Follow-the-Leader models can be viewed as a numerical approximation to the Lighthill-Whitham-Richards model for traffic flow
FFOLLOW-THE-LEADER MODELS CAN BE VIEWED AS ANUMERICAL APPROXIMATION TO THELIGHTHILL–WHITHAM–RICHARDS MODEL FOR TRAFFICFLOW
HELGE HOLDEN AND NILS HENRIK RISEBRO
Abstract.
We show how to view the standard Follow-the-Leader (FtL) modelas a numerical method to compute numerically the solution of the Lighthill–Whitham–Richards (LWR) model for traffic flow. As a result we offer a simpleproof that FtL models converge to the LWR model for traffic flow when trafficbecomes dense. The proof is based on techniques used in the analysis ofnumerical schemes for conservation laws, and the equivalence of weak entropysolutions of conservation laws in the Lagrangian and Eulerian formulation. Introduction
There are two paradigms in the mathematical modeling of traffic flow. One isbased on an individual modeling of each vehicle with the dynamics governed bythe distance between adjacent vehicles. The other is based on the assumption ofdense traffic where the vehicles are represented by a density function, and individualvehicles cannot be identified. The dynamics is governed by a local velocity functiondepending solely on the density. The first model is denoted the Follow-the-Leader(FtL) model, and the second is called the Lighthill–Whitham–Richards (LWR)model [13, 14] for traffic flow. Further refinements and extensions of these modelsare available. Intuitively, it is clear that the the FtL model should approach orapproximate the LWR model in the case of heavy traffic, and that is what is provedhere. This problem has been extensively studied, see [1, 2, 3, 5, 6, 7, 8, 9, 10, 12, 15].Using numerical methods for scalar conservation laws we show that FtL modelsappear naturally as a numerical approximation of the LWR model. Thus we offera short and direct proof that the FtL model converges to the LWR model, and ouranalysis is based on a careful study of the relationship between weak solutions inLagrangian and Eulerian variables.In the LWR model vehicles are described by a density ρ = ρ ( t, x ) where x isthe position along the road, and t as usual denotes time. Locally, one assumesthat the velocity is given by a function v that depends on the density only, that is, v = v ( ρ ). If we consider unidirectional traffic on a homogenous road without exitsor entries, conservation of vehicles requires that the dynamics is governed by thescalar conservation law ρ t + (cid:0) ρv ( ρ ) (cid:1) x = 0 , which constitutes the LWR model. It is often denoted as “traffic hydrodynamics”due to its resemblance with fluid dynamics. Date : September 22, 2017.2010
Mathematics Subject Classification.
Primary: 35L02; Secondary: 35Q35, 82B21.
Key words and phrases.
Follow-the-Leader model, Lighthill–Whitham–Richards model, trafficflow, continuum limit.Research was supported by the grant
Waves and Nonlinear Phenomena (WaNP) from theResearch Council of Norway. The research was done while the authors were at Institut Mittag-Leffler, Stockholm. a r X i v : . [ m a t h . A P ] S e p HOLDEN AND RISEBRO
