Discrete velocity models and relaxation schemes

Discrete velocity models and relaxation schemes for traffic flows M. Herty∗

L. Pareschi†

M. Sea¨ıd‡

Abstract We present simple discrete velocity models for traffic flows. The novel feature in the corresponding relaxation system is the presence of non negative velocities only. We show that in the small relaxation limit the discrete models reduce to the Lighthill-Whitham-Richards equation. In addition we propose second order schemes combined with IMEX time integrators as proper discretization of the relaxation-type system. Numerical tests are carried out on various situations in traffic flow. The results show that the proposed models are capable to describe correctly the formation of backward waves induced by traffic jam.

Keywords. Discrete velocity models, Relaxation schemes, IMEX schemes, ENO schemes, Lighthill-Whitham-Richards equation.

Contents 1 Introduction

2

2 Discrete velocity models of traffic flows

4

2.1

Two speeds models . . . . . . . . . . . . . . . . . . . . . . . . . .

5

2.2

Three speeds models . . . . . . . . . . . . . . . . . . . . . . . . .

6

∗

FB Mathematik, TU Darmstadt, 64289 Germany ([email protected]) Department of Mathematics, University of Ferrara, 44100 Italy ([email protected]) ‡ FB Mathematik, TU Darmstadt, 64289 Germany ([email protected]) †

1

3 Relaxation schemes

8

3.1

Space discretization . . . . . . . . . . . . . . . . . . . . . . . . .

3.2

Time integration . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.3

Relaxed schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4 Numerical examples

8

12

4.1

Free-flow traffic situation

. . . . . . . . . . . . . . . . . . . . . . 12

4.2

Jam situation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4.3

Bottleneck situation . . . . . . . . . . . . . . . . . . . . . . . . . 13

5 Conclusions

13

References

15

1

Introduction

Macroscopic modeling of vehicular traffic started with the work of Lighthill and Whitham [12]. They considered the continuity equation for the density ρ and approximated the mean velocity u by an equilibrium value ue (ρ) ∂t ρ + ∂x (ρue (ρ)) = 0.

(1)

The function ue (ρ) is also called fundamental diagram and ρue (ρ) is the flux. For a comparison with experimental data we refer to [13]. Including additional momentum equations for u, we can derive so called “higher order” models for traffic flow. Examples and mathematical studies of “second order” models can be found in [19, 3, 1]. Another type of mathematical models used in traffic dynamics are kinetic models. Therein, a kinetic car density f is introduced. Kinetic equations can be found, for example, in [20, 21, 9, 4]. Procedures to derive macroscopic traffic equations from underlying kinetic models have been performed in different ways by various authors, see [18, 9]. In this paper we focus on simple discrete velocity models of traffic flows which share the common properties that in the small relaxation limit reduce to a macroscopic description of traffic flow by the single conservation law (1). 2

In [7], Jin and Xin proposed the relaxation system ∂t ρ + ∂x q = 0, ∂t q + a2 ∂x ρ = −

 1 q − F (ρ) , ε

(2)

as an approximation to the scalar hyperbolic equation ∂t ρ + ∂x F (ρ) = 0.

(3)

Formally, solution of the relaxation system (2) approximates solution to the original equation (3) by order O(ε) if the constant a satisfies the subcharacteristic condition [7, 6, 2, 14] a2 − F ′ (ρ)2 ≥ 0,

∀ ρ.

(4)

At a numerical level, the main advantage in considering a relaxation system is the fact that the nonlinear conservation law (3) is replaced by a semi-linear system (2) with linear characteristic variables given by v ± au. Consequently nonlinear Riemann problems, characteristic decompositions and linear iterations are avoided in its numerical solution. The idea of developing relaxation methods to approximate numerical solutions to partial differential equations has a long tradition. This field of research is very active for hyperbolic systems of conservation laws, where a vast number of relaxation schemes have been designed based on high-order reconstructions and shock capturing techniques. We refer the reader to [10, 7, 5, 22] and further references can be found therein. All of these methods are easy to formulate and to implement. Moreover, there is strong links between relaxation methods and central schemes, see for instance [24, 16]. The relaxation system (2) is strictly connected to discrete velocity models in kinetic theory. In fact it can be reformulated by considering “kinetic” variables f and g as follows   1 1 q ∂t f + a∂x f = − , f− ρ+ ε 2 a   (5) 1 1 q ∂t g − a∂x g = − g− ρ− , ε 2 a where ρ = f + g and q = a(f − g). The equations (5) can be viewed as an evolution system of particles with speeds ±a and interaction terms given by the right-hand side, compare [15, 8, 17]. Note that in (5) the kinetic variables f and g propagate with positive and negative speed, respectively. This fits well with the fact that gas particles can move in either direction. However, if we restrict to a situation where all particles have the same (nonnegative) propagation direction, then the above formulation will not describe the correct dynamics of the relaxed flow. If we consider the 3

case of particles travelling with speeds b > a ≥ 0 easy computations lead to the subcharacteristic condition

