for all κ ∈ R.
The proof of convergence of ρ ∆x to an entropy weak solution in the sense of Definition 1.1 follows closely [5, Section 4] and is detailed in [12, Section 2.5].

Concerning the regularity of the solutions of (1), (2), we recover the same result as in [4, Lemma 5.5].

Proposition 4. (Lipschitz regularity) For any T > 0, the numerical solution generated by the scheme (7) converges to a Lipschitz continuous function ρ, provided ρ 0 is also Lipschitz continuous.

The proof is detailed in [12, Section 2.6].

3.

A finite volume central scheme. In this section, we describe a central scheme for (1) inspired by [START_REF] Kurganov | Non-oscillatory central schemes for a traffic flow model with Arrehenius look-ahead dynamics[END_REF], which is an extension of the second-order Nessyahu-Tadmor scheme [START_REF] Nessyahu | Non-oscillatory central differencing for hyperbolic conservation laws[END_REF]. At each time step, the solution is first approximated by a piece-wise linear function, then evolved to the next time step according to the integral form of conservation law.

Denoting by

R(t, x) := x+η x ρ(t, y)w η (y -x) dy ,
the flux function in (1) can be written as

F (ρ, R) := ρv(R) = ρ(t, x)v x+η x
ρ(t, y)w η (y -x) dy , so that (1) rewrites

∂ t ρ(t, x) + ∂ x F (ρ, R) = 0, t ∈ R + , x ∈ R . ( 10
)
With the notations introduced at the beginning of Section 2, let us assume that at time t = t n we have the solution approximated by its cell averages

ρn j := 1 ∆x Cj ρ(x, t n ) dx .
We construct the piecewise linear interpolant approximation given by ρn (x) := ρn

j + s n j (x -x j ), x ∈ x j-1/2 , x j+1/2 . (11)
Formally, we have a second-order approximation provided the slopes s n j are at least first-order approximations of the derivatives ρ x (t n , x j ). We use a generalized minmod reconstruction (as in [START_REF] Kurganov | Non-oscillatory central schemes for a traffic flow model with Arrehenius look-ahead dynamics[END_REF]) with:

s n j = minmod θ ρn j -ρn j-1 ∆x , ρn j+1 -ρn j-1 2∆x , θ ρn j+1 -ρn j ∆x , θ ∈ [1, 2], (12)
where the minmod function is defined by

minmod(a 1 , a 2 , a 3 ) :=      min j {a j }, if a j > 0, j = 1, 2, 3, max j {a j }, if a j < 0, j = 1, 2, 3, 0, otherwise,
and the parameter θ influences the numerical viscosity of the scheme: larger values of θ correspond to less dissipative but more oscillatory approximations, see [START_REF] Lie | On the artificial compression method for second-order nonoscillatory central difference schemes for system of conservation laws[END_REF].

We now evolve the approximated solution [START_REF] Garavello | Traffic Flow on Networks[END_REF] to the next time level t = t n+1 by integrating equation [START_REF] Crippa | Existence and uniqueness of measure solutions for a system of continuity equations with non-local flow[END_REF] over the space-time volumes [x j , x j-1 ] × [t n , t n+1 ], which gives:

ρn+1 j+ 1 2 = 1 ∆x xj+1 xj ρn (x)dx - 1 ∆x t n+1 t n [F (ρ(t, x j+1 ), R(t, x j+1 )) -F (ρ(t, x j ), R(t, x j ))] dt. (13)
The first integral on the right-hand side of ( 13) can be computed exactly:

1 ∆x xj+1 xj ρn (x)dx = 1 ∆x x j+ 1 2 xj [ρ n j + s n j (x -x j )]dx + 1 ∆x xj+1 xj + 1 2 [ρ n j+1 + s n j+1 (x -x j+1 )]dx = ρn j + ρn j+1 2 + ∆x 8 (s n j -s n j+1 ).
The flux integral in (13) should be computed using the approximate solution of the initial value problem [START_REF] Crippa | Existence and uniqueness of measure solutions for a system of continuity equations with non-local flow[END_REF], [START_REF] Garavello | Traffic Flow on Networks[END_REF]

-λ F (ρ(t n+ 1 2 , x j+1 ), R(t n+ 1 2 , x j+1 )) -F (ρ(t n+ 1 2 , x j ), R(t n+ 1 2 , x j )) , (14)
where we approximate the intermediate time level values of ρ and R with the corresponding Taylor expansions, namely:

