Models for dense multilane vehicular traffic

We study vehicular traffic on a road with multiple lanes and dense, unidirectional traffic following the traditional Lighthill-Whitham-Richards model where the velocity in each lane depends only on the density in the same lane. The model assumes that the tendency of drivers to change to a neighboring lane is proportional to the difference in velocity between the lanes. The model allows for an arbitrary number of lanes, each with its distinct velocity function. The resulting model is a well-posed weakly coupled system of hyperbolic conservation laws with a Lipschitz continuous source. We show several relevant bounds for solutions of this model that are not valid for general weakly coupled systems. Furthermore, by taking an appropriately scaled limit as the number of lanes increases, we derive a model describing a continuum of lanes, and show that the $N$-lane model converges to a weak solution of the continuum model.


Introduction
The Lighthill-Whitham-Richards (LWR) model for unidirectional traffic on a single road, see [13,16], reads (1.1) u t + (uv(u)) x = 0, where u = u(t, x) denotes the density of vehicles at the position x and time t, and v = v(u) is a given velocity function. The LWR-model expresses conservation of vehicles and is a well-established model for dense unidirectional single lane vehicular traffic on a homogeneous road without exits and entries. Furthermore, it serves as the standard textbook example to gain intuition regarding the behavior of solutions of scalar one-dimensional hyperbolic conservation laws, see, e.g., [10]. Given the importance of vehicular traffic modeling in modern society, it is no wonder that the LWR-model has been generalized to describe several important scenarios in dense traffic flow. Indeed, "traffic hydrodynamics" has become a research field in its own right, where the flow of vehicles is modeled by conservation laws or balance equations. In the general context, the LWR-model is the simplest model among the many hydrodynamic traffic models. Among the other models often used is the Aw-Rascle model [1], which is a system of conservation laws where the velocity v is not a given function of u, but satisfies a second conservation law. It is thus considerably more complicated than the simple LWR-model. For a general introduction to how conservation laws are used in traffic modeling, see [9,3] and the many references therein.
In this paper we introduce a new model for multilane dense vehicular traffic where the underlying model for each lane remains the LWR-model. Our basic assumption is that drivers prefer to drive faster, and that the tendency of a vehicle to change lane is proportional to the difference in velocity between neighboring lanes. If (1.1) describes the density of vehicles in a particular lane, the multilane behavior is described by a source term, accounting for lane changes. The result is thus a system of weakly coupled scalar conservation laws.
More precisely, consider two lanes denoted 1 and 2, the model we study, reads ∂ t u 1 + ∂ x (u 1 v 1 (u 1 )) = −S(u 1 , u 2 ), where the change of lanes is codified in Here u i denotes the density in lane i. The system constitutes a weakly coupled 2×2 system of one-dimensional hyperbolic conservation laws, and there is ample theory available for systems of this type, see Section 2. The system readily generalizes to an arbitrary number of lanes, see Section 3. We show that the general system with N lanes has a unique entropy solution, and that the solution is well-posed in the sense that one has a surprising L 1 stability for two solutions u i andū i , see Theorems 3.2 and 3.3. Note that the L 1 stability does not hold in general for systems of balance laws, that is, hyperbolic conservation laws with source. The models invites for considering the continuum limit where the number of lanes increases to infinity. We organize the parallel lanes along the x-axis, and measure the distance between the lanes along the y-axis. The distance between the lanes is scaled as ∆y = 1/N , where N denotes the number of lanes. For simplicity we assume that the velocity function is given by v i (u) = −k(y i )g(u) where y i = i∆y, and −g(u) is the velocity function. We scale the function such that g(0) = −1 and g(1) = 0. We need to scale the constant K as K = 1/∆y 2 . We consider given initial data u 0 : R × [0, 1] → [0, 1], where the initial data for lane i is u i,0 is given by (4.20) and with solution u i . We interpolate this function to u ∆y where u ∆y : [0, ∞) × R × [0, 1] → [0, 1]. We assume that k is smooth and positive with k (0) = k (1) = 0. In Theorem 4.2 we show that u ∆y → u where u is a weak solution of where the flux function f is defined as f (u) = uv(u). This equation is an interesting anisotropic and degenerate parabolic equation with non-trivial boundary conditions in the y-direction.
There is a plethora of approaches to the modeling of multilane dense traffic, and the most relevant to our approach here can be found in [5,11,12,14], using either kinetic models or the Aw-Rascle model or variations thereof, or [2,4] where more involved source terms modeling the change of lanes, are employed. See [15] for a survey of various models for lane changing.
The rest of this paper is organized as follows: In Section 2 we detail the two-lane case, and show that u i ∈ [0, 1] is an invariant region. In Section 3 we state the Nlane model, and prove a number of estimates on the solution. Finally, in Section 4, we study the limit as N → ∞. See [6] for a model for two-dimensional traffic flow on highways. Analogously to the analysis of numerical schemes for degenerate parabolic equations, we establish enough estimates on the solution, enabling us to conclude that a limit exists, and that this limit is a weak solution of a degenerate convection-diffusion equation. All sections are illustrated by numerical examples.

