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NETWORKS AND HETEROGENEOUS MEDIA doi:10.3934/nhm.2020032

American Institute of Mathematical Sciences

PROPERTIES OF THE LWR MODEL WITH TIME DELAY

Simone G¨

ottlich

University of Mannheim, Department of Mathematics

68131 Mannheim, Germany

Elisa Iacomini∗

University of Mannheim, Department of Mathematics

68131 Mannheim, Germany

Thomas Jung

Fraunhofer Institute ITWM

67663 Kaiserslautern, Germany

(Communicated by Raimund B¨urger)

Abstract. In this article, we investigate theoretical and numerical properties

of the ﬁrst-order Lighthill-Whitham-Richards (LWR) traﬃc ﬂow model with

time delay. Since standard results from the literature are not directly applicable

to the delayed model, we mainly focus on the numerical analysis of the proposed

ﬁnite diﬀerence discretization. The simulation results also show that the delay

model is able to capture Stop & Go waves.

1. Introduction. Nowadays traﬃc models have become an indispensable tool in

the urban and extraurban management of vehicular traﬃc. Understanding and

developing an optimal transport network, with eﬃcient movement of traﬃc and

minimal traﬃc congestions, will have a great socio- economical impact on the so-

ciety. This is why in the last decades an intensive research activity in the ﬁeld of

traﬃc ﬂow modelling ﬂourished.

Literature about traﬃc ﬂow is quite large and many methods have been developed

resorting to diﬀerent approaches. Starting from the natural idea of tracking every

single vehicle, several microscopic models, based on the idea of Follow-the-Leader,

grew-up for computing positions, velocities and accelerations of each car by means

of systems of ordinary diﬀerential equations (ODEs) [1,8,19,21,44]. Other ways

go from kinetic [28,34,45] to macroscopic ﬂuid-dynamic and measures approaches

[2,11,12,24,33,38], focusing on averaged quantities, such as the traﬃc density and

the speed of the traﬃc ﬂow, by means of systems of hyperbolic partial diﬀerential

equations (PDEs), in particular conservation laws. In this way we loose the detailed

level of vehicles’ description, indeed they become indistinguishable from each other.

The choice of the scale of observation mainly depends on the number of the involved

vehicles, the size of the network and so on.

2020 Mathematics Subject Classiﬁcation. 35L65, 90B20, 65M06.

Key words and phrases. Macroscopic traﬃc ﬂow models, hyperbolic delay partial diﬀerential

equation, Lax-Friedrichs discretization, numerical simulations, stop & go waves.

∗Corresponding author: Elisa Iacomini.

1

2 SIMONE G ¨

OTTLICH, ELISA IACOMINI AND THOMAS JUNG

In this paper we deal with the macroscopic scale, in particular we will focus on

ﬁrst order macroscopic models. The most relevant model in this framework is the

LWR model, introduced by Lighthill, Whitham [31] and Richards [36] in the ’50.

The main idea underlying this approach is that the total mass has to be preserved,

since cars can not disappear. Moreover, in this model the mean velocity is supposed

to be dependent on the density, thus is closing the equation. On the other hand the

lacks of the LWR model are well-known. For example, it fails to generate capacity

drop, hysteresis, relaxation, platoon diﬀusion, or spontaneous congestions like Stop

& Go waves, that are typical features of traﬃc dynamics. These drawbacks are due

to the fact that the LWR model represents a simpliﬁcation of the reality, assuming

that accelerations are instantaneous and traﬃc is described only at the equilibrium.

In order to overcome these issues, second order models have been proposed,

see [1,2,46]. They take into account the non-equilibria states, assuming that

accelerations are not instantaneous. To do this, the equation that describes the

variation of the velocity in time has to be added to the system, replacing the typical

given law of the ﬁst order models. Other ways are also possible to improve ﬁrst order

models, just considering phase transition models [6,14], non-local traﬃc models

[5,15,23,25,29,43] or multi-scale approaches [16,18]. Instead of switching to

second order models, we propose a ﬁrst order macroscopic model with a time delay

term in the ﬂux function, for taking into account that the velocity can not change

instantaneously. In this framework the delay represents the reaction time of both

drivers and vehicles.

At a microscopic level, a model with time delay appears for the ﬁrst time in

the work done by Newell [32], then similar models are presented in [3,13]. The

mathematical tools needed in this framework are not systems of ODEs anymore,

but systems of delay diﬀerential equations (DDEs), particular diﬀerential equations

in which the derivative of the unknown function at a certain time is given in terms

of the values of the function at previous times. Macroscopic models can be derived

from microscopic description following a well-known procedure described in [1,17,

21]. Depending on how to treat the delay term, one can recover diﬀerent macroscopic

models, as in [10] or [42], in which a Taylor’s approximation is applied to the delay

term and the obtained model is a diﬀusive LWR type model. On the other hand,

we want to keep the delay in the explicit form, and therefore avoid the diﬀusion

approximation. The model derived in [9] will be studied in details in the following,

investigating carefully its theoretical and numerical features.