ρ(t n+ 1 2 , x j ) ≈ ρn (x j ) + ∆t 2 ρ t (t n , x j ), (15)
R(t n+ 1 2 , x j ) ≈ R(t n , x j ) + ∆t 2 R t (t n , x j ). (16)
By [START_REF] Garavello | Traffic Flow on Networks[END_REF] we have: ρn (x j ) = ρn j and by [START_REF] Crippa | Existence and uniqueness of measure solutions for a system of continuity equations with non-local flow[END_REF] we evaluate the time derivative ρ t as:

ρ t (t n , x j ) = -F (ρ(t n , x j ), R(t n , x j )) x . (17)
The space derivative F x in ( 17) is computed using the minmod limiter:

F (ρ(t n , x j ), R(t n , x j )) x = = minmod θ F (ρ n j , R n j ) -F (ρ n j-1 , R n j-1 ) ∆x , F (ρ n j+1 , R n j+1 ) -F (ρ n j-1 , R n j-1 ) 2∆x , θ F (ρ n j+1 , R n j+1 ) -F (ρ n j , R n j ) ∆x .
We compute the terms in ( 16) by the composite trapezoidal and mid-point rules, noting that the first and the last intervals are halved. Recalling that N := η/∆x we get

R(t n , xj) = x j +η x j ρ(
Rt(t n , xj) = x j +η x j ∂tρ(t n , y)w(y -xj) dy = - x j +η x j ∂yF (ρ(t n , y), R(t n , y)) w(y -xj) dy = x j +η x j F (ρ(t n , y), R(t n , y)) w (y -xj) dy -F (ρ(t n , y), R(t n , y)) w(y -xj) y=x j +η y=x j = F (ρ(t n , xj), R(t n , xj))w(0) -F (ρ(t n , xj + η), R(t n , xj + η))w(η) + x j +η x j F (ρ(t n , y), R(t n , y))w (y -xj) dy ≈ F (ρ(t n , xj), R(t n , xj))w(0) -F (ρ(t n , xj + η), R(t n , xj + η))w(η) + ∆x 2 F (ρ(t n , xj), R(t n , xj))w (0) + F (ρ(t n , xj+N ), R(t n , xj+N ))w (η) + ∆x N -1 k=1 F (ρ(t n , x j+k ), R(t n , x j+k ))w (k∆x).
Since the derived scheme uses alternating, staggered grids, we have to distinguish between the odd and even time steps. Formulas ( 14), ( 11)-( 12), ( 15)-( 16) describe the odd steps. The even steps are obtained by shifting the indexes in the aforementioned equations by 1 2 and the computational domain should be extended by ∆x 2 on both sides. Concerning boundary conditions, in our numerical tests the solution is constant near the boundaries, so we use these constant values on an extended domain.

4. Numerical tests. In this section, we perform numerical simulations with different choices for the speed law v and the convolution kernel w η . More precisely, we consider the following velocity functions, see [START_REF] Garavello | Traffic Flow on Networks[END_REF]:

Greenshield: v(ρ) = v max 1 - ρ ρ max n , n ∈ N , Greenberg: v(ρ) = v max ln ρ max ρ , Underwood: v(ρ) = v max exp - ρ ρ max ,
with v max = 1 and ρ max = 1, and the kernels

w η ∈ C 1 ([0, η]; R + ): constant: w η (x) = 1 η , linear decreasing: w η (x) = 2 η -x η 2 ,
We recall that in [START_REF] Blandin | Well-posedness of a conservation law with non-local flux arising in traffic flow modeling[END_REF] the authors considered the Greenshield model with n = 1 (i.e. linear decreasing velocity).

For the tests presented below, the space domain is given by the interval [-1, 1], the space discretization mesh ∆x = 0.002 and η = 0.1, where not specified otherwise. Absorbing conditions are imposed at the boundaries. At the right boundary, we add N = η/∆x ghost cells and define

ρ n j = ρ n 2 ∆x
for every j = 2 ∆x +1, . . . , 2 ∆x +N , thus extending the solution constantly equal to the last value inside the domain. At the left boundary, we just need to add one ghost cell, as in classical problems.

Monotonicity of the solution.