A continuum model for two-lane vehicular traffic
Consider a road with two lanes, each with its own velocity function. The lanes are homogeneous, and traffic on the road is unidirectional. We assume that the vehicular traffic is dense, allowing for a continuum formulation. Let u i and v i = v i (u i ) denote the density and velocity, respectively, in lane i.
In this paper we focus on the interaction between the two lanes. We assume that drivers prefer to drive in the faster lane, and the tendency of a vehicle to change lane is proportional to the difference in velocity. Thus the flow from lane 1 to lane 2 equals where x is the position along the road and t denotes time. This 2 × 2 system of hyperbolic conservation laws is weakly coupled with a Lipschitz continuous source term.
The velocities v i = v i (u i ) are strictly decreasing positive functions, and we assume that they are scaled such that v 1 (1) = v 2 (1) = 0. For simplicity, we scale space and time such that K = 1.
It is well-known that this system in general only allows for weak solutions u i ∈ L 1 (R) ∩ BV (R), the set of integrable functions of finite total variation, see, e.g., [10]. Furthermore, the issue of uniqueness of the solution is non-trivial and one needs to require that the solution satisfies an entropy condition.
Remark 2.3. The existence and uniqueness of entropy solutions to (2.2) follows by Theorem 3.2 below.
We will throughout the paper use the following notation: We shall also employ the convention that C denotes a "generic" finite positive constant, independent of critical parameters, whose actual value may change from one occurrence to the next. Similarly, we use C α to denote a positive function c(α) < ∞ for α < ∞. This model (2.2) has the natural invariant region u ∈ [0, 1]. This is the content of the following lemma.
Adding these two equations and using that ( Similarly, by using the convex entropy η(u) = (u − 1) + we get the set of distributions. By the same argument as before, we arrive at if u 1 and u 2 are non-negative.
Remark 2.5. There are also other invariant regions for this equation. If

Multilane model
The model (2.2) can be generalized to an arbitrary number of lanes. Consider a road with N lanes. Traffic is unidirectional and dense. Each lane has its specific velocity function v i depending only on the density in that lane, Assume that drivers prefer to drive in the faster lane, and this tendency increases with the velocity difference with adjacent lanes. Thus the flow from lane i to lane i + 1 equals where we have scaled time such that the constant of proportionality is one. We then get, in the analogous manner to the derivation of (2.2), that coupled with the boundary conditions be Lipschitz continuous functions, and assume that for all compactly supported test functions ϕ ∈ C ∞ ([0, ∞) × R).
It is an entropy solution if for all convex functions η, and for all non-negative test functions The wellposedness of the system of equations (3.1) is ensured by the following general theorem from [8], see also [7].
is another entropy solution with initial data A fundamental property of hyperbolic conservation law is the L 1 contractivity of solutions in the sense that the spatial L 1 -norm of the difference between two entropy solutions at a specific time does not increase in time. This property is in general lost for weakly coupled systems, or for scalar conservation laws with a source. The general bound (3.4) does not imply L 1 contractivity. However, for the system (3.1), the special form of the source yields L 1 contractivity for the whole solution, as the next theorem shows.
We also note the following useful estimates. Define f i (u) = uv i (u) and ∆ − i a i = a i − a i−1 , divide by h and let h ↓ 0 to find that (3.10) If we assume that the quantity on the left is bounded by C, then we get Furthermore, we have the useful observation 3.1. An example. We also here include an example. For i = 1, . . . , 8 we set u i,0 (x) = sin 2 (πx/2), and define