The FtL model can be described as follows. Consider N vehicles with length (cid:96) and position z ( t ) < · · · < z N ( t ) on the real axis with dynamics given by˙ z i = v (cid:16) (cid:96)z i +1 − z i (cid:17) for i = 1 , . . . , N − z N = v max Here v denotes a given velocity function with maximum v max , perhaps the speedlimit. Our proofs are considerably simpler when we have a uniform bound on z i +1 ( t ) − z i ( t ). Having empty road ahead of the first car would mean that “ z N +1 − z N = ∞ ”. This is the same as imposing ˙ z N = v max , and in this case z i +1 ( t ) − z i ( t )would not be bounded by a constant independent of time. Therefore we will in thispaper assume that we model one of two alternatives: Periodic case:
We are in the periodic case in which z i ∈ [ a, b ] for some interval[ a, b ], and ˙ z N ( t ) = v (cid:18) (cid:96)b − z N ( t ) − a + z ( t ) (cid:19) . Non-periodic case:
We imagine that there are infinitely many vehicles to theright of z N , the distance between each of these vehicles is M (cid:96) , for a finite, butarbitrary, constant
M >
1. In this case˙ z N ( t ) = v (cid:18) M (cid:19) . Introduce y i = ( z i +1 − z i ) /(cid:96) for i = 1 , . . . , N −
1, to obtain˙ y i ( t ) = 1 (cid:96) (cid:0) v (1 /y i +1 ( t )) − v (1 /y i ( t )) (cid:1) . In this paper we analyze the limit of this system of ordinary differential equationswhen N → ∞ . There are two ways to proceed.We may analyze this system directly, in what we call the semi-discrete case, seeSection 2.1. By using methods from the theory of numerical methods for scalarconservation laws we show that the sequence { y i ( t ) } N − i =1 converges, as (cid:96) → N → ∞ , to a function y ( t, x ) that satisfies the equation(1.1) (cid:40) y t − V ( y ) x = 0 t > , x ∈ [0 , ,y (0 , x ) = y ( x ) x ∈ [0 , , where V ( y ) = v (1 /y ), and with boundary condition (cid:40) y ( t,
1) = y ( t,
0) in the periodic case, y ( t,
1) = M else.Note that x is the Lagrangian mass coordinate, so that the integer part of x/(cid:96) measures how many cars there are to the left of x .Equation (1.1) is an example of a hyperbolic conservation law. It is well-knownthat solutions develop singularities, denoted shocks, in finite time independent ofthe smoothness of the initial data. Thus one needs to study weak solutions, anddesign so-called entropy conditions to identify the unique weak physical solution.For a scalar conservation law u t + f ( u ) x = 0 with initial data u | t =0 = u , the uniqueweak entropy solution u = u ( t, x ), which is an integrable function of boundedvariation, satisfies the Kruˇzkov entropy condition(1.2) (cid:90) (cid:90) ∞ (cid:0) | u − k | φ t + sign ( u − k )( f ( u ) − f ( k )) φ x (cid:1) dtdx + (cid:90) | u − k | φ | t =0 dx ≥ k ∈ R , and all non-negative test functions φ ∈ C ∞ ( R × [0 , ∞ )).See [11]. TL2LWR 3
As an alternative approach, see Section 2.2, we may discretize the time derivativeby a small positive ∆ t and write z nj ≈ z j ( n ∆ t ), y nj ≈ y j ( n ∆ t ), we have that z n +1 j = z nj + ∆ tV nj , and y n +1 j = y nj + ∆ t(cid:96) (cid:16) V nj +1 − V nj (cid:17) , where V nj = V ( y nj ). The key observation is that this is an approximation of thehyperbolic conservation law y t − V ( y ) x = 0 by a monotone scheme, and from theclassical result of Crandall–Majda [4], see also [11, Thm. 3.9], we know that thisscheme converges, as (cid:96) → N → ∞ , and ∆ t →
