NON-LOCAL MULTI-CLASS TRAFFIC FLOW MODELS

. We prove the existence for small times of weak solutions for a class of non-local systems in one space dimension, arising in traﬃc modeling. We approximate the problem by a Godunov type numerical scheme and we provide uniform L ∞ and BV estimates for the sequence of approximate solutions, locally in time. We ﬁnally present some numerical simulations illustrating the behavior of diﬀerent classes of vehicles and we analyze two cost functionals measuring the dependence of congestion on traﬃc composition.

1. Introduction. Macroscopic traffic flow models based on fluid-dynamics equations have been introduced in the transport engineering literature since the midfifties of last century, with the celebrated Lighthill, Whitham [11] and Richards [13] (LWR) model. Since then, the engineering and applied mathematical literature on the subject has considerably grown, addressing the need for more sophisticated models better capturing traffic flow characteristics. Indeed, the LWR model is based on the assumption that the mean traffic speed is a function of the traffic density, which is not experimentally verified in congested regimes. To overcome this issue, the so-called "second order" models (e.g. Payne-Whitham [12,15] and Aw-Rascle-Zhang [3,16]) consist of a mass conservation equation for the density and an acceleration balance law for the speed, thus considering the two quantities as independent.
More recently, "non-local" versions of the LWR model have been proposed in [5,14], where the speed function depends on a weighted mean of the downstream vehicle density to better represent the reaction of drivers to downstream traffic conditions.
Another limitation of the standard LWR model is the first-in first-out rule, not allowing faster vehicles to overtake slower ones. To address this and other traffic heterogeneities, "multi-class" models consist of a system of conservation equations, one for each vehicle class, coupled in the speed terms, see [4] and references therein for more details.
In this paper, we consider the following class of non-local systems of M conservation laws in one space dimension: and we assume: (H1) The convolution kernels ω i ∈ C 1 ([0, η i ]; R + ), η i > 0, are non-increasing functions such that (H3) ψ : R + → R + is a smooth non-increasing function such that ψ(0) = 1 and ψ(r) = 0 for r ≥ 1 (for simplicity, we can consider the function ψ(r) = max{1 − r, 0}). We couple (1) with an initial datum Model (1) is obtained generalizing the n-populations model for traffic flow described in [4] and it is a multi-class version of the one dimensional scalar conservation law with non-local flux proposed in [5], where ρ i is the density of vehicles belonging to the i-th class, η i is proportional to the look-ahead distance and J i is the interaction strength. In our setting, the non-local dependence of the speed functions v i describes the reaction of drivers that adapt their velocity to the downstream traffic, assigning greater importance to closer vehicles, see also [7,9]. We allow different anisotropic kernels for each equation of the system. The model takes into account the distribution of heterogeneous drivers and vehicles characterized by their maximal speeds and look-ahead visibility in a traffic stream. Due to the possible presence of jump discontinuities, solutions to (1), (5) are intended in the following weak sense.
The main result of this paper is the proof of existence of weak solutions to (1), (5), locally in time. We remark that, since the convolution kernels ω i are not smooth on R, the results in [1] cannot be applied due to the lack of L ∞ -bounds on their derivatives.
In this work, we do not address the question of uniqueness of the solutions to (1). Indeed, even if discrete entropy inequalities can be derived as in [5,Proposition 3], in the case of systems this is in general not sufficient to single out a unique solution.
The paper is organized as follows. Section 2 is devoted to prove uniform L ∞ and BV estimates on the approximate solutions obtained through an approximation argument based on a Godunov type numerical scheme, see [8]. We have to point out that these estimates heavily rely on the monotonicity properties of the kernel functions ω i . In Section 3 we prove the existence in finite time of weak solutions applying Helly's theorem and a Lax-Wendroff type argument, see [10]. In Section 4 we present some numerical simulations for M = 2. In particular, we consider the case of a mixed flow of cars and trucks on a stretch of road, and the flow of mixed autonomous and non-autonomous vehicles on a circular road. In this latter case, we analyze two cost functionals measuring the traffic congestion, depending on the penetration ratio of autonomous vehicles. The final Appendix contains alternative L ∞ and BV estimates, based on approximate solutions constructed via a Lax-Friedrichs type scheme, which is commonly used in the framework of non-local equations, see [1,2,5].
2. Godunov type approximate solutions. First of all, we extend ω i (x) = 0 for x > η i . For j ∈ Z and n ∈ N, let x j+1/2 = j∆x be the cell interfaces, x j = (j − 1/2)∆x the cells centers and t n = n∆t the time mesh. We aim at constructing a finite volume approximate solution x j−1/2 , x j+1/2 ] and i = 1, ..., M. To this end, we approximate the initial datum ρ 0 i for i = 1, ..., M with a piecewise constant function Similarly, for the kernel, we set We consider the following Godunov-type scheme adapted to (1), which was introduced in [8] in the scalar case: where we have set λ = ∆t ∆x .
2.1. Compactness estimates. We provide here the necessary estimates to prove the convergence of the sequence of approximate solutions constructed via the Godunov scheme (7).
Proof. Let us assume that ρ n i,j ≥ 0 for all j ∈ Z and i ∈ 1, ..., M . It suffices to prove that ρ n+1 i,j in (7) is non-negative. We compute under assumption (8).
Corollary 1. (L 1 -bound) For any n ∈ N, under the CFL condition (8) the approximate solutions constructed via the scheme (7) satisfy where ρ n i 1 := ∆x j ρ n i,j denotes the L 1 norm of the i-th component of ρ ∆x .
.., M , and (8) holds, then the approximate solution ρ ∆x constructed by the algorithm (7) is uniformly bounded on [0, T ] × R for any T such that Proof. Letρ = max{ρ n i,j−1 , ρ n i,j }. Then we get and where (11) and (12) we get where the maximum is attained for K = e ρ 0 ∞ .
Iterating the procedure, at time t m , m ≥ 1 we set K = e m ρ 0 ∞ and we get that the solution is bounded by K until t m+1 such that Therefore, the approximate solution remains bounded, uniformly in ∆x, at least for Remark 1. Figure 1 shows that the simplex is not an invariant domain for (1), unlike the classical multi-population model [4]. Indeed, let us consider the system where r and v i are as in (2) and (3), respectively. We have the following: , for any initial datum ρ 0 ∈ S the approximate solutions to (13) computed by the upwind scheme with F(ρ n j , ρ n j+1 ) = ρ n j ψ(r n j+1 ), satisfy the following uniform bounds: ρ n j ∈ S ∀j ∈ Z, n ∈ N.
Proof. Assuming that ρ n j ∈ S for all j ∈ Z, we want to prove that ρ n+1 j ∈ S. Rewriting (14), we get Defining the following function of ρ n we observe that . This proves that r n+1 j ≤ 1. To prove the positivity of (14), we observe that (8) holds, then the approximate solution ρ ∆x (t, ·) constructed by the algorithm (7) has uniformly bounded total variation for t ∈ [0, T ], for any T such that where and setting ∆ n i,j+1/2 = ρ n i,j+1 − ρ n i,j , we get Now, we can write Observe that assumption (8) guarantees the positivity of (19). The term (20) can be estimated as with ξ j ∈ I ∆x +∞ k=0 ω k i r n j+k , ∆x +∞ k=0 ω k i r n j+k+1 andξ j+1/2 ∈ I (ξ j , ξ j+1 ), where we set I(a, b) = [min{a, b}, max{a, b}]. For some ϑ, µ ∈ [0, 1], we compute By monotonicity of ω i we have Taking the absolute values we get Let now K 1 > 0 be such that j ∆ β,j ≤ K 1 for β = 1, . . . , M , = 0, . . . , n. Taking the absolute values and rearranging the indexes, we have that we can rewrite as TV (ρ ∆x i )(n∆t, ·) ≤ e Gn∆t TV (ρ 0 i ) + e HK1n∆t − 1 ≤ e HK1n∆t TV (ρ 0 i ) + 1 − 1 , since H ≥ G and it is not restrictive to assume K 1 ≥ 1. Therefore, we have that where the maximum is attained for some K 1 < e TV (ρ 0 i ) + 1 − 1 such that Therefore the total variation is uniformly bounded for .
Iterating the procedure, at time t m , m ≥ 1 we set K 1 = e m TV (ρ 0 i ) + 1 − 1 and we get that the solution is bounded by K 1 until t m+1 such that Therefore, the approximate solution has bounded total variation for t ≤ T with . Corollary 2. Let ρ 0 i ∈ (BV ∩ L ∞ ) (R; R + ). If (8) holds, then the approximate solution ρ ∆x constructed by the algorithm (7) has uniformly bounded total variation on [0, T ] × R, for any T satisfying (16).
Let us assume now that T > ∆t. Let n T ∈ N\{0} such that n T ∆t < T ≤ (n T + 1)∆t. Then We then need to bound the term

