A non-local macroscopic model for traffic flow

In this work, we propose a non-local Hamilton–Jacobi model for traffic flow and we prove the existence and uniqueness of the solution of this model. This model is justified as the limit of a rescaled microscopic model. We also propose a numerical scheme and we prove an estimate error between the continuous solution of this problem and the numerical one. Finally, we provide some numerical illustrations.


Introduction
Traffic flow modelling is an important challenge and has known an important development in the last decades. The goal of this paper is to propose a new non-local macroscopic model for traffic flow. Macroscopic models consider quantities that describe the collective behaviour of traffic flow. At the macroscopic scale, the most popular model is the LWR model (see [28,31]). This model, expressed in the Eulerian coordinates, describes the dynamics of the density of vehicles. Since these pioneering works, a lot of models have been proposed and we refer to [17] for an overview of these models. More recently other approaches have been proposed. First, using the lagrangian coordinates, the LWR model can be reformulated to describe the dynamics of the spacing (see [27]). Moreover, all these models could be reformulated using a Hamilton-Jacobi equation. Indeed, the link between conservation laws and Hamilton-Jacobi equations has been known to mathematicians for decades [21,25,30], but was brought up to the attention of the traffic flow theory community just recently by [10,11] (see also [24]). At the Hamilton-Jacobi level, the link between eulerian and lagrangian coordinates is given in [20] (see also [26]).
More recently, non-local LWR models have been proposed in order to take into account the action of drivers to the surrounding density of other vehicles, see [3]. We also refer to [4-8, 12, 18].
In the present paper, we propose a new non-local macroscopic model. This model is expressed in the lagrangian coordinates at the Hamilton-Jacobi level. It then describes the dynamics of the position of the vehicles. In our model, drivers will adapt their velocity to the downstream traffic, assigning greater importance to close vehicles.
Our model is obtained by rescaling a microscopic model, which describes the dynamics of each vehicle individually. We recall in particular that the main advantage of microscopic models is that they are easily justifiable but, with these models, it is difficult to model the traffic at the scale of a road or a city since the number of vehicles becomes too large.

Description of the model and assumptions
The non-local macroscopic model we propose is the following: (0, ) = 0 ( ) on R. (1.1) In this model, expressed in the Lagrangian coordinates, represents the position, at time of the car . The velocity of the car is given by while the spacing is given by . Moreover, using the link between Lagrangian and Eulerian coordinates, the density of cars is given by 1/ . In this model, the function , called the weight, is decreasing and takes into account the fact that drivers will adapt their velocity to the downstream traffic, assigning a greater importance to close vehicles. The non-local term ( , + ) − ( , ) represents the average spacing between vehicles and + . Finally, the function is called an optimal velocity function and is non-negative, non-decreasing and bounded. Precise assumptions on and are given below: is non-decreasing on R. (H3) There exists ℎ 0 ∈ (0, +∞) such that for all ℎ ≤ ℎ 0 , (ℎ) = 0.

Main results
The first main result of the paper is an existence and uniqueness result for the macroscopic non local model (1.1). Theorem 1.1 (Existence and uniqueness for the macroscopic model). Assume (H) and let 0 be a Lipschitz continuous and non-decreasing function. Then Problem (1.1) admits a unique solution which is Lipschitz continuous and non-decreasing.
The second main result is a justification of the macroscopic model. In fact, we will show that our macroscopic model can been obtained by rescaling a microscopic model. More precisely, we consider the following microscopic model:˙( where ( ) = ( ). The rescaling in the function takes into account that the weight of the closer cars is more and more important as goes to zero. In this model, the term + ( ) − ( ) represents the average distance of two successive vehicles comprised between vehicles and + . Then the velocity of the vehicle depends on the weighted average of these average distances.
Concerning the initial condition, let us assume that, at initial time, vehicles satisfy (0) = −1 0 ( ) for some > 0 and where 0 is a Lipschitz continuous function (we denote by 0 its Lipschitz constant). From a traffic point of view, we also assume that 0 is non-decreasing. Then, if we define by we have the following result.
Theorem 1.2 (Micro-macro limit). Under assumption (H), if moreover 0 is Lipschitz continuous, then the function defined in (1.3) converges to the unique solution of (1.1).
In fact, the idea of the proof of this theorem is inspired by [15,16] and consists in injecting the microscopic models into a partial differential equation. More precisely, we have that the function is solution of the following problem the result is then obtained by homogenization (i.e. a passage as → 0) of this non-local PDE. We refer to [13,29] for homogenization results in the local case and to [14][15][16]22] for the non-local case.
In the last Section of the paper, we also consider the numerical analysis of the macroscopic model (1.1). We propose a finite difference numerical scheme and we prove an error estimate between the solution of the macroscopic model (1.1) and its numerical approximation. We also provide some numerical simulations.
Organization of the paper. This article is organized as follow. In Section 2, we present the microscopic problem. The existence and uniqueness of the solution of this problem are proved. Section 3 is devoted to the study of the macroscopic problem where a comparison principle is proved. The proof of the homogenization result is given in Section 4. Finally in Section 5, we present a numerical scheme with an error estimate result. We also present some numerical simulations.