0, to the entropy solution ofequation (1.1), namely y t − V ( y ) x = 0. Thus in both cases we obtain convergenceto the same hyperbolic conservation law in Lagrangian coordinates.Next we have to transform the result of the two approaches, both in Lagrangiancoordinates, to Eulerian coordinates. For smooth solutions this is nothing but asimple exercise in calculus, but for weak entropy solutions this is a deep resultdue to Wagner [16]. To be specific, we introduce the Eulerian space coordinate z = z ( t, x ), with z x = y and z t = V ( y ). A straightforward (but formal) calculationreveals that the Eulerian functions satisfy y t = − ρ (cid:0) ρ t + ρ z v (cid:1) , V ( y ) x = 1 ρ v (cid:48) ( ρ ) ρ z , and hence ρ t + (cid:0) ρv ( ρ ) (cid:1) z = 0 , which is nothing but the LWR model. These formal transformations are not validin general for weak entropy solutions. However, thanks to the fundamental result ofWagner [16], weak entropy solutions in Lagrangian coordinates transform into weakentropy solutions in Eulerian variables. The approach here bears some resemblanceto the approach in [12], where the proof is obtained in a grid-less manner, and itdoes not depend on the use of Crandall–Majda and Wagner.2. The model
Let us first introduce the FtL model. Consider N vehicles moving on a one-dimensional road. Their position is given as a function of time t as z ( t ) , . . . , z N ( t ).For the moment (we shall actually show that this is so below) we assume that z ( t ) < z ( t ) < · · · < z N ( t ). We introduce the “local inverse density” by y i = 1 (cid:96) (cid:0) z i +1 − z i (cid:1) , i = 1 , . . . , N − , where (cid:96) is the length of each vehicle. The velocity of the vehicle at z i is assumedto be a function of the distance to the vehicle in front, at z i +1 . This means that(2.1) ˙ z i ( t ) = v (cid:16) (cid:96)z i +1 ( t ) − z i ( t ) (cid:17) , i = 1 , . . . , N − . Regarding the first vehicle, located at z N , we either assume that there are infinitelymany equally spaced vehicles in front of it, i.e., y N = M , or that we are in theperiodic setting in an interval [ a, b ], so that the distance from the vehicle at z N tothe vehicle at z is b − z N + z − a , i.e., y N = ( b − z N + z − a ) /(cid:96) . We have(2.2) ˙ z N ( t ) = v (cid:16) y N ( t ) (cid:17) . Regarding the velocity function v , we assume it to be a decreasing Lipschitz con-tinuous function such that(2.3) v (0) = v max and v ( ρ ) = 0 for ρ ≥ v ( ρ ) = v max max { , − ρ } . We define the velocityin Lagrangian variables by V ( y ) = v (1 /y ). Observe that V is globally bounded, HOLDEN AND RISEBRO
Lipschitz continuous and increasing for y ≥
1, with a bounded Lipschitz constant L v .Rewriting (2.1) in terms of { y i } we get(2.4) ˙ y i = 1 (cid:96) ( V ( y i +1 ) − V ( y i )) , i = 1 , . . . , N − , and(2.5) y N = (cid:40) M, non-periodic case, (cid:96) ( b − z N + z − a ) , periodic case.Let us also define the Lagrangian grid (cid:8) x i − / (cid:9) Ni =1 by x i − / = ( i − (cid:96) . We shallalso assume throughout that there is a constant 1 ≤ K < ∞ , K independent of N and (cid:96) , such that(2.6) 1 ≤ y j (0) ≤ K, N − (cid:88) j =1 | y j +1 (0) − y j (0) | ≤ K. The semi-discrete case.
In this section we show that the solution of thesystem (2.4) of ordinary differential equations converges to an entropy solution of(2.20) as (cid:96) →
0, and that “1 /y ” converges to an entropy solution of (2.12).Concretely, we define the piecewise constant function(2.7) y (cid:96) ( t, x ) = y j ( t ) , x ∈ ( x j − / , x j +1 / ] . We shall also use the notation D + h j = 1 (cid:96) (cid:0) h j +1 − h j (cid:1) for the forward difference. Let y + = max { y, } and y − = − min { y, } , and let H denote the Heaviside function H ( y ) = (cid:40) y ≤ , y > . Lemma 2.1.