F ′ (ρ) − a b − F ′ (ρ) ≥ 0, (6)

which is not satisfied unless a ≤ F ′ (ρ) ≤ b. A physical field where such situation takes place is the case of traffic flow models where cars can have only positive velocities but a traffic jam moves also backwards. Therefore, it is our goal here to derive simple relaxation-type models with a clear “kinetic” interpretation since we allow car to propagate only with nonnegative speeds but still able to capture the backward propagation of the solution. Our objective in this paper is twofold: on the one hand, we develop simple discrete velocity models for traffic flow problems, on the other hand we develop relaxation schemes which are able to capture the correct asymptotic limit even when only nonnegative positive speeds are used. The accuracy of the method is illustrated by numerical examples for traffic flow in presence of both smooth and shock solutions. In the following section, we present the traffic flow equations used in the paper to develop relaxation-type model. In section 3, the relaxation scheme for the model problem is detailed. This includes the space and time discretizations. The numerical results are presented in section 4. We apply our scheme to common situations like free flow, bottleneck and traffic jam. In section 5, some concluding remarks are listed.

2

Discrete velocity models of traffic flows

In traffic flow theory the time evolution of a vehicle density in a freeway is governed by the Lighthill-Whitham-Richards model [12] ∂t ρ + ∂x V (ρ) = 0,

(7)

where ρ(t, x) is the vehicle density at location x and time t, and V (ρ) is the flow function given by V (ρ) = ρ min{u0 , ue (ρ)}, (8) where ue (ρ) is the fundamental diagram, i.e. an equilibrium velocity, and u0 the minimum speed. A typical choice for ue (ρ) is for example   ρ , (9) ue (ρ) = um 1 − ρm with ρm and um are the maximum density and the maximum speed, respectively. Remark 1 More general flux functions as well as general traffic flow models have been considered in the literature, see [9, 1]. For inhomogeneous traffic flow situations we may have an additional dependence on x, i.e. V (ρ) = V (x, ρ). 4

2.1

Two speeds models

Let us consider the following simple discrete velocity model ∂ t f + v1 ∂ x f

= −

∂ t g + v2 ∂ x g =

 K(ρ)  v1 f + v2 g − V (ρ) , 2ε

 K(ρ)  v1 f + v2 g − V (ρ) , 2ε

(10) (11)

where K(ρ) is a suitable function such that  K(ρ) ≥ 0, if ρ ≤ ρ∗ , K(ρ) < 0, if

ρ > ρ∗ ,

with 0 < ρ∗ < ρm is a constant.

The system (10)-(11) offers the following interpretation. We have two species of cars driving with velocity v2 > v1 ≥ 0. The functions f and g are probability distributions of cars with speeds v1 and v2 , respectively. Depending on the density, K(ρ) changes sign. Usually in traffic flow models, the flux function V : ρ → ρue (ρ) is a concave function. Hence K(ρ) ≥ 0 in low density regimes, whereas K(ρ) should become large and negative in high density regimes so that there is gain for f and a loss for g by the interaction terms. Therefore, more cars will attain the speed v1 . This fits well to the observation, that in areas of high densities the cars tend to slow down. The opposite phenomena is also governed by the equations (10)-(11). In the sequel we assume that
K(ρ) = H V ′ (ρ) − v1 ,

(12)

where H(·) is a function that preserves the sign of the argument (for example a Heavyside or a hyperbolic tangent function, see Figure 1). Introducing the density and the flux of cars defined as ρ=f +g

and

J = v1 f + v2 g,

(13)

we obtain the following macroscopic system ∂t ρ + ∂x J

= 0,

∂t J + (v1 + v2 )∂x J − v1 v2 ∂x ρ = −

(14)   K(ρ) (v2 − v1 ) J − V (ρ) . ε

(15)

In the small relaxation limit (ε → 0), the equation (15) gives to leading order  

ε V ′ (ρ) − v1 V ′ (ρ) − v2 ∂x ρ . J = V (ρ) + K(ρ) (v2 − v1 ) 5

By using this variable in the equation (14) we obtain ∂t ρ + ∂x V (ρ) + ∂x



ε V ′ (ρ) − v1 V ′ (ρ) − v2 ∂x ρ K(ρ) (v2 − v1 )

!