Several delayed-systems are presented in literature, since many phenomena need

some transient to become visible or eﬀective: the study of the evolution of the HIV in

medicine [20,39], cell population dynamics in biology [26,35], the feedback control

loops in control engineering [30], and many applications in mechanics and economics

[4], but to the authors best knowledge, they are closer to delayed parabolic partial

diﬀerential equations or to delayed ordinary diﬀerential equations, i.e. they are

studied only at a microscopic level.

In this work instead we deal with a delayed hyperbolic partial diﬀerential equa-

tion. We will point out similarities and diﬀerences with the undelayed model in

order to catch the eﬀect of the delay on traﬃc dynamics, both from theoretical

and numerical points of view. Moreover, since we are interested in reproducing real

traﬃc phenomena, the numerical tests are mainly focused on traﬃc instabilities.

In particular we investigate the phenomenon of Stop & Go (S&G) waves, which

are a typical feature of congested traﬃc and represent a real danger for drivers.

PROPERTIES OF THE LWR MODEL WITH TIME DELAY 3

They lead not only to safety hazard, but they also have a negative impact on fuel

consumption and pollution. Indeed a S&G wave is detected when vehicles stop

and restart without any apparent reason, generating a wave that travels backward

with respect to the cars’ trajectories. Since modeling properly this phenomenon is

crucial for developing techniques aimed at reducing it, a considerable literature is

growing up on this topic. This means that a lot of models have been developed in

the last years, i.e. [7,22,28,37,42], and also several real experiments took place,

just see [41,47].

In this framework, our aim is to investigate if our delayed model is able to

capture the S&G phenomena and, therefore, to present an easy to use algorithm

able to reproduce S&G waves at a macroscopic level. Indeed from the numerical

point of view, just an altered Lax Friedrichs numerical scheme will be employed

to compute the evolution of the density. In order to validate our model, several

numerical tests will be provided for comparing our delayed model with the existing

ones.

Paper organization. In Section 2, we introduce the delayed model and investigate its

theoretical properties, as the conservation of mass, the positivity and the bound-

edness of the solution. After that, we focus on the numerical aspects, presented

in Section 3, proposing a suitable numerical scheme and checking the theoretical

features still hold. Section 4 is completely devoted to the numerical tests.

2. The delayed traﬃc ﬂow model. In macroscopic models [24], traﬃc is de-

scribed in terms of macroscopic variables such as density ρ=ρ(x, t), that is the

number of vehicles per unit length, and the mean velocity V=V(x, t) at the point

x∈Rat time t > 0.

The LWR model, introduced by Lighthill, Whitham [31] and Richards [36], is

one of the oldest and still most relevant ﬁrst order macroscopic models for traﬃc

ﬂow. The natural assumption that the total mass is conserved along the road is

closed by the assumption that the velocity V=V(ρ) is given as function of the

density ρ:

(∂tρ(x, t) + ∂x(ρ(x, t)V(ρ(x, t))) = 0

ρ(x, 0) = ρ0(x).(1)

A lot of possible choices for the function V(ρ) are available in the literature, i.e. the

Greenshields function [27] which proposes a linear velocity function:

V(ρ) = Vmax1−ρ

ρmax .(2)

In order to simplify the notation, we will consider the normalized quantities ρmax =

Vmax = 1. Aiming to overcome the drawbacks of LWR model 1presented in the

introduction, in 3we propose a ﬁrst order macroscopic model with a time delay

term in the ﬂux function, for taking into account that the velocity can not change

instantaneously.

In this framework the delay represents the reaction time of both drivers and

vehicles. Such a model has been recovered from a delayed microscopic model, as

shown in [9] keeping the delay in the explicit form. Assuming T≥0 as the time

delay, we consider:

∂tρ(x, t) + ∂x(ρ(x, t)V(ρ(x, t −T))) = 0.(3)

4 SIMONE G ¨

OTTLICH, ELISA IACOMINI AND THOMAS JUNG

We will call this model delayed LWR model. Note that in the limit case of T= 0

the classical LWR model is recovered and therefore, it can be interpreted as a

generalization of the LWR model. On the other hand, if the delay is too large

and there are suitable initial conditions, cars can overtake or crash each other, for

example when a vehicle suddenly brakes and the following car is not reacting in

time to slow down.

Note that in order to guarantee the well-posedness of the problem, we have to

provide an initial history function as initial data deﬁned on [−T, 0], thus we need

ρ0(x, t) deﬁned on t∈[−T, 0] when starting at t= 0.

As far as it concerns the existence and the uniqueness of the solution for time

delayed model, it might be possible to apply the results presented in [29] for non-

local conservation laws with time delay. Indeed, as the authors in [29] said, the

existence of solutions as well as uniqueness can only be obtained for smooth initial

datum and only on a signiﬁcantly small time horizon. However, the main diﬀerence

with the model presented here is that we consider a delayed model of local type and

convergence results for non-local to local traﬃc ﬂow models are still missing.