Let y j ( t ) solve the system (2.4) . Then ddt ( y j − k ) + ≤ D + [ H ( y j − k ) ( V ( y j ) − V ( k ))] , (2.8a) ddt ( y j − k ) − ≤ D + [ − H ( k − y j ) ( V ( y j ) − V ( k ))] , (2.8b) for any constant k .Proof. Throughout we use the notation V j = V ( y j ). We have that ddt ( y j − k ) + = 1 (cid:96) H ( y j − k ) ( V j +1 − V j )= 1 (cid:96) [ H ( y j +1 − k )( V j +1 − V ( k )) − H ( y j − k )( V j − V ( k ))] − (cid:96) ( H ( y j +1 − k ) − H ( y j − k )) ( V j +1 − V ( k ))= D + [ H ( y j − k )( V j − V ( k ))] − (cid:96) ( H ( y j +1 − k ) − H ( y j − k )) ( V j +1 − V ( k )) . Now ( H ( y j +1 − k ) − H ( y j − k )) ( V j +1 − V ( k )) TL2LWR 5 = y j , y j +1 ≥ k or y j , y j +1 ≤ k , V j +1 − V ( k ) y j < k < y j +1 ,V ( k ) − V j +1 y j +1 < k < y j , ≥ , since y (cid:55)→ V ( y ) is increasing. This proves (2.8a); estimate (2.8b) is proved similarly. (cid:3) Now define y j ( t ) = y ( t ) for j < y j ( t ) = y N − ( t ) for j > N − y j ( t ) by periodic extension. Tosave space, we also use the convention that in the non-periodic case, sums over j range over all j ∈ Z , while in the periodic case, sums range over j = 1 , . . . , N − Lemma 2.2. If ≤ y j (0) ≤ K , then ≤ y j ( t ) ≤ K for t > .Proof. From (2.8a) and (2.8b) we have ddt (cid:88) j ( y j ( t ) − k ) ± ≤ . Thus if y j (0) ≤ K for all j , then y j ( t ) < k for any constant k > K . Similarly y j ( t ) > k for any constant k < y j (0) ≥ j . (cid:3) Lemma 2.3. If { ˜ y j ( t ) } N − j =1 is another solution of (2.4) and (2.5) with initial data ˜ y j (0) , then (2.9) (cid:88) j | y j ( T ) − ˜ y j ( T ) | ≤ (cid:88) j | y j (0) − ˜ y j (0) | , for T > .Proof. Adding (2.8a) and (2.8b), and observing that( y − k ) + + ( y − k ) − = | y − k | and H ( y − k ) − H ( k − y ) = sign ( y − k ) , we find that(2.10) ddt | y j − k | ≤ D + [sign ( y j − k ) ( V ( y j ) − V ( k ))] . Set q ( y, k ) = sign ( y − k ) ( V ( y ) − V ( k )). Choosing k = ˜ y j ( τ ) in the inequality for y j ( t ) and k = y j ( t ) in the inequality for ˜ y j ( τ ), and adding the two inequalities, give (cid:18) ddt + ddτ (cid:19) | y j ( t ) − ˜ y j ( τ ) | ≤ ( D + , + D + , ) q ( y j ( t ) , ˜ y j ( τ )) , where D + , denotes the difference with respect to the first (second) argument.Summing over j , multiplying with a non-negative test function ϕ ( t, τ ), where ϕ ∈ C ∞ ((0 , ∞ ) ), and integrating by parts yield (cid:90) ∞ (cid:90) ∞ ( ϕ t + ϕ τ ) (cid:88) j | y j ( t ) − ˜ y j ( τ ) | dτ dt ≥ , since taking the sum makes the right-hand side “telescope”. Now we can useKruˇzkov’s trick, see [11, Sec. 2.4], and choose ϕ ( t, τ ) = ψ (cid:18) t + τ (cid:19) ω ε ( t − τ ) , where ψ ∈ C ∞ ((0 , ∞ )), ψ ≥ ω ε is a standard mollifier, to obtain, as ε → (cid:90) ∞ ψ (cid:48) ( t ) (cid:88) j | y j ( t ) − ˜ y j ( t ) | dt ≥ . HOLDEN AND RISEBRO
Choose ψ to be a smooth approximation to the characteristic function of the interval( t , t ) ⊂ (0 , T ), to get (cid:88) j | y j ( t ) − ˜ y j ( t ) | ≤ (cid:88) j | y j ( t ) − ˜ y j ( t ) | . The lemma follows by letting t ↓ t ↑ T . For details, see [11, Sec. 2.4]. (cid:3) Lemma 2.4.