= 0. (16)

To ensure the dissipative nature of (16), it is necessary that (V ′ (ρ) − v1 ) (V ′ (ρ) − v2 ) ≤ 0. K(ρ) (v2 − v1 )

(17)

This condition is satisfied by the assumption (12) on the kernel K(ρ) if v1 ≤ um ≤ v2 . Formally, when ε → 0, the relaxation system (14)-(15) converges to the original equation (7) provided the subcharacteristic condition (17) is satisfied. Note that the equilibrium states for f and g are defined by     ρ V (ρ) V (ρ) ρ Ef (ρ) = v2 − − v1 . , Eg (ρ) = v2 − v1 ρ v2 − v1 ρ

(18)

Nonegativity of the equilibrium states is guaranteed if v1 ≤

V (ρ) ≤ v2 , ρ

(19)

which is satisfied under the natural requirements u0 ≥ v1 and um ≤ v2 . Remark 2 It is remarkable that when the density in the system becomes critical, i.e. ρ → ρm , a desirable feature would be that all cars reduce their speed to the lowest speed v1 without allowing f + g > ρm . This behavior is obtained naturally for small values of ε in agreement with the kinetic interpretation of ε as the mean free path between car-particles. As an alternative, in order to have this feature even for large values of ε we can ask that K(ρ) → −∞ as ρ → ρm to recover the critical equilibrium states f=

ρm (v2 − u0 ) , v2 − v1

g=

ρm (u0 − v1 ) , v2 − v1

(20)

which correspond to a traffic jam situation. Clearly taking u0 = v1 we have f = ρm and g = 0.

2.2

Three speeds models

In this section we extend the previous idea by considering models with three different speeds v2 , v1 , 0 s.t. v2 > v1 > 0. The model describing the traffic interaction will be given by

6

∂ t f + v1 ∂ x f

 K(ρ)(v2 − v1 )  Ef (ρ) − f , 2ε

=

 K(ρ)(v2 − v1 )  Eg (ρ) − g , 2ε  K(ρ)(v2 − v1 )  Eh (ρ) − h , 2ε

∂ t g + v2 ∂ x g = ∂t h =

where the equilibrium states are     V (ρ) um − v1 V (ρ) v2 − um , Eg (ρ) = , Ef (ρ) = um v2 − v1 um v2 − v 1

Eh (ρ) = ρ −

(21) (22) (23)

V (ρ) . um

In (21)-(23), K(ρ) is a kernel as in the two speeds model. Nonnegativity of the equilibrium states is guaranteed under the natural assumption v2 ≥ um ≥ v1 . Introducing the macroscopic variables ρ = f + g + h,

J = v1 f + v2 g,

z = f + g,

(24)

we obtain the following set of equations ∂ t ρ + ∂x J

= 0,   K(ρ) (v2 − v1 ) J − V (ρ) , ε  V (ρ)  K(ρ) (v2 − v1 ) z − . = − ε um

∂t J + (v1 + v2 )∂x J − v1 v2 ∂x z = − ∂t z + ∂x J

(25) (26) (27)

In the small relaxation limit (ε → 0), equation (27) gives z = V (ρ)/um and thus equation (26) gives to leading order    v1 v2 ε ′ ′ V (ρ) − (v1 + v2 ) + J = V (ρ) + V (ρ)∂x ρ . K(ρ) (v2 − v1 ) um By using this variable in the equation (25) we obtain ∂t ρ+∂x V (ρ)+∂x

εV ′ (ρ) K(ρ) (v2 − v1 )

!   v1 v 2 ′ ∂x ρ = 0. (28) V (ρ) − (v1 + v2 ) + um

To ensure the dissipative nature of (28), it is necessary that   v1 v2 V ′ (ρ) V ′ (ρ) − (v1 + v2 ) + ≤ 0. K(ρ)(v2 − v1 ) um

(29)

The condition is satisfied by assuming
K(ρ) = H V ′ (ρ) ,

where H(·) is a function that preserves the sign of the argument. 7

(30)

Remark 3 Again, when the density in the system becomes critical, i.e. ρ → ρm , if we ask that K(ρ) → −∞ we obtain the critical equilibrium states     u 0 v2 − u m u 0 u m − v1 u0 Ef (ρ) = , Eg (ρ) = , Eh (ρ) = ρ − , u m v2 − v1 u m v2 − v1 um corresponding to the traffic jam situation. Clearly now we can take u0 = 0 to get f = 0, g = 0 and h = ρm .

3

Relaxation schemes

We present a numerical method to approximate the discrete velocity models in such a way that the resulting scheme is asymptotic preserving (AP) and high order accurate in space and time (we refer to [16] for more details on AP schemes). The numerical method we construct is based on the method of lines approach. We use a high order central WENO scheme as space discretization of the kinetic system. The necessity to use a central scheme comes from the fact that the limiting equation (7) allows solutions ρ(t, x) which can propagate with either positive or negative speed (see the Appendix). For the time integration we apply a third-order IMEX Runge-Kutta method. For simplicity we will describe the numerical schemes for the two speeds discrete velocity model. The extension to the three speeds model is straightforward.