2.1. Properties of the model. After introducing the delayed model, we want to

investigate its properties. Since this model can be seen as a generalization of the

classical LWR model, i.e. when T= 0, it is natural to investigate how its properties

diﬀer from the undelayed model.

2.1.1. Conservation of mass. In the framework of conservation laws and traﬃc ﬂow

models the conservation of the total mass is a crucial property which has to be guar-

anteed. For the LWR model 1, we have one equation and one conserved quantity,

i.e. ρ. Indeed cars do not appear or disappear, they can only enter and leave at the

boundaries. We note that the structure of the equation in the delayed and unde-

layed model stays the same and that we have a ﬂux function that is now dependent

on two variables.

Lemma 2.1. The delayed LWR model 3conserves the quantity ρ(x, t).

Proof. We integrate the equation 3over an arbitrary space interval [a, b] and get

d

dt Zb

a

ρ(x, t)dx =−Zb

a

∂x(ρ(x, t)V(ρ(x, t −T)))dx

=ρ(a, t)V(ρ(a, t −T)) −ρ(b, t)V(ρ(b, t −T)).

Since Rb

aρ(x, t)dx is the amount of density in the interval [a, b], d

dt Rb

aρ(x, t)dx de-

notes the change over time for the density. Therefore, the density only changes due

to the ﬂux at the boundaries aand bfor every space interval.

We see that the density is still conserved in the delayed model, which is very

important for its reliability. The introduction of an explicit time delay therefore

does not destroy this property.

2.1.2. Positivity. Another property one would ensure is the positivity of the solu-

tion. Indeed we want the density to stay positive, as negative densities have no

physical meaning.

Lemma 2.2. Assume we have initial data with non-negative density ρ. For the

delayed LWR model, then the density stays non-negative.

PROPERTIES OF THE LWR MODEL WITH TIME DELAY 5

Proof. We rewrite 3as

∂tρ(x, t) = −(ρ(x, t)∂xV(ρ(x, t −T)) + V(ρ(x, t −T))∂xρ(x, t)) .(4)

For the density to become negative, we need to have ρ= 0 and ∂tρ < 0.

We show now that this is not possible. Let us ﬁx a time t∗such that the density

is positive for t<t∗. We can assume that due to the hypothesis the initial density

is positive. Then, at time t∗, there exists a point x∗in which ρ(x∗, t∗) = 0. Since

ρ(x, t)>0 for t<t∗, the density has a minimum in (x∗, t∗), therefore ∂xρ(x∗, t∗) =

0.

Plugging ρ(x∗, t∗) = 0 and ∂xρ(x∗, t∗) = 0 into 4, we obtain:

∂tρ(x∗, t∗) = −0∂xV(ρ(x∗, t∗−T)) + V(ρ(x∗, t∗−T)) 0= 0.

We have therefore shown that ρcan not become negative.

Remark 1. The velocity Vin this model is a function of ρand can be chosen

and altered depending on the needs. The properties regarding the velocity in the

ﬁrst order model can therefore be acquired by choosing a suitable function V. For

example, we can have lower and upper bounds for the velocity by deﬁning Vto be

cut at the bounds. This means that inﬁnite (as in the Greenberg model) or negative

velocities should be truncated.

2.1.3. Upper bound. The last property we want to investigate is the boundedness

of the solution. In particular, we want to know if there is a maximal density. For

the undelayed model, this is guaranteed. For the delayed model, we need to check

if this still true.

Lemma 2.3. Assume Vis monotone decreasing and V(ρmax )=0for ρmax ,the

maximal density in the classical LWR model. The delayed ﬁrst order model 3has

no maximal density ρmax.

Proof. Assume we have a maximal density ρmax . The velocity function Vis chosen

in such a way that V(ρmax ) = 0 and monotone decreasing. Then, for an arbitrary

point (x∗, t∗) where ρ(x∗, t∗) = ρmax we have

∂tρ(x∗, t∗) = −ρ(x∗, t∗)∂xV(ρ(x∗, t∗−T)) −V(ρ(x∗, t∗−T))∂xρ(x∗, t∗).

Since ρ(x∗, t∗) is the maximal density, ∂xρ(x∗, t∗) = 0 if the derivative exists and

we have

∂tρ(x∗, t∗) = −ρ(x∗, t∗)∂xV(ρ(x∗, t∗−T))

left. We know ρ(x∗, t∗)≥0 and this means that the sign of ∂tρ(x∗, t∗) is only

dependent on ∂xV(ρ(x∗, t∗−T)).

In the undelayed case, we know that ∂xV(ρ(x∗, t∗)) >0, since Vis monotone

decreasing and ρ(x∗, t∗) is the maximal ρ.