Infinitely many lanes -the continuum limit
It is natural, at least mathematically, to consider the case where the lanes increase in number while at the same time get closer. Our aim in this section is therefore to investigate limit as N → ∞ in the system in the previous section.
To this end we let (the number of lanes) N be a positive integer and set ∆y = 1/N . Let y i = (i − 1/2)∆y for i = 1, . . . , N . We shall also use the "divided difference" notation D ± a i = ± a i±1 − a i ∆y .
For simplicity, we restrict our presentation to the case where v i (u) = −k(y i )g(u) where g is a differentiable function with g (u) > 0, g(0) = −1 and g(1) = 0. Define f (u) = −ug(u). Throughout we will use the notation f i = f (u i ), g i = g(u i ) and k i = k(y i ). Now we reintroduce the scaling constant K in (2.1), and set K = κ/∆y 2 . For the convenience of the reader we set κ = 1. Thus, for i = 1, . . . , N , u i is the unique entropy (in the sense of Definition 3.1) solution of the balance equation , with the boundary conditions It is also useful to define the function u ∆y (t, x, y) by We shall investigate whether the family {u ∆y } ∆y=1/N , N ∈ N is compact and characterize the limit lim ∆y→0 u ∆y . To this end we must show a number of estimates. The right-hand side of (4.1) equals 1 Thus (4.1) reads . . , N , and we have the boundary values Remark. Observe that the above term b i is an upwind discretization of the transport term corresponding to au y , with a = (kg) y .
Similarly to (4.3), we also get the expression Recall (3.3) with η(u) = u 2 /2 and ϕ an approximation to 1 [0,T ] . That gives where Π T = [0, T ] × R. We can sum this for i = 1, . . . , N , multiply with ∆y and do a summation by parts to get It will be useful to lower bound the last two terms on the left-hand side.
Recall first that for some constant C independent of ∆y. Using this and the fact that max u∈[0,1] |g(u)| is bounded, as well as we have that Furthermore, note that the same argument yields and then use the inequality (a + b) − ≥ a − − |b|. Thus, since g > 0, where 0 < c ≤ min i k i min u g (u). Similarly, and therefore Note that due to the monotonicity of g we have for someũ between u i and u 1−1 , We can now estimate the last two terms of the left-hand side of (4.7) from below. More precisely, which we can rewrite as using (4.10) and (4.11). This implies that Observe that by (4.9), (4.12) follows from (4.13), viz.
By the same procedure, starting with (4.4) but using the alternate form (4.6) of the right-hand side, we arrive at the bounds (4.14) and (4.15) ∆y Combining the two bounds (4.12) and (4.14) we get In a similar manner, we find The other two bounds, (4.13) and (4.15) can be used for a continuity estimate. Write u i−1/2 = (u i + u i−1 )/2 and compute for ≥ m Squaring and integrating over [0, T ] × R gives By direct computations we have that which gives Multiplying with ∆y summing over i and integrating in x, t, gives the bound, using (4.18) with m = i − 1, = i and (4.17), Note that this also follows from (4.13), using that u i ∈ [0, 1].
Hence, the limit u is a weak solution.
We can sum up the result of our arguments in the following theorem.
The limit u is a weak solution according to Definition 4.1.
We have used ∆y = 1/60 (i.e., 60 lanes) and solved (4.1) using the Engquist-Osher scheme with 800 grid points in the interval [0, 2]. Figure 3 shows the computed density u at four different times. It is illuminating to compare this figure with Figures 1 and 2.