3.1

Space discretization

Let the spatial interval discretized in equally spaced grids with stepsize ∆x, and let wi denotes the point-value of an arbitrary function w at gridpoint xi . Then a semi-discretization of the equation (10) can be written as fi+ 1 − fi− 1 dfi 2 2 + v1 = G+ i . dt ∆x

(31)

For simplicity in presentation and in order to show how a space discretication can be reconstructed we consider only the Lax-Friedrichs method. Extension to higher order reconstructions can be developed using similar formulation. Thus, the first order Lax-Friedrichs numerical flux fi+ 1 is given by 2

fi+ 1 = 2

1 2

  ∆x fi+1 + fi − (fi+1 − fi ) . v1 ∆t

8

(32)

Similarly, the semi-discretization of equation (11) can be written as gi+ 1 − gi− 1 dgi 2 2 + v2 = G− i . dt ∆x

(33)

where gi+ 1 is defined analogously by 2

gi+ 1

2

1 = 2

  ∆x (gi+1 − gi ) . gi+1 + gi − v2 ∆t

(34)

The source terms G± i are given by G± i =±

 K(ρi )  v1 fi + v2 gi − V (ρi ) . 2ε

(35)

dρi Ji+ 21 − Ji− 12 + = 0, dt ∆x

(36)

Using (18) in (31) and (33) gives a first-order semi-discretization of the relaxation system (14)-(15)

Ji+ 1 − Ji− 1 ρi+ 1 − ρi− 1 dJi 2 2 2 2 + (v1 + v2 ) − v 1 v2 = dt ∆x ∆x − where ρi+ 1 = fi+ 1 + gi+ 1 2

2

2

and

  K(ρi ) (v2 − v1 ) Ji − V (ρi ) , 2ε

(37)

Ji+ 1 = v1 fi+ 1 + v2 gi+ 1 , 2

2

2

with fi+ 1 and gi+ 1 are given by (32) and (34), respectively. 2

2

Higher order spatial discretizations can be derived following the ENO reconstructions, compare [23, 24, 11] among others. For instance, a second order reconstruction is obtained by substituting the numerical fluxes fi+ 1 and gi+ 1 2 2 using the convex ENO scheme [11]. Thus,     and gi+ 1 = R v2 , gi+ 1 , (38) fi+ 1 = R v1 , fi+ 1 2

2

2

2

where the reconstruction flux function R is defined as     1 ∆x R α, wi+ 1 = (wi+1 − wi ) + wi+1 + wi − 2 2 α∆t  
∆x 1 α wi+1 − wi + (wi+1 − wi ) φ θ+ − 4 α∆t  
1 ∆x α wi+2 − wi+1 − (wi+2 − wi+1 ) φ θ− , 4 α∆t

(39)

α and θ α are given by with the slopes θ+ − α θ+ =

wi − wi−1 + wi+1 − wi +

∆x α∆t ∆x α∆t

(wi − wi−1 ) (wi+1 − wi )

,

α θ− =

9

wi+1 − wi − wi+2 − wi+1 −

∆x α∆t ∆x α∆t

(wi+1 − wi ) (wi+2 − wi+1 )

,

and φ is a slope limiter function. A simple choice is the minmod limiter φ(θ) = max (0, min (1, θ)) . Note that if we set φ = 0 in (39), the reconstruction (38) reduces to the first order Lax-Friedrichs discretization (32) and (34). We should mention that any discretization that requires characteristics information will generate instabilities in the computed solution. This is due to the structure of the relaxation system which allows propagation along the characteristics in one direction only.

3.2

Time integration

The fully discretization of system (14)-(15) can be obtained by the well established IMEX methods, see for instance [16, 10]. The special structure of the nonlinear terms in (37) makes it trivial to evolve the flux terms explicitly and the source term implicitly. Denote by vi+ 1 − vi− 1

2 . ∆x With ∆t being the time step and win denotes the approximate of a function w at time t = n∆t and gridpoint xi , the implementation of the third order Runge-Kutta method to solve (36)-(37) can be carried out in the following steps:

D x vi =

2

ρ∗i = ρni , Ji∗ = Jin − 3

(1)

ρi

(1)

Ji

∆t K(ρ∗i ) (v2 − v1 ) (Ji∗ − V (ρ∗i )) − 2ε

∆t K(ρni ) (v2 − v1 ) (Jin − V (ρni )) , 2ε = ρ∗i − ∆tDx Ji∗ , = Ji∗ − ∆t(v1 + v2 )Dx Ji∗ + ∆tv1 v2 Dx ρ∗i , (1)

= ρi , ρ∗∗ i (1)

Ji∗∗ = Ji

+2

∆t ∗∗ ∗∗ K(ρ∗∗ i ) (v2 − v1 ) (Ji − V (ρi )) + 2ε

∆t K(ρ∗i ) (v2 − v1 ) (Ji∗ − V (ρ∗i )) − 2ε ∆t K(ρni ) (v2 − v1 ) (Jin − V (ρni )) , 2ε 3 n 1 ∗∗ ρ + (ρi − ∆tDx Ji∗∗ ) , 4 i 4 3 n 1 ∗∗ J + (Ji ∆t(v1 + v2 )Dx Ji∗∗ + ∆tv1 v2 Dx ρ∗∗ i ), 4 i 4 (2) ρi , ∆t (2) ∗∗∗ − V (ρ∗∗∗ K(ρ∗∗∗ Ji − i )) − i ) (v2 − v1 ) (Ji 2ε 3∆t ∗∗ ∗∗ K(ρ∗∗ i ) (v2 − v1 ) (Ji − V (ρi )) − 8ε 10