In the delayed case, we do not have knowledge if ρ(x∗, t∗−T) is maximal, so

we can in general say nothing about ∂xV(ρ(x∗, t∗−T)). This means, in general,

ρ > ρmax is possible.

Remark 2. Regarding the positivity, we claimed that the choice of Vin the ﬁrst

order model is a key to guarantee a positive velocity. We here see, due to the fact

that ρovershoots any ρmax, that the classical choices for Vmust be altered to

avoid negative velocities, i.e. we need to cut the function, for example assuming

that V(ρ) = 0 when ρ>ρmax.

6 SIMONE G ¨

OTTLICH, ELISA IACOMINI AND THOMAS JUNG

Remark 3. If the density ρ>ρmax, the model is not reliable any more. On the

other hand this situation could not be avoided since rear-end collisions are actually

possible in real situations.

3. Numerical discretization. After the investigations on the analytical proper-

ties of the delayed LWR model, let us focus on its numerical counterpart.

Since 3is a hyperbolic partial diﬀerential equation, we can employ the Lax-

Friedrichs method for the numerical approximation. To do that, we ﬁrst introduce

space and time steps ∆x, ∆t > 0 and a grid in space {xj=j∆x, j ∈Z}and time

{tn=n∆t, n ∈N}. Since we are in the framework of ﬁnite diﬀerence approximation,

discretized variables are expressed by ρn

j=ρ(xj, tn), where iis the space and nthe

time index. Also, we have ∆t≤Tto be able to treat the delay.

The Lax-Friedrichs numerical scheme for 1is stated by:

ρn+1

j=1

2(ρn

j+1 +ρn

j−1)−∆t

2∆x(f(ρn

j+1)−f(ρn

j−1)).

Now using the structure of 3, we can identify a ﬂux function f(ρ(x, t−T), ρ(x, t)) =

V(ρ(x, t −T))ρ(x, t). Plugging this into the Lax-Friedrichs numerical scheme, we

end up with an altered Lax-Friedrichs numerical scheme

ρn+1

j=1

2(ρn

j+1 +ρn

j−1)−∆t

2∆x(f(ρn−T∆

j+1 , ρn

j+1)−f(ρn−T∆

j−1, ρn

j−1)),(5)

where j∈Z, n ∈Nand T∆is the number of steps that make up the time delay T.

In order to guarantee the well-posedness of the discrete problem, we have to provide

an initial history function as initial data deﬁned on [−T, 0], as we said above for

the continuous problem. The simplest choice one can do is to consider ρ0(x, t) as

a constant function on t∈[−T, 0] when starting at t= 0. In the following we will

assume that ρ0(x, 0) is constant in tfor t∈[−T, 0]. The Lax-Friedrichs numerical

scheme has a CFL condition in the classical case, which is given as ∆t≤∆x

maxk(λk),

where λkare the eigenvalues of the jacobian matrix of f. We also expect to ﬁnd a

CFL condition in the delayed case, but a priori it is not clear how this condition may

look like. In the following, we want to investigate some properties of this method,

and in this process we will ﬁnd an appropriate CFL condition. For the sake of the

calculations, we assume the velocity function Vto be the Greenshields function, or

a cut variation of it, where we have |V(ρ)| ≤ ρmax with Vmax = 1.

3.1. Properties of the discretization.

3.1.1. Conservation of mass. First, we check if the conservation property is pre-

served from the numerical scheme. Here, we assume the density to be on a compact

support, so we do not have inﬁnite density initially. We get

∆xX

j

ρn+1

j= ∆xX

j

1

2(ρn

j+1 +ρn

j−1)−∆t

2∆x(V(ρn−T∆

j+1 )ρn

j+1 −V(ρn−T∆

j−1)ρn

j−1),(6)

where the part

X

j

1

2(ρn

j+1 +ρn

j−1) = X

j

ρn

j

and the part

X

j

−∆t

2∆x(V(ρn−T∆

j+1 )ρn

j+1 −V(ρn−T∆

j−1)ρn

j−1)

PROPERTIES OF THE LWR MODEL WITH TIME DELAY 7

is a telescope sum and equals zero due to the compact support. This gives us

∆xX

j

ρn+1

j= ∆xX

j

ρn

j

and therefore conservation.