Assume that ≤ y j (0) ≤ K and that (cid:80) j | y j +1 (0) − y j (0) | ≤ K forsome constant K independent of (cid:96) . Then there is a sequence { (cid:96) i } , where (cid:96) i → as i → ∞ , and there exists a function y ∈ C ([0 , T ]; L ([0 , such that y (cid:96) i convergesto y in C ([0 , T ]; L ([0 , .Proof. Lemma 2.2 shows that { y (cid:96) } (cid:96) is bounded independently of (cid:96) ; choosing ˜ y j = y j +1 and using Lemma 2.3 yields the BV bound on { y (cid:96) ( t ) } (cid:96) uniformly in (cid:96) and t .Choosing ˜ y j ( t ) = y j ( t − σ ) in Lemma 2.3 for some 0 < σ < t gives (cid:107) y (cid:96) ( t, · ) − y (cid:96) ( t − σ, · ) (cid:107) L = (cid:96) (cid:88) j | y j ( t ) − y j ( t − σ ) |≤ (cid:96) (cid:88) j | y j ( σ ) − y j (0) |≤ (cid:88) j (cid:90) σ | V ( y j +1 ( ξ )) − V ( y j ( ξ )) | dξ ≤ (cid:107) V (cid:107) Lip (cid:88) j (cid:90) σ | y j +1 ( ξ ) − y j ( ξ ) | dξ ≤ (cid:107) V (cid:107) Lip σ (cid:88) j | y j +1 (0) − y j (0) |≤ (cid:107) V (cid:107) Lip σK.
Hence the map t (cid:55)→ y (cid:96) ( t, · ) is L Lipschitz continuous, with a Lipschitz constantindependent of (cid:96) . Thus by [11, Thm. A.11], the family { y (cid:96) } (cid:96)> is compact in C ([0 , ∞ ); L ([0 , (cid:3) Furthermore we assume that as N increases, the initial position of the vehiclesare such that there is a function y ( x ) such that(2.11) lim (cid:96) → y (cid:96) (0 , · ) = y ( · ) , and that this convergence is in L ([0 , (cid:107) y (cid:107) L ∞ ([0 , ≤ K ,without loss of generality we can then also assume that (cid:107) y (cid:96) (0 , · ) (cid:107) L ∞ ([0 , ≤ K .It is now straightforward, starting from the discrete entropy inequality (2.10), toshow that any limit of { y (cid:96) } (cid:96)> is the unique entropy solution to (2.20) by followinga standard Lax–Wendroff argument, see [11, Thm. 3.4]. Thus the whole sequence { y (cid:96) } converges, and the unique entropy solution to (1.1) is the limit y = lim (cid:96) → y (cid:96) . Introduce the Eulerian spatial coordinate z , given by the equations ∂z∂x = y, ∂z∂t = V ( y ) , and the variable ρ = 1 /y . We can now proceed following the argument of Wag-ner [16] to obtain that ρ is the unique weak entropy solution to the LWR model(2.12) (cid:40) ρ t + (cid:0) ρv ( ρ ) (cid:1) z = 0 , t > ,ρ (0 , z ) = ρ ( z ) . TL2LWR 7