(2)

=

(2)

=

ρi Ji

ρ∗∗∗ = i Ji∗∗∗ =

10

∆t 2∆t K(ρ∗i ) (v2 − v1 ) (Ji∗ − V (ρ∗i )) + K(ρni ) (v2 − v1 ) (Jin − V (ρni )) , 2ε 4ε 1 n 2 ∗∗∗ ρ + (ρi − ∆tDx Ji∗∗∗ ) , ρn+1 = i 3 i 3 1 n 2 ∗∗∗ n+1 J + (Ji − ∆t(v1 + v2 )Dx Ji∗∗∗ + ∆tv1 v2 Dx ρ∗∗∗ Ji = i ). 3 i 3 Note that in the above IMEX scheme neither linear algebraic equations no nonlinear source terms can arise. Furthermore, the obtained relaxation scheme is stable independently of ε such that the selection of ∆t is based only on the usual CFL condition ∆t CFL = max {v1 , v2 } ≤ 1. (40) ∆x

3.3

Relaxed schemes

Using a Hilbert expansion for (36)-(37) and neglecting terms of higher order in ε we obtain Ji = V (ρi ), and

  1 ∆x V (ρi+1 ) + V (ρi ) − (V (ρi+1 ) − V (ρi )) . 2 2 ∆t This yields a first order Lax-Friedrichs discretization of the flux for the semidiscrete scheme dρi Ji+ 21 − Ji− 12 + = 0. (41) dt ∆x Similarly, when ε −→ 0, the reconstruction (38) yields to the second order convex ENO scheme for the relaxed equation (41) with   1 ∆x V (ρi+1 ) + V (ρi ) − (V (ρi+1 ) − V (ρi )) + Ji+ 1 = 2 2 ∆t   1 ∆x (V (ρi+1 ) − V (ρi )) φ (θ+ ) − V (ρi+1 ) − V (ρi ) + 4 ∆t   1 ∆x (V (ρi+2 ) − V (ρi+1 )) φ (θ− ) , V (ρi+2 ) − V (ρi+1 ) − 4 ∆t Ji+ 1 =

where θ+ = θ− =

∆x ∆t (V (ρi ) − V (ρi−1 )) , ∆x (ρi+1 ) − V (ρi ) + ∆t (V (ρi+1 ) − V (ρi )) V (ρi+1 ) − V (ρi ) − ∆x ∆t (V (ρi+1 ) − V (ρi )) . ∆x (ρi+2 ) − V (ρi+1 ) − ∆t (V (ρi+2 ) − V (ρi+1 ))

V (ρi ) − V (ρi−1 ) + V V

For equilibrium initial data the leading behavior of relaxation scheme is governed by the associated relaxed scheme. Hence, one can recover all physical properties of the full conservation law (7) by our relaxation-type system with positive velocities (10)-(11). In addition, at the limit when ε → 0, the time integration converges to an explicit TVD Runge-Kutta scheme, see [10] for details. 11

4

Numerical examples

In this section we illustrate the performance of our relaxation method for different test cases on traffic flow. Throughout the numerical tests the relaxation rate ε = 10−1 or ε = 10−10 and the time step ∆t is chosen according to the CFL condition (40). The kernel K(ρ), and the characteristic speeds v1 and v2 are chosen

v1 = 0.1, v2 = 1, (42) K(ρ) = tanh 5 V ′ (ρ) − v1 , for the two speeds model, and


K(ρ) = tanh 5V ′ (ρ) ,

v1 = 0.5,

v2 = 1,

(43)

for the three speeds model. In all the results reported in this section for traffic flow, the spatial domain is the road interval [0, 1] discretized into 200 gridpoints and a CFL = 0.5 is used. We display results for the two speeds model at ε = 10−1 and ε = 10−10 . For the three speeds model, results are shown only for ε = 10−1 , since for small values of ε these results overlap those obtained by the two speeds model. The following examples are selected:

4.1

Free-flow traffic situation

We turn to the traffic flow model (7)-(9) in the simplest case where cars moves freely without jams. Here we assume ρm = um = 1 and initially the cars are normally distributed around the point x = 0.4 (Gaussian-like initial data). The propagation of the density in the space-time domain for ε = 10−1 and ε = 10−10 is given in figure 2 whereas figure 3 shows the contour lines of density ρ. Results obtained by the three speeds model are shown in figure 4 at ε = 10−1 . For comparison reason, the contour lines of the flux J are also included in figure 2. As expected, the relaxation scheme evolves the free traffic in the correct way. Note that because of the different choice of the speeds we made in (42)-(43) cars have the tendency to drive faster in the three speeds model. In figure 5 we plot the density at t = 0.6 for different relaxation limits. The asymptotic convergence of the solution as ε → 0 is clearly preserved by our discretizations. All these results are computed using the second order ENO reconstruction.