3.1.2. Positivity. We show that Lemma 2.2 holds also at the discrete level under

a certain CFL condition. Starting with 5, we see that 1

2(ρn

j+1 +ρn

j−1) is always

positive, since we assume ρnto be positive. Indeed, if ρn= 0, ρn+1

jwill also be

zero. So to guarantee positivity, we need to guarantee

∆t

2∆x(V(ρn−T∆

j+1 )ρn

j+1 −V(ρn−T∆

j−1)ρn

j−1)≤1

2(ρn

j+1 +ρn

j−1).(7)

We compute a CFL-condition, namely ∆t≤∆x

max{|ρn|,|ρn−T∆|} . Therefore, we get

for the left-hand-side of 7

1

2 max{|ρn|,|ρn−T∆|}(V(ρn−T∆

j+1 )ρn

j+1 −V(ρn−T∆

j−1)ρn

j−1)

=V(ρn−T∆

j+1 )

2 max{|ρn|,|ρn−T∆|}ρn

j+1 −V(ρn−T∆

j−1)

2 max{|ρn|,|ρn−T∆|}ρn

j−1

≤max{|ρn|,|ρn−T∆|}

2 max{|ρn|,|ρn−T∆|}ρn

j+1 +max{|ρn|,|ρn−T∆|}

2 max{|ρn|,|ρn−T∆|}ρn

j−1

=1

2(ρn

j+1 +ρn

j−1),(8)

which shows the positivity for this CFL-condition. Here we use that |V(ρ)| ≤

max{|ρn|,|ρn−T∆|} if ρis positive.

The crucial role played by the new CFL condition is explained at the beginning

of the Section 4.

3.1.3. L∞-Bound. Focusing on the boundedness of the discrete solution, we look

for an estimate in the norm |·|L∞. Since we have positivity, only an upper bound

for ρin 5is required. We assume that the data at time tnhas an upper bound

which we denote with ρn

max. We further denote V(ρn−T∆

j)ρn

j=f(ρn−T∆

j, ρn

j) and

get for a special ξ= (ξ1, ξ2) using the mean value theorem:

|ρn+1

j|=

1

2(ρn

j+1 +ρn

j−1)−∆t

2∆x(f(ρn−T∆

j+1 , ρn

j+1)−f(ρn−T∆

j−1, ρn

j−1))

=

1

2(ρn

j+1 +ρn

j−1) + ∆t

2∆x∇f(ξ1, ξ2)ρn

j−1−ρn

j+1

ρn−T∆

j−1−ρn−T∆

j+1

≤

1

2(ρn

j+1 +ρn

j−1)+∆t

2∆x(V(ξ2)(ρn

j−1−ρn

j+1) + ξ1(ρn−T∆

j+1 −ρn−T∆

j−1))

CF L

≤1

2ρn

j+1 +ρn

j−1+1

2ρn

j−1−ρn−T∆

j−1+ρn−T∆

j+1 −ρn

j+1

≤1

22ρn

j−1−ρn−T∆

j−1+ρn−T∆

j+1

≤ρn

max +1

2ρn−T∆

j+1 −ρn−T∆

j−1

≤2 max{|ρn|,|ρn−T∆|}.(9)

8 SIMONE G ¨

OTTLICH, ELISA IACOMINI AND THOMAS JUNG

We use again the positivity and the CFL-condition, estimating the velocity by

the maximal density value and V0(ξ1)≤1.

Remark 4. We can ﬁnd a diﬀerent estimate for ρn+1

j, depending on T. Note that

the estimate in 9is more accurate. We start in the same way as above, and have

|ρn+1

j| ≤ 1

2|2ρn

j−1−ρn−T∆

j−1+ρn−T∆

j+1 |.

Rearranging gives us

1

2(ρn

j−1+ρn

j−1−ρn−T∆

j−1+ρn−T∆

j+1 ) = 1

2(ρn

j−1−ρn−T∆

j−1+ρn−T∆

j+1 +ρn

j−1),

and by again using the mean value theorem for ρj−1(t) we get

ρn+1

j≤1

2(T ∂tρj−1+ρn

j−1+ρn−T∆

j+1 )≤max{|ρn|,|ρn−T∆|} +1

2T||∂tρ||L∞.(10)

3.1.4. TV Bound. Let us look for an estimate on the Total Variation for the method

5. The ﬁrst thing we check is the diﬀerence between the velocity function in two

cells.

V(ρn−T∆

j+1 )−V(ρn−T∆

j)=(1 −ρn−T∆

j+1 )−(1 −ρn−T∆

j) = ρn−T∆

j−ρn−T∆

j+1 =−∆n−T∆

j+1

2

.

(11)

Then, we look at the diﬀerence between two neighboring cells, where we use 5,11

and the notation ∆n

j+1

2

=ρn

j+1 −ρn

j:

∆n+1

j+1

2

=ρn+1

j+1 −ρn+1

j

=1

2(∆n

j+3

2+ ∆n

j−1

2)

−∆t

2∆x(V(ρn−T∆

j+2 )∆n

j+3

2−V(ρn−T∆

j)∆n

j−1

2−ρn

j+1∆n−T∆

j+3

2

+ρn

j−1∆n−T∆

j−1

2

).

(12)

The Total Variation at time tn+1 (denoted by T V (ρn+1

∆)) is given as Pj|∆n+1

j+1

2

|.