In Figure 1 we compare the solutions of the Riemann problem with initial datum

ρ 0 (x) = 0.2, if x < 0, 0.8, if x > 0, (18)
with constant kernel w η = 1/η. Numerical simulations confirm that the scheme generally is not monotonicity preserving, as shown in the case of the Greenberg's and the Underwood's velocities, which are non-linear, see the Appendix for more details on this aspect. As a consequence, the corresponding total variations (Figure 2) are not constant. 4.2. Limit η 0. We consider equation ( 1) equipped with Greenshield's velocity (n = 1) in the situation studied in [18, Section 3, Example 1]: a red traffic light is located at x = -0.1 and it turns green at the initial time t = 0, so the initial datum is:

ρ 0 (x) = 0.8, if -0.5 < x < -0.1, 0, otherwise. (19)
First of all, we compare the approximate solutions obtained with the Lax-Friedrichs and the Nessyahu-Tadmor schemes at time T = 0.5 for the mesh sizes ∆x = 0.01 and ∆x = 0.001, see Figure 3. In all cases, we remark the higher precision obtained by the Nessyahu-Tadmor scheme.

We now investigate the convergence to the solution of the classical conservation law

∂ t ρ + ∂ x (ρ(1 -ρ)) = 0
as η 0. To this end, we compare the solutions of (1) obtained using scheme [START_REF] Herrera | Incorporation of lagrangian measurements in freeway traffic state estimation[END_REF] with different values of η = 0.1, 0.01, 0.001 with the local solution at time T = 0.5 for ∆x = 0.0001 (Figure 4). Contrarily to the results obtained in the case of the Arrhenius look-ahead model, the solutions to the non-local LWR model seem to converge to the solution of the classical LWR equation as the support of the convolution kernel shrinks to a point, as shown in Figure 4. In particular, we remark that the solutions of the non-local model (1) display sharp discontinuities in the density profile (one of which being a decreasing jump that could look unphysical, but satisfies the entropy condition (3)). On the contrary, the corresponding velocities are Lipschitz continuous, with a Lipschitz constant that depends on the L ∞ -norm of w η . Indeed, it is easy to show that there holds the estimate

∂ x V (t, •) ∞ ≤ 2w η (0) v ∞ ρ(t, •) ∞ , see also [5, Proof of Theorem 2].
We also compute numerical convergence orders and L 1 -errors of non-local solutions ρ η computed with scheme ( 14), θ = 1, for η 0, with respect to the local solution ρ computed with classical Lax-Friedrichs method as reference solution, as for η = 0.1, 0.01, 0.001 (see Table 1). 1. Convergence orders and L 1 -errors for η 0. The data are related to the cases of Figure 4, with linear decreasing velocity v(ρ) = 1 -ρ at final time T = 0.5 corresponding to the initial datum [START_REF] Li | Shock formation in a traffic flow model with Arrhenius look-ahead dynamics[END_REF].

e(η) = ρ η (T, •) -ρ(T, •) L 1 ,
w η (x) = 1/η w η (x) = 2(η -x)/η 2 η γ(η) L 1 -error γ(η) L 1 -
Appendix: Sketch of the proof of Proposition 2.

Proof. Setting ∆ n j+k-1/2 = ρ n j+k -ρ n j+k-1 for k = 0, . . . , N + 1, from [START_REF] Colombo | A class of nonlocal models for pedestrian traffic[END_REF] we get

∆ n+1 j+ 1 2 = λα 2 ∆ n j-1 2 + (1 -λα) ∆ n j+ 1 2 + λα 2 ∆ n j+ 3
+ λ 2 V n j ∆ n j-1 2 + ρ n j-1 V n j -V n j-1 -V n j+2 ∆ n j+ 3 2 + ρ n j+1 V n j+1 -V n j+2 = λ 2 α + V n j ∆ n j-1 2 + (1 -λα)∆ n j+ 1 2 + λ 2 α -V n j+2 ∆ n j+ 3 2 + λ 2 ρ n j-1 ∆x v (ξ) N -1 k=0 w k η ∆ n j+k-1 2 - λ 2 ρ n j+1 ∆x v (ξ ) N -1 k=0 w k η ∆ n j+k+ 3 2
for some ξ between

N -1 k=0 w k η ρ n j+k and N -1 k=0 w k η ρ n j+k-1 and ξ between N -1 k=0 w k η ρ n j+k+2 and N -1 k=0 w k η ρ n j+k+1 .
After some manipulations we get:

∆ n+1 j+ 1 2 = λ 2 α + V n j + ρ n j-1 ∆x v (ξ)w 0 η -∆x v (ξ ) N -2 k=2 w k-1 η ∆ n j+k+ 1 2 ∆ n j-1 2 (20a) + 1 -λα + λ 2 ρ n j-1 ∆x v (ξ)w 1 η - λ 2 ∆x v (ξ ) N -2 k=2 w k-1 η ∆ n j+k+ 1 2 ∆ n j+ 1 2 (20b) + λ 2 α -V n j+2 + ρ n j-1 ∆x v (ξ)w 2 η -ρ n j+1 ∆x v (ξ )w 0 η ∆ n j+ 3 2 (20c) + λ 2 ∆x N -2 k=2 ∆ n j+k+ 1 2 ρ n j-1 v (ξ)(w k+1 η -w k-1 η ) + w k-1 η ρ n j-1 v (ξ) -v (ξ ) (20d) - λ 2 ρ n j+1 ∆x v (ξ ) w N -2 η ∆ n j+N -1 2 (20e) - λ 2 ρ n j+1 ∆x v (ξ ) w N -1 η ∆ n j+N + 1 2 . ( 20f
)
The coefficients of (20e) and (20f) are non-negative. The assumption α ≥ 2A∆x w η (0)

guarantees the positivity of (20a). Similarly for (20c) we get α ≥ v max + ∆xAw η (0)

and for (20b) we get the CFL condition ∆t ≤ 1 α + 2A ∆x w η (0) ∆x.

We observe here that the coefficient in (20d) is not positive in general, hence the scheme is not monotonicity preserving unless v = 0 as was the case in [START_REF] Blandin | Well-posedness of a conservation law with non-local flux arising in traffic flow modeling[END_REF].

Taking the absolute values in [START_REF] Lie | On the artificial compression method for second-order nonoscillatory central difference schemes for system of conservation laws[END_REF], summing over j ∈ Z and rearranging the indexes we obtain: Details of the above calculations can be found in [START_REF] Goatin | The Lighthill-Whitham-Richards traffic flow model with nonlocal velocity: analytical study and numerical results[END_REF].

Figure 1 .

 1 Figure 1. Density profiles that solve (1), (18) at time t = 0.01 with kernel w η (x) = 1/η. Cases (c) (d) show that the corresponding numerical scheme is not monotonicity preserving.

  (a) Greenshield's velocity, n = 1 (b) Greenshield's velocity, n = 5 (c) Greenberg's velocity (d) Underwood's velocity

Figure 2 .

 2 Figure 2. Total Variation TV(ρ(t, •); [-1, 1]) for t ∈ [0, 0.3] corresponding to the Riemann-like initial datum with ρ L = 0.2, ρ R = 0.8 for w(x) = 1/η.

  and γ(η) = log 10 e(η) e(η/10) , (a) ∆x = 0.01, wη = 1/η (b) ∆x = 0.01, wη = 2(η -x)/η 2 (c) ∆x = 0.001, wη = 1/η (d) ∆x = 0.001, wη = 2(η -x)/η 2

Figure 3 .

 3 Figure 3. Density profiles that solve (1), (19) at time t = 0.5 computed by the Lax-Friedrichs (LF) and the Nessyahu-Tadmor (NT) schemes.

2 (

 2 a) density (left) and velocity (right) profiles for linear velocity with constant kernel, θ = 1 (b) density (left) and velocity (right) profiles for linear velocity with linear kernel, θ = 1

Figure 4 .

 4 Figure 4. Density and velocity profiles at time t = 0.5 for the non-local equation with η = 0.1, 0.01, 0.001 and the local equation.

1 η 2 η-. ( 22 ) 2 .

 12222 n j+1 + ρ n j ∆x v (ξ)w 0 η -∆x v (ξ ) +2 ∆x v (ξ ) w N -2 +1 ∆x v (ξ ) w N -η v (ξ) -v (ξ ) + ρ n j-1 v (ξ)w 1 η + ρ n j-2 v (ξ)w -1 v (ξ) -v (ξ ) -v (ξ )ρ n j-N +2 w N -2 η -v (ξ )ρ n j-N +1 w N -1 η Rewriting v (ξ) -v (ξ ) as v (β)(ξ -ξ ), with β a point between ξ and ξ , we get the estimate|v (ξ) -v (ξ )| ≤ 7 v ∞ w η (Therefore, we recover the following estimate for the total variation TV(ρ ∆x (T, •)) ≤ 1

Acknowledgments. This research was conducted while S. Scialanga was visiting Inria Sophia Antipolis -Méditerranée, partially supported by the Faculty of Science of the University of Rome "La Sapienza".

Partially supported by the European Research Council under the European Union's Seventh Framework Program (FP/2007-2013) / ERC Grant Agreement n. 257661. Received xxxx 20xx; revised xxxx 20xx.