4.2

Jam situation

Next we consider a similar situation as before but with an additional traffic jam at x = 0.75. In realistic vehicular traffic flow this is equivalent to relaxed traffic upstream (x < 0.75) and crowded traffic downstream (x ≥ 0.75). In 12

such situation one expect that the Gaussian-like cars distribution start moving into the jam, and once they reach the jam a backwards wave should be formed. By examinating the evolution of the density in figure 6 for ε = 10−10 , we can observe that the mentioned physical features are well captured by our relaxation scheme. The contour plots of density ρ and flux J are displayed in figure 7. It is clear that the Gaussian density distribution is moving into the jam and then propagating backwards as expected. Figure 9 shows the results for the three speeds model at ε = 10−1 . As for the two speed model there is an overshoot of the density at the jam due to the large value of ε. This suggest to take ε small when the value of the density is close to critical. All these results are computed using the second order ENO reconstruction. Figure 8 represents a comparison of density plots at t = 0.6 using the first order Lax-Friedrichs and the second order ENO discretizations. It is clear that excessive diffusion was introduced by the first order Lax-Friedrichs discretization, while the second order ENO reconstruction sharply captures the shock.

4.3

Bottleneck situation

Our final example is the bottleneck traffic taken from [25]. The problem statement is given by equation (7)-(9) where the flux function V (ρ, x) depends also on the space coordinate x subject to discontinuous maximum density and velocity   ρ V (x, ρ) = um (x)ρ 1 − , ρm (x) with

ρm (x) =

 4, if 

2, if

x < 0.3, and x ≥ 0.3,

um (x) =

 1,

if

x < 0.3,

 0.6, if

x ≥ 0.3.

Initially the road has a constant density ρ(x, 0) = 0.2. The evolution of density is given in figure 10 and the contour plots of density ρ and flux J are given in figure 11 at ε = 10−1 and ε = 10−10 . Our relaxation scheme performs well for this discontinuous situation in traffic flow. In figure 12, we display the results using the three speeds model at ε = 10−1 . Again the difference in the results can be explained by the different choice we made for the velocities in (42) and (43).

5

Conclusions

We have presented a relaxation-type formulation for hyperbolic traffic flow problems. The advection part of these relaxation systems has only nonnegative velocities so that they can be interpreted as simple discrete velocity models for 13

traffic flow problems. We showed that high order central discretizations are an essential feature in order to simulate also backward traffic waves. The numerical results show that the performance of our second order numerical schemes is very attractive without solving Riemann problems or requiring special front tracking.

Acknowledgements: The authors acknowledge the financial support by HYKE network of the European Union, contract HPRN-CT-2002-00282. M. Herty and M. Sea¨ıd acknowledge the hospitality of the University of Ferrara in Italy where most of their research was done.

Appendix: Instability of upwind schemes In this short appendix we show that the most natural approach for the discretization of a system like (10)-(11) becomes unstable in the small relaxation limit due to the presence of backward waves. Similar arguments applies also to the three speeds model and can be extended to higher order upwind methods. The most popular way to solve system (10)-(11) is to consider in two separate steps an advection problem ∂ t f + v1 ∂ x f

= 0, (44)

∂t g + v2 ∂x g = 0, and a stiff relaxation problem ∂t f

= −

∂t g =

 K(ρ)  v1 f + v2 g − V (ρ) , 2ε

 K(ρ)  v1 f + v2 g − V (ρ) . 2ε

(45)

Since the advection part contains only nonnegative speeds we can apply a simple first order upwinding scheme. If we write the upwind scheme in terms of the macroscopic variables we get ρn+1 = ρni − i Jin+1

=

Jin


∆t n n Ji − Ji−1 , ∆x


∆t
∆t n − (v1 + v2 ) Jin − Ji−1 v1 v2 ρni − ρni−1 . + ∆x ∆x

(46)

For small values of ε the relaxation step yields the equilibrium projections     V (ρni ) ρni V (ρni ) ρni n n v2 − , gi = − v1 , fi = (47) v2 − v1 ρni v2 − v1 ρni 14

corresponding to Jin = V (ρni ) which substituted into the first equation in (46) gives
∆t ρn+1 = ρni − V (ρni ) − V (ρni−1 ) . (48) i ∆x Clearly (48) is an unstable numerical method for the Lighthill-Whitham-Richards equation when ρ > ρm /2 and thus V ′ (ρ) < 0.