For the next step and with 12 as well as shifting the indices, we get

X

j

|∆n+1

j+1

2

|=X

j

|ρn+1

j+1 −ρn+1

j|

=X

j

(1 + 2|∆t

2∆xρn−T∆

j+1 |+ 2|∆t

2∆xρn

j+1|) max{|∆n

j+1

2|,|∆n−T∆

j+1

2

|}.(13)

If we introduce a CFL-condition of the type ∆t≤2∆x

max{|ρn−T∆

j|,|ρn

j|} (i.e. not as

strong as above), the dependency on jdisappears and we get

(5 + 1

max{|ρn−T∆

j|,|ρn

j|})X

j

max{|∆n

j+1

2|,|∆n−T∆

j+1

2

|}.

By assumption, T V (ρn

∆) and T V (ρn−T∆

∆) are ﬁnite, so we can estimate further

T V (ρn+1

∆)≤2(5 + 1

max{|ρn−T∆

j|,|ρn

j|}) max{T V (ρn

∆), T V (ρn−T∆

∆)}.(14)

PROPERTIES OF THE LWR MODEL WITH TIME DELAY 9

For the estimate in time, we look at

X

j

|ρn+1

j−ρn

j|.

By simply plugging in 5and using the same CFL-condition, we can write

X

j

|ρn+1

j−ρn

j| ≤ X

j

1

2(|ρn

j+1|+|ρn

j−1|) + ∆t

2∆x(|V(ρn−T∆

j+1 )||ρn

j+1|+|V(ρn−T∆

j−1)||ρn

j−1|) + |ρn

j|

≤X

j

2 max{|ρn−T∆

j|,|ρn

j|} + 2 + 2 max{|ρn−T∆

j|,|ρn

j|}

=X

j

4 max{|ρn−T∆

j|,|ρn

j|} + 2.

We now have a BV-Bound in space as well as in time, which gives us all the desired

BV estimates.

3.1.5. Time span. With the CFL-condition we introduced, the time step can be-

come smaller every step, since we have no maximum principle. Here, we want to

see if we can actually reach every time horizon. Therefore, we plug our L∞-bound

into the CFL and get

∆t≤∆x

max{|ρn|,|ρn−T∆|} ≤∆x

(3

2)n||ρ0||L∞

.

Now, the time horizon we reach with ntime steps is given by

tn=

n

X

i=1

∆t=∆x

||ρ0||L∞

n

X

i=1 2

3i

.

So for inﬁnite time steps n→ ∞, we end up with a geometric series which converges

to 3 ∆x

||ρ0||L∞. That is the time horizon we can guarantee with this estimate.

With the alternative estimate that depends on T, the time horizon we can reach is

dependent also on T. With ntime steps, we can now reach the time horizon

tn= ∆x

n

X

i=1

1

ρ0

max +i1

2||∂tρ||T.

This is basically a harmonic series shifted and with a factor, but it is divergent to

∞if the factor is not zero. This also means, that for small T, the steps can be

larger. Furthermore, with this second estimate, we can guarantee to reach every

time horizon.

4. Numerical results. This section is devoted to the numerical simulation results

for the model presented above, focusing in particular on the S&G waves phenome-

non, a typical feature of congested traﬃc, detected when vehicles stop and restart

without any apparent reason, generating a wave that travels backward with respect

to the cars’ trajectories.

Starting from empirical observations and the work done in [18,47], let us assume

the velocity function as follow:

V(ρ) =

Vmax ρ≤ρf

α(1

ρ−1

ρc)ρf< ρ < ρc

0ρ≥ρc

(15)

where α > 0 is a parameter, ρc∈(0, ρmax] and ρf∈[0, ρmax) are two density

thresholds. In particular ρcrepresents the so-called safe distance at the macroscopic

10 SIMONE G ¨

OTTLICH, ELISA IACOMINI AND THOMAS JUNG

level: if the density is higher than ρc, vehicles do not respect the safe distance so

the desired velocity has to be 0, indeed they should stop.

On the other hand, if the density is very low, which means that vehicles are far

enough from each others, the desired velocity is the maximum one.

Note that 15 respects the hypothesis |V(ρ)| ≤ ρmax choosing Vmax = 1. More-

over, depending on the choice made for α, the velocity function can be discontinuous.

For the discretization, let us assume the space interval [a, b] = [0,1], ∆x= 0.02

and periodic boundary conditions. Moreover, the time step ∆tis chosen in such

a way the CFL condition is satisﬁed. We denote with Tfthe ﬁnal time of the

simulation. The density thresholds are ρc= 0.75 and ρf= 0.2 as real data suggests.

The delay term depends on the CFL condition and the initial data, for this reason

each numerical test has its time delay interval which ensures the reliability of the

model, i.e. ρ≤ρmax . However, in general, one has to assume T∆one order of

magnitude greater than ∆tto see the eﬀect, i.e. T∆≈(10∆t, 20∆t).