We can also study the convergence in Eulerian coordinates directly by defininga discrete version of the transformation from Lagrangian to Eulerian coordinates.To define the discrete version of ρ , we need the approximate Eulerian coordinate; z (cid:96) ( t, x ). Define z (cid:96) ( t, x ) = 1 (cid:96) (cid:0) x j +1 / − x (cid:1) z j ( t ) + 1 (cid:96) (cid:0) x − x j − / (cid:1) z j +1 ( t ) , for x ∈ [ x j − / , x j +1 / ],where { z j ( t ) } solves (2.1). Then ∂z (cid:96) ∂t = 1 (cid:96) (cid:0) x j +1 / − x (cid:1) V j + 1 (cid:96) (cid:0) x − x j − / (cid:1) V j +1 ,∂z (cid:96) ∂x = y j , for x ∈ ( x j − / , x j +1 / ). The sequence { z (cid:96) } (cid:96)> is uniformly Lipschitz continuous.Hence by the Arzel`a–Ascoli theorem, it converges uniformly to a Lipschitz contin-uous limit z ( t, x ) satisfying z t = V ( y ) and z x = y almost everywhere. Furthermorethe map x (cid:55)→ z (cid:96) ( t, x ) is invertible, with inverse x (cid:96) ( t, z ). In the periodic case we set(2.13a) z l,(cid:96) ( t ) = a, z r ( t ) = b, otherwise we define(2.13b) z l,(cid:96) ( t ) = z (cid:96) ( t, , z r ( t ) = b + tV ( M ) = z ( t,
1) = z (cid:96) ( t, . Observe that z l,(cid:96) ( t ) = z (cid:96) ( t, → z ( t,
0) = z l ( t ) as (cid:96) →
0. Define(2.14) ρ (cid:96) ( t, z ) = 1 y (cid:96) ( t, x (cid:96) ( t, z )) for z ∈ [ z l,(cid:96) ( t ) , z r ( t )].In the periodic case, we define ρ (cid:96) by periodic continuation, while in the non-periodiccase we define ρ (cid:96) ( t, z ) = (cid:40) z < z l,(cid:96) , /M z > z r . Next we claim that(2.15) ρ (cid:96) ( t, z ) → ρ ( t, z ) = ˜ ρ ( t, x ( t, z ))in L ([ z l , z r ]) as (cid:96) →
0. To see this, define ˜ ρ (cid:96) ( t, x ) = 1 /y (cid:96) ( t, x ), and compute (cid:107) ρ ( t, · ) − ρ (cid:96) ( t, · ) (cid:107) L = (cid:107) ˜ ρ ( t, x ( t, · )) − ˜ ρ (cid:96) ( t, x (cid:96) ( t, · )) (cid:107) L ≤ (cid:107) ˜ ρ ( t, x ( t, · )) − ˜ ρ ( t, x (cid:96) ( t, · )) (cid:107) L (cid:124) (cid:123)(cid:122) (cid:125) A + (cid:107) ˜ ρ ( t, x (cid:96) ( t, · )) − ˜ ρ (cid:96) ( t, x (cid:96) ( t, · )) (cid:107) L (cid:124) (cid:123)(cid:122) (cid:125) B . We have that A = (cid:90) z r z l | ˜ ρ ( t, x ( t, z )) − ˜ ρ ( t, x (cid:96) ( t, z )) | dz ≤ (cid:16)(cid:90) max { z l ,z l,(cid:96) } min { z l ,z l,(cid:96) } + (cid:90) z r max { z l ,z l,(cid:96) } (cid:17) | ˜ ρ ( t, x ( t, z )) − ˜ ρ ( t, x (cid:96) ( t, z )) | dz. Since ˜ ρ (cid:96) and ˜ ρ are both bounded by 1, and z l,(cid:96) → z l as (cid:96) →
0, the first of theseintegrals tend to zero. Since x (cid:96) → x uniformly, the integrand tends to zero almosteverywhere, and is bounded by 2. Hence by the dominated convergence theorem,the last integral tends to zero. The same argument applies to B . Thus the claim(2.15) is justified.Summing up, we have shown the following result. HOLDEN AND RISEBRO
Theorem 2.5.