References [1] A. Aw, M. Rascle: Resurrection of “second order” models of traffic flow? SIAM J. Appl. Math. 60 916–938 (2000) [2] G. Q. Chen, C. D. Levermore, T. P. Liu: Hyperbolic conservation laws with stiff relaxation terms and entropy, Comm. Pure Appl. Math. 47 787–830 (1994) [3] C. Daganzo: Requiem for second order fluid approximations of traffic flow, Transportation Res. B. 29 277–286 (1995) [4] R. Illner, A. Klar, M. Materne: Vlasov-Fokker-Planck models for multilane traffic flow, Comm. Math. Sci. 1 1-12 (2003) [5] G. S. Jiang, D. Levy, C. T. Lin, S. Osher, E. Tadmor: High-resolution nonoscillatory central schemes with nonstaggered grids for hyperbolic equations, SIAM J. Numer. Anal. 35 2147–2168 (1998) [6] S. Jin: Runge-Kutta Methods For Hyperbolic Conservation Laws with Stiff Relaxation Terms, J. Comp. Physics, 122 51–67 (1995) [7] S. Jin, Z. Xin: The relaxation schemes for systems of conservation laws in arbitrary space dimensions. Comm. Pure Appl. Math. 48 235–277 (1995) [8] A. Klar: Relaxation Schemes for a Lattice-Boltzmann type discrete velocity model and numerical Navier-Stokes limit, J. Comp. Phys., 148 1–17 (1999) [9] A. Klar, R. Wegener: A Hierarchy of Models for Multilane Vehicular Traffic I: Modeling, SIAM J. Appl. Math. 59 983–1001 (1999) [10] X.-G. Li, X.-J. Yu, G.-N. Chen: The third-order relaxation schemes for hyperbolic conservation laws, J. Comp. Appl. Math. 138 93–108 (2002) [11] X. Liu, S. Osher,: Convex ENO High Order Multi-Dimensonal Schemes without Field by Field Decomposition or Staggered Grids, J. Comp. Physics. 141 1–27 (1998) [12] M. J. Lighthill, J. B. Whitham: On kinematic waves, Proc. Royal Soc. Edinburgh A229 281–345 (1955)

15

[13] A. May: Traffic Flow Fundamentals, Prentice Hall, Englewood Cliffs, NJ (1992) [14] R. Natalini: Convergence to equilibrium for relaxation approximations of conservation laws, Comm. Pure Appl. Math., 49 795–823 (1996) [15] R. Natalini: A discrete kinetic approximation of entropy solutions to multidimensional scalar conservation laws, J. Differential Equations, 148 292– 317 (1998) [16] L. Pareschi, G. Russo: Implicit-Explicit Runge-Kutta schemes for stiff systems of differential equations. J. Scientific Computing (2004) to appear [17] L. Pareschi, M. Sea¨ıd: A New Monte-Carlo Approach for Conservation Laws and Relaxation Systems, Lecture Notes in Computer Science 3037 276–283 (2004) [18] S. L. Paveri-Fontana: On Boltzmann like treatments for traffic flow, Transportation Res. 9 225 (1975) [19] H. Payne: FREFLO: A macroscopic simulation model of freeway traffic, Transportation Res. Record 722 68–75 (1979) [20] I. Prigogine, F. Andrews: A Boltzmann like approach for traffic flow, Operations Res. 8 789 (1960) [21] I. Prigogine, R. Herman: Kinetic Theory of Vehicular Traffic, American Elsevier, NY (1971) [22] M. Sea¨ıd: Non-oscillatory relaxation methods for the shallow water equations in one and two space dimensions, J. Int. Num. Meth. Fluids 46 457–484 (2004) [23] C.W. Shu: Essentially Non-Oscillatory and Weighted Essentially NonOscillatory Schemes for Hyperbolic Conservation Laws, In Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, Editor A. Quarteroni, Lecture Notes in Mathematics, Springer 1697 352–432 (1998) [24] C. W. Shu, S. Osher: Efficient implementation of essential non-oscillatory shock-capturing schemes, J. Comp. Phys. 77 439–471 (1988) [25] P. Zhang, R. Liu: Generalization of Runge-Kutta discontinuous Galerkin method to LWR traffic flow model with inhomogeneous road conditions, Numer. Methods for PDEs 21 80–88 (2004)

16

1 Heaviside(V’(ρ)) tanh(10(V’(ρ)))’ 0.8

0.6

0.4

k(ρ)

0.2

0

−0.2

−0.4

−0.6

−0.8

−1

0

0.1

0.2

0.3

0.4

0.5 ρ/ρm

0.6

0.7

0.8

0.9

1

Figure 1: Examples of kernels for the discrete velocity models (here v1 = 0 for the two speeds model).

17

Density ρ

0.8 0.6 0.4 0.2 0

0.9 0.8

1

0.7 0.8

0.6 0.5

0.6

0.4

Space x0.4

Time t

0.3 0.2

0.2

0.1

Density ρ

0.8 0.6 0.4 0.2 0

0.9 0.8

1

0.7 0.8

0.6 0.5

0.6

0.4

Space x0.4

0.3

Time t

0.2

0.2

0.1

Figure 2: Two speeds model: Density evolution for the free-flow traffic situation at ε = 10−1 (top) and ε = 10−10 (bottom).

18

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.1

0.8

0.7

0.7

0.6

0.6

0.5

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0.5

0.6

0.7

0.8

0.9

0.1

Time t

0.6

0.7

0.8

0.9

0.6

0.7

0.8

0.9

0.5

0.4

0.4

0.5

Flux J

0.8

0.3

0.4

Density ρ

0.9

0.2

0.3

Time t

0.9

0.1

0.2

Time t

Space x

Space x

Flux J

Space x

Space x

Density ρ

0.2

0.3

0.4

0.5

Time t

Figure 3: Two speeds model: Contour plots of density ρ and flux J for the free-flow traffic situation at ε = 10−1 (top) and ε = 10−10 (bottom).