Remark 5. Let us point out the relevance of the new CFL condition. If we compute

the solution of 5with the classical CFL condition, we will end up with spurious

oscillations as in Fig.1(left)-2(left). Applying the computed CFL condition, these

oscillations disappear, Fig.1(right)-2(right). The parameters are ∆x= 0.1, Tf= 10

T∆= 1 and the time step is chosen ∆t= 0.5∆x

2 max{|ρn|,|ρn−T∆|} in the latter and

∆t= 1.5∆x

2ρmax for the former case. In this way we can focus only on the CFL

condition, without any inﬂuence from the delay term, since the value of the delay

is ﬁxed and not depending on the time step.

0 10 20 30 40 50 60 70 80 90 100

Space

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Density

Comparison with and without CFL Initial Data

Delayed LWR without CFL

0 10 20 30 40 50 60 70 80 90 100

Space

0.4

0.45

0.5

0.55

0.6

0.65

0.7

Density

Comparison with and without CFL Initial Data

Delayed LWR without CFL

Figure 1. Comparison between density proﬁles computed with

(right) and without(left) CFL condition in case of a rarefaction

wave.

4.1. Backward propagation of stop & go waves. In order to be more compre-

hensive as possible in reproducing S&G waves, let us describe ﬁrst the backward

propagation of the perturbation. After that we will focus also on the triggering of

this phenomenon.

Test 0. In order to point out the crucial role played by the delay in this framework,

let us compare the evolution of the density obtained with the delayed model and

the classical LWR model, or, in other words, when T∆= 0.

Assume as initial data ρ0(x) = 5

8+1

8sin (2πx). Moreover the time step is ∆t=

0.01, the delay T∆= 15∆tin the delayed case and the ﬁnal time of the simulation

is Tf= 10.

PROPERTIES OF THE LWR MODEL WITH TIME DELAY 11

0 10 20 30 40 50 60 70 80 90 100

Space

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Density

Comparison with and without CFL Initial Data

Delayed LWR without CFL

0 10 20 30 40 50 60 70 80 90 100

Space

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

Density

Comparison with and without CFL Initial Data

Delayed LWR with CFL

Figure 2. Comparison between density proﬁles computed with

(right) and without (left) CFL condition in a shock framework.

Figure 3. Test 0: Comparing the density evolution computed by

the delayed model (left) and the LWR model (right).

It is evident how the LWR model smears out the perturbations in the initial data

and after a certain time the density becomes constant on the whole road, see Fig.

3(right). On the other hand, the delayed model preserves the perturbations and

also makes them increase as usually happens in traﬃc evolution, Fig. 3(left).

Remark 6. Unfortunately, we have no deﬁnition of weak solutions for the continu-

ous delayed problem. Therefore, we can not show the convergence of the numerical

scheme 5to any weak solution of 3. On the other hand, we provide a comparison

with the reference solution to show the numerical convergence. For reference solu-

tion we mean the solution computed with the same numerical scheme but with a

high resolution of the mesh, i.e. with ∆xand ∆tvery small. In Fig. 4, we show

the comparison between three density proﬁles corresponding to diﬀerent grid steps

size, i.e. ∆x= 10−2,10−3,10−4and ∆t= 0.5·10−2,10−3,10−4respectively, at the

same time, with T∆= 5∆t. We can note that the diﬀerence between the density

proﬁles is decreasing as the grid is more and more reﬁned.

Remark 7. We have to be very careful in choosing the delay term. Indeed if the

delay is too small, we recover a situation very similar to the LWR model, but, on

the other hand, if the delay is too high, i.e. T∆= 18∆t, the hypothesis on the

model are no longer satisﬁed and the density grows more than 1, so the model has

no sense anymore, see Fig. 5.

12 SIMONE G ¨

OTTLICH, ELISA IACOMINI AND THOMAS JUNG

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Space

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Density

10-2" t

10-3" t

10-4" t

Figure 4. Test 0: Comparison between density proﬁles corre-

sponding to diﬀerent grid steps size.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Figure 5. Test 0: Density evolution and proﬁle, at time T=1

3Tf,

in case of a too high delay.

In the following, we will consider the delay as the maximum allowed by the model

feasibility.

Test 1. In this numerical simulation we want to reproduce with our model the tests

presented in [7]. Starting from the same initial data our aim is to recover a similar

behaviour for the density. In [7], a nonlinear 2-equations discrete velocity model is

implemented to compute the density evolution:

∂tρ+∂xq= 0

∂tq+Hq

1−ρ∂xρ+1−Hq

1−ρ∂xq=−1

ε(q−F(ρ)),

where qis the ﬂux, Fis the fundamental diagram and His a measure for the look

ahead and the nonlocality of the equations. Such a model converges to the LWR

type equations in the relaxation limit but shows also similarities to the Aw-Rascle

model. The main diﬀerence to the delayed model is that at the macroscopic level

they are composed of a system of two PDEs while the delayed model is described

only by a conservation law with a time delay in the velocity term.