Well-posedness of the microscopic problem
The goal of this section is to give some preliminary results for the following problem (0, ) = 0 ( ) on R. (2.1) We will show in particular a comparison principle as well as an existence result. We shall recall first the definition of a viscosity solution.
We say that is a viscosity sub-solution of (2.1) if (0, ·) ≤ 0 and if, for all ∈ 1 (]0, [ × R) such that − attends a maximum point in We say that is a viscosity super-solution of equation ( Finally, we say that is a solution of (2.1) if * is a sub-solution and * is a super-solution of (2.1).
Remark 2.2. We recall that * and * are respectively the upper and lower semi-continuous envelope of defined by * ( , ) = lim sup We now give the comparison principle for (2.1).
Theorem 2.3 (Comparison principle for (2.1)). Let and be respectively a sub-and a super-solution of (2.1). We assume that there exists a constant 0 such that We assume, by contradiction, that > 0. For , , > 0, we duplicate the variable by considering

Let
(︀˜,˜,˜,˜)︀ be a point of maximum of , . By assumption, we have, for small enough, that , > 0. This implies in particular that︀ where is a positive constant. We then deduce that︀ → 0, |̃︀−̃︀| → 0, |̃︀ −̃︀| → 0 respectively as , and go to zero. We now claim that̃︀ ,̃︀ > 0 for and small enough. Indeed, by contradiction, assume that̃︀= 0 (the proof for̃︀ is similar). We then have Taking and small enough, we get a contradiction.
We are now able to use the equation satisfied by and . We consider We can easily see that Φ 1 and Φ 2 belong to 1 (]0, [ × R) and that − Φ 1 has a maximum point in (︀˜,˜)︀ ∈]0, [×R and − Φ 2 has a minimum point in (˜,˜) ∈]0, [×R. Hence, by definition of viscosity solutions, we get that We subtract the two previous inequalities and we get that Using that (︀˜,˜,˜,˜)︀ is a point of maximum of , , we have, for all ( , , , ) that Choosing =˜+ , =˜+ , =˜and =˜, we get that Injecting this in (2.2) and using the monotonicity of , we get Using that and ↦ → ( ) are in 1 , we get a contradiction for small enough.
We also give a comparison principle in a bounded set that will be useful later. Since the proof is similar to the previous one, we skip it.

Theorem 2.4 (Comparison in a bounded set).
Let Ω be a subset of [0, ] × R and and be respectively a sub-solution and a super-solution of (2.1) in Ω. We assume that there exists a constant 0 such that We also assume that ≤ outside Ω. Then We can now give the existence and uniqueness result for problem (2.1) . For all > 0, there exists a unique solution of (2.1) such that there is a positive constant 0 (depending only on the Lipschitz constant of 0 ) such that Proof. We use Perron's method (see [23]) to prove the existence of , the uniqueness being a direct consequence of Theorem 2.3. It is sufficient to show that 0 ± 0 are respectively super-and sub-solution of (2.1) for a suitable choice of 0 . We start by verifying that 0 + 0 is a super-solution. Since 0 is assumed to be Lipschitz continuous with Lipschitz constant 0 , we have that Since is non-decreasing we get Choosing 0 = ( 0 ), we get that 0 + 0 is a super-solution of (2.1). Arguing in a similar way, we get that 0 − 0 is a sub-solution and the proof is complete.