FELISIA ANGELA CHIARELLO AND PAOLA GOATIN
From the definition of the numerical scheme (7), we obtain Taking the absolute values and using (12) we obtain that is bounded by Corollary 1, Lemma 2.2 and Lemma 2.4.
Finally, again by the L 1 loc convergence of ρ ∆x 4. Numerical tests. In this section we perform some numerical simulations to illustrate the behaviour of solutions to (1) for M = 2 modeling two different scenarios. In the following, the space mesh is set to ∆x = 0.001.

4.1.
Cars and trucks mixed traffic. In this example, we consider a stretch of road populated by cars and trucks. The space domain is given by the interval [−2, 3] and we impose absorbing conditions at the boundaries, adding N 1 = η 1 /∆x ghost cells for the first population and N 2 = η 2 /∆x for the second one at the right boundary, and just one ghost cell for both populations at the left boundary, where we extend the solution constantly equal to the last value inside the domain. The dynamics is described by the following 2 × 2 system with In this setting, ρ 1 represents the density of trucks and ρ 2 is the density of cars on the road. Trucks moves at lower maximal speed than cars and have grater view horizon, but of the same order of magnitude. Figure 2 describes the evolution in time of the two population densities, correspondent to the initial configuration In this case, we assume that autonomous and non-autonomous vehicles have the same maximal speed, but the interaction radius of CAVs is two orders of magnitude grater than the one of human-driven cars. Moreover, we assume CAVs have constant convolution kernel, modeling the fact that they have the same degree of accuracy on information about surrounding traffic, independent from the distance. In this case, model (1) reads with ω 1 (x) = 1 η 1 , η 1 = 1,  Above ρ 1 represents the density of autonomous vehicles, ρ 2 the density of nonautonomous vehicles and β ∈ [0, 1] is the penetration rate of autonomous vehicle. Figure 3 displays the traffic dynamics in the case β = 0.9. As a metric of traffic congestion, given a time horizon T > 0, we consider the two following functionals: wherex = x 0 ≈ 0. The functional J measures the integral with respect to time of the spatial total variation of the total traffic density, see [6]. The functional Ψ measures the integral with respect to time of the traffic flow at a given pointx, corresponding to the number of cars that have passed throughx in the studied time interval. Figure 4 displays the values of the functionals J and Ψ for different values of β = 0, 0.1, 0.2, . . . , 1. We can notice that the functionals are not monotone and present minimum and maximum values. The traffic evolution patterns corresponding these stationary values are reported in Figure 5, showing the (t, x)-plots of the total traffic density r(t, x) corresponding to these values of β.