Assume that the function v satisfies (2.3) . Let { y j } N − j =1 satisfy (2.4) , with either periodic boundary conditions; y N ( t ) = y ( t ) , or y N ( t ) = M for some fixed constant M > . Assume that the initial positions of the vehi-cles { z i (0) } Ni =1 are such the we can define a bounded function y by (2.11) , and that (2.6) holds, namely that the initial data are bounded with finite total variation.(i) The piecewise constant (in space) function y (cid:96) ( t, x ) defined by (2.7) converges in C ([0 , T ]; L ([0 , as (cid:96) → to the unique weak entropy solution y of (1.1) . Thefunction ρ = 1 /y satisfies the LWR model (2.12) in Eulerian variables.(ii) The function ρ (cid:96) defined by (2.14) converges in C ([0 , T ]; L ([0 , as (cid:96) → tothe unique weak entropy solution ρ of (2.12) . Analysis of the Euler scheme for (2.1) . The simplest numerical methodto approximate solutions of (2.1) is the forward Euler scheme, viz.,(2.16) z i (( n + 1)∆ t ) = z i ( n ∆ t ) + ∆ tv (cid:16) (cid:96)z i +1 ( n ∆ t ) − z i ( n ∆ t ) (cid:17) , where ∆ t is a (small) positive number.If we write the Euler scheme (2.16) in the y variable, we get(2.17) y n +1 i = y ni + λ (cid:0) V ni +1 − V ni (cid:1) , i = 1 , . . . , N − , where y ni = y i ( t n ), t n = n ∆ t , λ = ∆ t/(cid:96) and V ni = V ( y ni ). As a (right) boundarycondition we use(2.18) V nN = (cid:40) V ( M ) non-periodic, V n periodic.For t ≥ x ∈ [0 , ( N − (cid:96) ] define the function y (cid:96) ( t, x ) = y ni ( t, x ) ∈ [ t n , t n +1 ) × ( x i − / , x i +1 / ] . Observe that we can rewrite (2.17) as y n +1 i = (cid:16) − λθ ni +1 / (cid:17) y ni + λθ ni +1 / y ni +1 , where θ ni +1 / = − V ni +1 − V ni y ni +1 − y ni ≥ , and since V is Lipschitz continuous, θ ni +1 / ≤ L v . Hence if the CFL-condition(2.19) λL v ≤ , holds, then y n +1 i is a convex combination of y ni and y ni +1 . Thus the scheme (2.17)is monotone. In passing, we note that a consequence is that if 1 ≤ y i ≤ K for all i , then 1 ≤ y ni ≤ K for all i . Regarding the position of vehicles, this means that if z i (0) ≤ z i +1 (0) − (cid:96) , then z i ( t n ) ≤ z i +1 ( t n ) − (cid:96) . So from a road safety perspective,the model is rather optimistic.We are now interested in taking the limit as (cid:96) →
0. We do this by increasingthe number of vehicles such that ( N − (cid:96) = 1; furthermore we assume that (2.11)holds. Now the conditions are such that fundamental results of Crandall and Majda[4], see also [11, Thm. 3.9], can be applied. Thus we know that there is a function y : R +0 × [0 , → R , with y ∈ C ( R + ; L ([0 , y (cid:96) ( t, x ) → y ( t, x ) , with the limit being in C ( R + ; L ([0 , y is the unique entropy solutionto the Cauchy problem(2.20) (cid:40) y t − V ( y ) x = 0 , t > , x ∈ [0 , ,y (0 , x ) = y ( x ) . TL2LWR 9
If we do not have periodic conditions, this is supplemented with the boundarycondition y ( t,
1) =