Well-posedness of the macroscopic model
This section is devoted to useful results concerning problem (1.1). We begin by the definition of the viscosity solution. This definition and the main properties of the solution are inspired from [2].
In order to have the non-local term well defined and as well as for equation (2.1), we will assume that all the sub-and super-solutions (and hence solutions) satify the following property: (3.1) Definition 3.1 (Definition of -viscosity solution for (1.1)). Let be a positive constant. We say that an upper semi-continuous function : We say that a lower semi-continuous function : We say that is a -solution of (1.1) if it is a -sub-and super-solution.
Remark 3.2. For simplicity of presentation, given Φ : Proof. We only prove the result for sub-solution, the super-solution case being similar. We assume that is a 1sub-solution and we want to show that it is We then define a test function ∈ 1 (]0, [×R), which will be used for the 1 -sub-solution, by Since is 1 -viscosity sub-solution of (1.1) and ( 0 , 0 ) is maximum point of − , we have Using that ( 0 , 0 ) = ( 0 , 0 ), we then deduce that where for the last inequality, we have used the fact that is non-decreasing. Sending 1 → 0 and using that and ↦ → ( ) are in 1 , we get the result.
Theorem 3.4 (Comparison principle for (1.1)). Let be a sub-solution and be a super-solution of (1.1). We assume that and satisfy (3.1). Then ≤ .
Proof. We assume by contradiction that = sup We then duplicate the variable, by considering, for , , , > 0, We denote by ( , , , ) a point of maximum of , . For and small enough, we have , > 0 and we deduce as in the proof of Theorem 2.3 that respectively as , and go to zero, and that , > 0 for and small enough. We can then use the equations satisfied by and . We have that − 1 reaches a maximum in ( , ), with 1 given by So by definition of viscosity solution we obtain, for every , that Since − 2 reaches a minimum point in ( , ), with 2 given by We subtract the two viscosity inequalities, and we obtain Using that ( , , , ) is a maximum point we obtain We then deduce that We now compare 1, [ 1 ( , ·)]( ) and 1, [ 2 ( , ·)]( ). We have Injecting the previous estimate and (3.3) in (3.2) and using the fact that is non-decreasing, we get Sending , → 0, we get that and we obtain a contradiction since > 0. This ends the proof.
We are now able to give the proof of Theorem 1.1 Proof of Theorem 1.1. As in the proof of Theorem 2.5, we have that 0 ± 0 are respectively sub-and supersolution of (1.1), so the existence follows by Perron's method. The uniqueness is a direct consequence of Theorem 3.4. It just remains to show that is Lipschitz continuous.
To do that, let us define ℎ by ℎ ( , ) = ( , + ℎ) − 0 ℎ. In particular, we have Moreover, since equation (1.1) is invariant by translation in space and by addition of constant, we get that ℎ is a sub-solution of (1.1). Hence, the comparison principle yields that To show that is non-decreasing, we argue in a similar way by using the fact that ( , + ℎ) (recall that 0 is non decreasing) is a super-solution of (1.1). This shows that is Lipschitz continuous in space. The fact that is also Lipschitz continuous in times is a direct consequence of the fact that is bounded.

Homogenization result
This section is devoted to the proof of the following convergence result. We will see that Theorem 1.2 is a corollary of this result.
We are going to prove that is a sub-solution of (1.1) on [0, [×R. Similarly, we can prove that is a supersolution of the same equation. Then, using the comparison principle Theorem 3.4, we will get ≤ and so = = which implies the convergence of to . First, by (2.3), we have that (0, ·) = 0 . We argue by contradiction by assuming that is not a sub-solution on ]0, [×R. Then there exists ( 0 , 0 ) ∈]0, [×R and a test function Φ ∈ 1 such that − Φ reaches a strict maximum in ( 0 , 0 ) and such that We also assume that ( 0 , 0 ) = Φ( 0 , 0 ). We now apply the perturbed test function method introduced by Evans [13] in terms here of hull functions. We refer to [14,16] for the use of hull functions as correctors. We recall that in these papers, the perturbed test function is essentially defined by where ℎ is the hull function. In fact in our case, the hull function is simpler (since our Hamilton-Jacobi equation is independent of ( , )) and is given by ℎ = . So the perturbed test function reduces to if not where is chosen later and , ( 0 , 0 ) = ( 0 − , 0 + ) × ( 0 − , 0 + ). We want to prove that Φ is solution in , ( 0 , 0 ), of and that Φ ≥ outside , ( 0 , 0 ). In particular is chosen smaller that 1 so that , ( 0 , 0 ) ⊂ 1,1 ( 0 , 0 ). Let us first focus on the "boundary conditions". Since − Φ reaches a strict maximum at ( 0 , 0 ), we can ensure that Hence we conclude that Φ ≥ outside , ( 0 , 0 ). We now turn to the equation. Since Φ is smooth, we can check this property pointwise. Let ( , ) ∈ , ( 0 , 0 ). We have for small enough. Using that, by convergence of Riemann sum, and show that Φ is a super-solution in , ( 0 , 0 ). Recalling that ≤ Φ outside , ( 0 , 0 ) and using the comparison principle on bounded set Theorem 2.4, we get that ( , ) ≤ Φ ( , ).
Passing to the limit as goes to zero, and as ( , ) → ( 0 , 0 ) we get that which gives a contradiction with > 0. Therefore is a sub-solution of (1.1) on [0, [×R and this ends the proof of the theorem.
We are now able to give the proof of Theorem 1.2.
Proof of Theorem 1.2. We recall that the initial condition of is given by Using the comparison principle Theorem 3.4, we then get By convergence of to given in Theorem 4.1, we deduce the convergence of to . This ends the proof of the theorem.

