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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)
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
xRat 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) = Vmax1ρ
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 T0 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 tsuch 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 xin 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) = 0xV(ρ(x, tT)) + V(ρ(x, tT)) 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, tT)) V(ρ(x, tT))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, tT))
left. We know ρ(x, t)0 and this means that the sign of tρ(x, t) is only
dependent on xV(ρ(x, tT)).
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, tT) is maximal, so
we can in general say nothing about xV(ρ(x, tT)). 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=jx, j Z}and time
{tn=nt, 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 ∆tTto be able to treat the delay.
The Lax-Friedrichs numerical scheme for 1is stated by:
ρn+1
j=1
2(ρn
j+1 +ρn
j1)t
2∆x(f(ρn
j+1)f(ρn
j1)).
Now using the structure of 3, we can identify a ﬂux function f(ρ(x, tT), 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
j1)t
2∆x(f(ρnT
j+1 , ρn
j+1)f(ρnT
j1, ρn
j1)),(5)
where jZ, n Nand Tis 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 ∆tx
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
j1)t
2∆x(V(ρnT
j+1 )ρn
j+1 V(ρnT
j1)ρn
j1),(6)
where the part
X
j
1
2(ρn
j+1 +ρn
j1) = X
j
ρn
j
and the part
X
j
t
2∆x(V(ρnT
j+1 )ρn
j+1 V(ρnT
j1)ρn
j1)
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
j1) 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(ρnT
j+1 )ρn
j+1 V(ρnT
j1)ρn
j1)1
2(ρn
j+1 +ρn
j1).(7)
We compute a CFL-condition, namely ∆tx
max{|ρn|,|ρnT|} . Therefore, we get
for the left-hand-side of 7
1
2 max{|ρn|,|ρnT|}(V(ρnT
j+1 )ρn
j+1 V(ρnT
j1)ρn
j1)
=V(ρnT
j+1 )
2 max{|ρn|,|ρnT|}ρn
j+1 V(ρnT
j1)
2 max{|ρn|,|ρnT|}ρn
j1
max{|ρn|,|ρnT|}
2 max{|ρn|,|ρnT|}ρn
j+1 +max{|ρn|,|ρnT|}
2 max{|ρn|,|ρnT|}ρn
j1
=1
2(ρn
j+1 +ρn
j1),(8)
which shows the positivity for this CFL-condition. Here we use that |V(ρ)| ≤
max{|ρn|,|ρnT|} 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(ρnT
j)ρn
j=f(ρnT
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
j1)t
2∆x(f(ρnT
j+1 , ρn
j+1)f(ρnT
j1, ρn
j1))
=
1
2(ρn
j+1 +ρn
j1) + t
2∆xf(ξ1, ξ2)ρn
j1ρn
j+1
ρnT
j1ρnT
j+1 
1
2(ρn
j+1 +ρn
j1)+t
2∆x(V(ξ2)(ρn
j1ρn
j+1) + ξ1(ρnT
j+1 ρnT
j1))
CF L
1
2ρn
j+1 +ρn
j1+1
2ρn
j1ρnT
j1+ρnT
j+1 ρn
j+1
1
22ρn
j1ρnT
j1+ρnT
j+1
ρn
max +1
2ρnT
j+1 ρnT
j1
2 max{|ρn|,|ρnT|}.(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
j1ρnT
j1+ρnT
j+1 |.
Rearranging gives us
1
2(ρn
j1+ρn
j1ρnT
j1+ρnT
j+1 ) = 1
2(ρn
j1ρnT
j1+ρnT
j+1 +ρn
j1),
and by again using the mean value theorem for ρj1(t) we get
ρn+1
j1
2(T ∂tρj1+ρn
j1+ρnT
j+1 )max{|ρn|,|ρnT|} +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(ρnT
j+1 )V(ρnT
j)=(1 ρnT
j+1 )(1 ρnT
j) = ρnT
jρnT
j+1 =nT
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
j1
2)
t
2∆x(V(ρnT
j+2 )∆n
j+3
2V(ρnT
j)∆n
j1
2ρn
j+1nT
j+3
2
+ρn
j1nT
j1
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∆xρnT
j+1 |+ 2|t
2∆xρn
j+1|) max{|n
j+1
2|,|nT
j+1
2
|}.(13)
If we introduce a CFL-condition of the type ∆t2∆x
max{|ρnT
j|,|ρn
j|} (i.e. not as
strong as above), the dependency on jdisappears and we get
(5 + 1
max{|ρnT
j|,|ρn
j|})X
j
max{|n
j+1
2|,|nT
j+1
2
|}.
By assumption, T V (ρn
) and T V (ρnT
) are ﬁnite, so we can estimate further
T V (ρn+1
)2(5 + 1
max{|ρnT
j|,|ρn
j|}) max{T V (ρn
), T V (ρnT
)}.(14)
PROPERTIES OF THE LWR MODEL WITH TIME DELAY 9
For the estimate in time, we look at
X
j
|ρn+1
jρn
j|.
By simply plugging in 5and using the same CFL-condition, we can write
X
j
|ρn+1
jρn
j| ≤ X
j
1
2(|ρn
j+1|+|ρn
j1|) + t
2∆x(|V(ρnT
j+1 )||ρn
j+1|+|V(ρnT
j1)||ρn
j1|) + |ρn
j|
X
j
2 max{|ρnT
j|,|ρn
j|} + 2 + 2 max{|ρnT
j|,|ρn
j|}
=X
j
4 max{|ρnT
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
tx
max{|ρn|,|ρnT|} 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 ρρf
α(1
ρ1
ρc)ρf< ρ < ρc
0ρρc
(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. ρρmax . However, in general, one has to assume Tone 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= 0.5x
2 max{|ρn|,|ρnT|} in the latter and
t= 1.5x
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
0.4
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= 102,103,104and ∆t= 0.5·102,103,104respectively, 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
1ρxρ+1Hq
1ρxq=1
ε(qF(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
0.1x0.5.(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 ρρc, 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
1τxV(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 x1.342
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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... Some models consider multiple lanes, either modeled by distinguishing between those directly, e.g., [42,90,95,97,104,135], or considering a two dimensional space [96], while others concentrate on different vehicle classes [13,150]. Furthermore, moving bottlenecks taking slower vehicles into account have been included into the traffic dynamics [58,122] and time delays incorporating the reaction times of drivers have been introduced [21,88]. ...
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We present a Godunov type numerical scheme for a class of scalar conservation laws with non-local flux arising for example in traffic flow models. The proposed scheme delivers more accurate solutions than the widely used Lax-Friedrichs type scheme. In contrast to other approaches, we consider a non-local mean velocity instead of a mean density and provide $L^\infty$ and bounded variation estimates for the sequence of approximate solutions. Together with a discrete entropy inequality, we also show the well-posedness of the considered class of scalar conservation laws. The better accuracy of the Godunov type scheme in comparison to Lax-Friedrichs is proved by a variety of numerical examples.
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