Let us assume ρ0(x, k) = 5

8+1

8sin (2kπx) for k= 1,2, Tf= 10 and ∆t= 0.01.

We consider T∆= 16∆tand T∆= 22∆t, respectively if k= 1,2.

In Fig. 6, the density values on the (x, t)-plane are represented by the colors as

shown in the colorbar. We obtain persistent waves where the number of waves is

PROPERTIES OF THE LWR MODEL WITH TIME DELAY 13

Figure 6. Test 1: Reproducing the simulation presented in [7],

with ρ0(x, 1) on the left and ρ0(x, 2) on the right.

directly related to the perturbations in the initial condition, as in [7]. Numerically,

we recover this result if T∆∈(11∆t, 16∆t) for k= 1 and T∆∈(18∆t, 22∆t) for

k= 2.

Test 2. We consider the discrete delayed model 5with the velocity function 15 and

initial data:

ρ0(x) = (0.6x < 0.5

0.1x≥0.5.(16)

Let us assume the time delay as T∆= 10∆t,Tf= 3.5 and the time step ∆t= 0.01

such that the CFL condition is satisﬁed. Moreover, the coeﬃcient αis chosen in such

a way as to make the velocity function continuous. Looking at Fig. 7(left), one can

immediately note that the initial slowdown in the ﬁrst half of the road is increasing

in time until vehicles completely stop since ρ≥ρc, as the colorbar suggests. This is

a typical S&G wave behaviour, as we can see also in [42], where they compute the

density evolution considering another extension of the LWR model. Indeed they

assume that the velocity term depends on ρ, the reaction time τand the derivative

in space of the velocity itself ∂xV(ρ):

∂tρ+∂xρV ρ

1−τ∂xV(ρ)= 0.(17)

Investigating the density proﬁle at the end of the simulation, we are able to

recognize a well-deﬁned S&G wave proﬁle, see Fig. 7(right), as described in [22].

Moreover, let us note that if the delay is smaller, i.e. T∆= 4∆t, the model is no

longer able to preserve the perturbation and the density proﬁle becomes smoother,

see Fig. 8.

Choosing the delay term 7∆t<T∆<11∆t, we are able to recover the S&G

behaviour.

4.2. Triggering of stop & go waves. Let us now focus on the triggering of S&G

waves. Starting from a small perturbation, i.e. a slowdown, in the initial data, our

aim is to ﬁnd out if it is possible to recover a S&G wave. In this direction, let us

14 SIMONE G ¨

OTTLICH, ELISA IACOMINI AND THOMAS JUNG

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Figure 7. Test 2: Density values in the (x, t)-plane (left) and

density proﬁle at time T=Tf(right).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Figure 8. Test 2: Density values in the (x, t)-plane with low delay

term, T∆= 4∆t, (left) and density proﬁle at T=Tf(right).

consider the example presented in [18], in which the initial data is given by:

ρ0(x) = (0.35 1.34 ≤x≤1.342

0.2elsewhere, (18)

and Dirichlet boundary conditions. The initial data is modelling a small region,

a cell, where vehicles are moving slower than elsewhere and therefore the density

is higher in that cell. This slowdown can be caused by the presence of sags, road

sections in which gradient changes signiﬁcantly from downwards to upwards [37],

or the presence of a school zone in which the velocity has to be reduced, [40].

In [18], the density evolution is computed by coupling the LWR model with a

second order microscopic model, speciﬁcally conceived to reproduce S&G waves.

Instead of switching to multiscale models in which we have to manage the micro-

scopic model too, let us see if we can recover the density evolution with the delayed

model.

Assuming ∆t= 0.009, Tf= 5 and T∆= 21∆t, we note that the initial pertur-

bation increases and moves backward as the time increases, as it happens in the

multiscale case.

PROPERTIES OF THE LWR MODEL WITH TIME DELAY 15

Figure 9. Triggering of Stop & Go waves: Density values in the

(x, t)-plane.

Therefore, the delayed model is able to reproduce the triggering of S&G waves

too, not only their backward propagation.

Conclusion. In this paper, we have introduced theoretical and numerical prop-

erties of the delayed LWR traﬃc model. While the derivation of the model has

been done in [9], the numerical behavior of the delayed model has not been studied

intensively before.

Starting from the undelayed scenario, we investigated the theoretical features

of the delayed model to point out the numerical properties of the scheme and we

proposed an altered Lax-Friedrichs numerical scheme to compute the evolution of

the density. The key observation therefore is that the delayed model is really able

to reproduce Stop & Go waves for the right choice of parameters. Comparisons to

already existing results from the literature also underline this characteristic.

Future works will include the extension to networks as well as parameter estima-

tion techniques to determine the time delay.

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Received March 2020; 1st revision September 2020; 2nd revision October 2020.

E-mail address:goettlich@uni-mannheim.de

E-mail address:eiacomin@mail.uni-mannheim.de

E-mail address:thomas.jung@itwm.fraunhofer.de