Numerical tests
In this section, we present a numerical scheme in order to compute the solution of the macroscopic model. In a first subsection, we present the scheme. Then, we give some properties of the solution of the scheme and we prove an error estimate between the numerical solution and the solution of the macroscopic model. We end the paper with some numerical results in Section 5.3.

Discretization aspects and numerical scheme
Since problem (1.1) is defined on the unbounded line R, we need to consider a bounded computational domain [ , ] with < two real parameters. The treatment of the boundary condition on the right side, on = , will be detailed below. To implement this problem, we consider the spatial discretization = ∆ + , ∈ 0, · · · , and ∆ = ( − ) with ∈ N * a parameter, so that 0 = and = . For the time variable, we set = ∆ with ∆ = / and ∈ N * a second parameter. We will denote by the approximation of ( , ). Then, the main point is to handle the non-local integral term. To do so, the idea is to decompose the integral as follows: where ≥ ∆ is a small parameter (of order √ ∆ ) and is a big one. In the numerical implementation, we simply neglect the terms ( ( , ·))( ) to avoid the treatment of the division by 0, which, as we will see in the analysis of the scheme, induces an error of order , and the last term ∞ ( ( , ·))( ) taking > sufficiently large, which introduces a (small) consistency error. In particular, if the weight function is compactly supported, this term is exactly 0 by choosing properly .
For the remaining integral ( ( , ·))( ), we use a simple trapezoidal quadrature rule based on the discretization points , see Figure 1, in order to include directly the quantity . Yet, as we can see, + can be greater where ∞ = ( , ) > 0 represents the constant density in the outgoing direction > . Thus, we obtain the following approximation: Note that we approximate To sum up, the numerical scheme is then given by: and I(˜)( ) is given by (5.2).

Analysis of the scheme
This approximation of the integral term is justified by the following lemma We now prove, under the following CFL condition and is the Lipschitz constant associated to , that the scheme is monotone.
Remark 5.2. We can notice that the CFL condition does not explicitly depend on the space step ∆ . Indeed, we can remark that the term ℐ corresponds to the trapezoidal quadrature rule applied to the function ( ) on the interval ( , ). Also, it is worth recalling that for the "local" equation = ( ) discretized with the simple scheme the CLF condition is ∆ ≤ Δ . This condition is usually worse than the one obtained in the non local case if ( ∆ ) is small. Proof. The proof is made by induction. The initialization is true by hypothesis. Assume that for a certain , we have ≤ ∀ ∈ {0, . . . , }.
We have Using that I(˜)( ) ≤ I(˜)( ) − ( − )ℐ and since is monotone, we deduce that This ends the proof.
We now show that ( ) is non decreasing and "Lipschitz continuous".
Proof. The estimate (5.6) is a direct consequence of the definition of the scheme and the fact that 0 ≤ ≤ max . Let us show the second result by induction. The initialization is true by hypothesis. Assume that, for a given , we have We set = +1 and In particular, we have +1 = +1 +1 for all = 0, . . . , − 1. By monotonicity of the scheme, since ≥ , we deduce that +1 +1 = +1 ≥ +1 ∀ = 0, . . . , − 1.
In the same way, if we define = +1 − 0 ∆ and then we have +1 = +1 +1 − 0 ∆ for all = 0, . . . , − 1 and ≤ . By monotonicity of the scheme, we then get +1 +1 − 0 ∆ = +1 ≤ +1 . This ends the proof of the theorem.  1)). Assume that ≤ 1 and ∆ + ∆ ≤ 1. Assume also that the CFL condition (5.5) is satisfied. Then, there exists a constant > 0, depending only on the Lipschitz constant of 0 and on , such that the error estimate between the continuous solution of (1.1) and the discrete solution of (5.3) (with initial condition given by 0 ) is given by Remark 5.6. For example, if has compact support, B can be chosen large enough so that = 0 and we can take of the order of √ ∆ . In that case, we recover a estimate of order √ ∆ + ∆ . If, we take ( ) = − , then we can take = − ln(∆ ) and = √ ∆ and we get an estimate of order − ln(∆ ) √ ∆ + ∆ .
Proof. The proof is inspired by the one of Crandall Lions [9], revisited by Alvarez et al. [1] and is an adaptation of the comparison principle. We define by = sup We want to prove that is bounded by (∆ , ∆ ), being defined later. We first assume that 0 ( ) ≥ 0 and we set We define for 0 < , ≤ 1 and > 0, the function This function has a maximum denoted by , . Indeed, is Lipschitz continuous then and since, ≤ 1, In the same way, using Theorem 5.4, we also have that Let us underline that the constant Then by definition of -viscosity solution of (1.1), we have that, On the other hand, since ⋆ > 0, and ( ⋆ , ⋆ , ⋆ , ⋆ ) is a maximum point of , we have that for all ∈ Δ , Then, for all ∈ Δ , we obtain that, Since is a solution of the discrete problem (5.3), then we obtain, for )︂ .