M, t > . We remark that since the characteristic speeds of (2.20) are strictly negative, thisboundary condition can be enforced strictly.Note that the convergence of y (cid:96) and the bounds 1 ≤ y (cid:96) ≤ M , imply the conver-gence of ˜ ρ (cid:96) = 1 /y (cid:96) to some function ˜ ρ . We now proceed to show how ˜ ρ is relatedto the solution of the LWR model.We also define the discrete “Lagrange to Euler” map ˜ z (cid:96) as follows. Let˜ z ni +1 / = ˜ z ni − / + (cid:96)y ni , i.e., ˜ z ni +1 / = z ni +1 . Since z ni solves (2.16), we also have that˜ z n +1 i +1 / = ˜ z ni +1 / + ∆ tv ni +1 . Define ˜ z (cid:96) ( t n , x i +1 / ) = ˜ z ni +1 / , and by bilinear interpolation between these points.For later use we employ the notation for the value of ˜ z (cid:96) at the edges of the “ La-grangian grid ”,˜ z i +1 / ( t ) = 1∆ t (cid:16) ( t n +1 − t )˜ z ni +1 / + ( t − t n )˜ z n +1 i +1 / (cid:17) , for t ∈ [ t n , t n +1 ] , ˜ z n ( x ) = 1 (cid:96) (cid:16) ( x i +1 / − x )˜ z ni − / + ( x − x i − / )˜ z ni +1 / (cid:17) , for x ∈ [ x i − / , x i +1 / ].Observe that ˜ z i − / ( t ) coincides with the approximate trajectory of the vehiclestarting at z i (0) calculated by the Euler method (2.16). Since y (cid:96) is bounded, wecan invoke the Arzel`a–Ascoli theorem to establish the convergencelim (cid:96) → ˜ z (cid:96) ( t, x ) = z ( t, x ) , with the limit being in C ([0 , T ] × [0 , z ∈ C ([0 , T ] × [0 , ∂z∂x = y, ∂z∂t = V ( y ) , weakly. We have that the map x (cid:55)→ ˜ z (cid:96) ( t, x ) is invertible for each t , we denote theinverse map by x (cid:96) , so that x (cid:96) ( t, z (cid:96) ( t, x )) = x . Define z l,(cid:96) and z r as in (2.13) and ρ (cid:96) as in (2.14).Note that if z ∈ ( z i − / ( t ) , z i +1 / ( t )] and t ∈ [ t n , t n +1 ), then x (cid:96) ( t, z ) ∈ ( x i − / , x i +1 / ] , ρ (cid:96) ( t, z ) = ˜ ρ ni := 1 y ni . As before we have that ρ (cid:96) ( t, z ) → ρ ( t, z ) = ˜ ρ ( t, x ( t, z ))in L ([ z l , z r ]) as (cid:96) → Theorem 2.6.
Assume that the function v satisfies (2.3) . Let (cid:96) > and N ∈ N , let { z j } Nj =1 satisfy (2.16) , and assume that either we are in the periodic case z j ∈ [0 , ,or that z N satisfies the boundary condition (2.2) , with y N = M . Assume that theinitial positions of the vehicles { z i (0) } Ni =1 are such the we can define a boundedfunction y by (2.11) , and that (2.6) holds.Define the function ρ (cid:96) ( t, z ) by (2.14) . Let N and (cid:96) satisfy ( N − (cid:96) = 1 andassume that the CFL-condition (2.19) holds.As (cid:96) → , ρ (cid:96) converges in C ([0 , ∞ ); L ) to the unique entropy solution ρ of theconservation law (2.12) . To illustrate the ideas in this paper we show how the method works in a concreteexample. We have a periodic road in the interval z ∈ [ − , N vehicles in this interval so that ρ (cid:96) (0 , z ) ≈
12 (cos( πz ) + 1) . In Figure 1 we show the Lagrangian grid and the corresponding mapping to Euleriancoordinates for N = 20, and t ∈ [0 , x t Lagrangian coordinates z -1 1 t Eulerian coordinates
Figure 1.
Left: the Lagrangian grid (cid:8) ( t n , x i − / ) (cid:9) Ni =1 . Right:the Eulerian grid (cid:110) ( t n , z ni − / ) (cid:111) Ni =1 . In both cases n = 0 , . . . , z (cid:96) to the rectangular grid depicted in Lagrangiancoordinates on the left. In Figure 2, we show the approximate density ρ (cid:96) at t = 0and t = 2 in Eulerian coordinates. We see that the solution at t = 2 approximates z -1 -0.5 0 0.5 1 ρ ℓ Initial densityFinal density . Figure 2.
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Department of Mathematical Sciences, NTNU Norwegian University of Science andTechnology, NO–7491 Trondheim, Norway
E-mail address : [email protected] URL : (Risebro) Department of Mathematics, University of Oslo, P.O. Box 1053, Blindern, NO–0316Oslo, Norway
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