19

Density ρ

0.8 0.6 0.4 0.2 0

0.9 0.8

1

0.7 0.8

0.6 0.5

0.6

0.4

Space x0.4 0.2

0.1 Flux J

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

Space x

Space x

Density ρ

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0.1

0.2

0.3

0.4

0.5

Time t

0.3 0.2

0.6

0.7

0.8

0.9

0.1

Time t

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Time t

Figure 4: Three speeds model at ε = 10−1 : Density evolution for the free-flow traffic situation (top) and contour plots of density ρ and flux J (bottom).

20

0.5 −1

ε = 10 −2 ε = 10 −10 ε = 10

0.45

0.4

0.35

Density ρ

0.3

0.25

0.2

0.15

0.1

0.05

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Space x

Figure 5: Two speeds model: Plots of density for the free-flow traffic situation at t = 0.6 using different values of ε.

21

Density ρ

0.8 0.6 0.4 0.2 0

0.9 0.8

1

0.7 0.8

0.6 0.5

0.6

0.4

Space x0.4

Time t

0.3 0.2

0.2

0.1

Density ρ

0.8 0.6 0.4 0.2 0

0.9 0.8

1

0.7 0.8

0.6 0.5

0.6

0.4

Space x0.4

0.3

Time t

0.2

0.2

0.1

Figure 6: Two speeds model: Density evolution for the jam traffic situation at ε = 10−1 (top) and ε = 10−10 (bottom).

22

0.9

0.9

0.8

0.8

0.7

0.7

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0.1

Time t

0.6

0.7

0.8

0.9

0.6

0.7

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0.9

0.5

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0.5

Flux J

0.8

0.3

0.4

Density ρ

0.9

0.2

0.3

Time t

0.9

0.1

0.2

Time t

Space x

Space x

Flux J

Space x

Space x

Density ρ

0.2

0.3

0.4

0.5

Time t

Figure 7: Two speeds model: Contour plots of density ρ and flux J for the jam traffic situation at ε = 10−1 (top) and ε = 10−10 (bottom).

23

Initial First order LxF Second order ENO

0.9

0.8

0.7

Density ρ

0.6

0.5

0.4

0.3

0.2

0.1

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Space x

Figure 8: Two speeds model: Plots of density for the jam traffic situation at t = 0.6 and ε = 10−10 using the first order Lax-Friedrichs and the second order ENO discretizations.

24

Density ρ

0.8 0.6 0.4 0.2 0

0.9 0.8

1

0.7 0.8

0.6 0.5

0.6

0.4

Space x0.4 0.2

0.1 Flux J 1

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

Space x

Space x

Density ρ

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0.1

0.2

0.3

0.4

0.5

Time t

0.3 0.2

0.6

0.7

0.8

0

0.9

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Time t

Time t

Figure 9: Three speeds model at ε = 10−1 : Density evolution for the jam traffic situation (top) and contour plots of density ρ and flux J (bottom).

25

0.55 0.5

Density ρ

0.45 1

0.4 0.35

0.8

0.3 0.6

0.25

Time t

0.2

0.4

1 0.8 0.2

0.6

Space x

0.4 0.2 0

0.55 0.5

Density ρ

0.45 1

0.4 0.35

0.8

0.3 0.6

0.25

Time t

0.2

0.4

1 0.8 0.2

0.6

Space x

0.4 0.2 0

Figure 10: Two speeds model: Density evolution for the bottleneck traffic situation at ε = 10−1 (top) and ε = 10−10 (bottom).

26

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.1

0.8

0.7

0.7

0.6

0.6

0.5

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0.5

0.6

0.7

0.8

0.9

0.1

Time t

0.6

0.7

0.8

0.9

0.6

0.7

0.8

0.9

0.5

0.4

0.4

0.5

Flux J

0.8

0.3

0.4

Density ρ

0.9

0.2

0.3

Time t

0.9

0.1

0.2

Time t

Space x

Space x

Flux J

Space x

Space x

Density ρ

0.2

0.3

0.4

0.5

Time t

Figure 11: Two speeds model: Contour plots of density ρ and flux J for the bottleneck traffic situation at ε = 10−1 (top) and ε = 10−10 (bottom).

27

0.55 0.5

Density ρ

0.45 1

0.4 0.35

0.8

0.3 0.6

0.25

Time t

0.2

0.4

1 0.8 0.2

0.6 0.4

Space x

0.2 0 Flux J

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

Space x

Space x

Density ρ

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.1

Time t

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Time t

Figure 12: Three speeds model at ε = 10−1 : Density evolution for the bottleneck traffic situation (top) and contour plots of density ρ and flux J (bottom).

28