That is
Moreover, by (5.6), we have This implies that (with ℐ defined in (5.5)) (5.14) Subtracting inequality (5.14) to (5.13) and using the fact that is a Lipschitz continuous, we get Note that, by (5.10) and (5.11), we have Moreover, by (5.4), we have We now want to bound This implies that We then get Injecting this in (5.16) and using (5.17), we get , then we have a contradiction for all ≥ ⋆ , and then ⋆ = 0 or ⋆ = 0 for all such that ≥ ⋆ .
Remark 5.7. If ≥ 1, then since we have the estimation in each time interval of length 1, we then obtain that ≤ .

Some numerical illustrations
In the numerical experiments below, we take the following discretization parameters ∆ = 0.005, = 0.5, ∆ = 0.05, = − = 3 and = 10. For the velocity , inspired by [19] we consider the two following cases: where 0 > 0, max > 0 and max are real parameters and ∈ N * . For the interpretation, we recall that 0 corresponds to the minimal distance between two successive cars and max the distance up to which a driver will not increase his speed. One can easily check that these two examples satisfy assumptions (H). In the sequel, we take 0 = 0.2, max = 10, = 1 and max = 90. For the weighting function , we take ( ) = 1 − with > 0 so that = 1, see equation (5.3). It can be also verified that it satisfies assumptions (H).
Riemann initial data. The initial data is given by where > 0 and > 0 are the initial density on left and on the right respectively. On Figure 2, we represent the density = 1 at time = 0.2 for different values of ∈ {0.5, 1, 5} taking = 0.2 and = 0.8. We also represent in dotted line the "local" case corresponding to the situation where we solve = ( ) (meaning that the driver at position take only into account the car directly in front of him to adapt his speed). As we  can see, and as we can expect, taking into account downstream traffic on a wider distance (meaning big ) leads to a smoother repartition of the density and somehow delays the progression of the front.
Similarly, we represent on Figure 3 the comparison for ∈ {0.5, 1, 5} of the density 1 at time = 0.2 taking this time = 0.8 and = 0.2. We can see that, even thought the downstream density is lower, taking into account downstream traffic on large distance has a strong impact on the result. In particular for large , the "decompression" is more uniform in the sense that the density on the left is almost flat.
Oscillating initial data. Let us consider a last situation with an oscillating initial density:  The associated initial state 0 is recovered by simple integration. On Figure 4, we represent the density profile also at time = 0.2 for ∈ {0.5, 1, 5} and in dotted line the "local" case. Once again, we can remark that the bigger is, the "smoother" the density is.

Concluding remarks
In this paper, we proposed and study a non-local traffic flow model starting from the microscopic model to build the homogenized macroscopic model. Using Hamilton-Jacobi framework, we are able to prove the convergence of the homogenization process, and existence and uniqueness of the solution for the macroscopic problem. We believe that this approach is relevant in the context of traffic flow modeling since it enables to consider more general assumptions (such as the regularity of ) compared to what we can find for models based on conservation laws for which the analysis is more complicated and particular. Finally, we proposed and analyzed a numerical scheme to solve the proposed non local macro-model, and illustrated with some results the impact of the non-local aspect of the model. In future works, we would like to extend this model to traffic flow on network.