Lattanzio2010traffic

Math. Models Methods Appl. Sci. 2010.20:2349-2370. Downloaded from www.worldscientific.com
by NATIONAL AUTONOMOUS UNIVERSITY OF MEXICO (UNAM) on 04/02/20. Re-use and distribution is strictly not permitted, except for Open Access articles.
Mathematical Models and Methods in Applied Sciences
Vol. 20, No. 12 (2010) 23492370
#
.c World Scienti¯c Publishing Company
DOI: 10.1142/S0218202510004945
COUPLING OF MICROSCOPIC AND MACROSCOPIC
TRAFFIC MODELS AT BOUNDARIES
CORRADO LATTANZIO
Dipartimento di Matematica Pura ed Applicata,
Universit
a degli Studi dell'Aquila,
Via Vetoio, Coppito, I67010 L'Aquila, Italy
[email protected]
BENEDETTO PICCOLI
Department of Mathematical Sciences,
Rutgers University Camden,
311 N 5th Street, Camden, NJ 08102, USA
[email protected]
Received 1 October 2009
Revised 26 February 2010
Communicated by T. P. Liu
In this paper, we investigate boundary coupling of microscopicmacroscopic models in the study
of tra±c °ows. We propose a de¯nition of solution through coupling of initial boundary value
problems for the discrete Follow the Leader model and the °uidodynamic second-order
AwRascleZhang model in the two quadrants t > 0; x 7 0. Moreover, new connections
between these two models are pointed out, showing another °uidodynamical interpretation of
the microscopic Follow the Leader model.
Keywords: Microscopic tra±c °ow models; °uid dynamic tra±c °ow models; interface conditions.
AMS Subject Classi¯cation: 35L65, 34A34
1. Introduction
The aim of this paper is the investigation of boundary coupling between microscopic
(discrete) and macroscopic (°uidodynamic) models for vehicular tra±c °ow.
Microscopic models for vehicular tra±c consist of a ¯nite set of ordinary di®erential equations, describing the motion of each vehicle in the tra±c °ow. One of the
most known is the Follow the Leader model,12,21 where each driver adjusts his
velocity to the vehicle in front. More precisely, the velocity change is proportional to
di®erence in velocities and inversely proportional to vehicles' distance.
2349
Math. Models Methods Appl. Sci. 2010.20:2349-2370. Downloaded from www.worldscientific.com
by NATIONAL AUTONOMOUS UNIVERSITY OF MEXICO (UNAM) on 04/02/20. Re-use and distribution is strictly not permitted, except for Open Access articles.
2350
C. Lattanzio & B. Piccoli
The use of °uid dynamic models started in the '50s with the seminal work of
LighthillWitham19 and Richards.22 In those papers a single conservation law for car
density is considered, while successively systems of two equations were used, prescribing also conservation of momentum. The latter are known in transport engineering as \second-order" models. After the well-known criticism of Daganzo,9 Aw and
Rascle proposed in Ref. 3 the conservation of \modi¯ed" momentum. The same
model was independently suggested by Zhang in Ref. 27, thus we refer to it as the
AwRascleZhang (or brie°y ARZ) model. Since then, many other papers were
devoted to °uid dynamic approaches, see for instance Refs. 15, 8, 5, 4, 6, 14, 25, 26,
13, 7 and 11. We do not deal here with mesoscopic models, which are also extensively
used in tra±c modeling, see for instance Refs. 21, 18, 4 and 10.
The presence of many alternative models at various scales, on one hand gives more
opportunities to researchers to ¯t real data, but on the other hand makes the modeling choice more delicate. It is a quite shared opinion that each model has its
advantages and disadvantages, thus an e±cient approach can be based on the
possibility of combining them. Along this idea some results were already obtained.
For example, the relations among microscopic and (°uid dynamic) macroscopic
models were studied in some papers. In Ref. 2 the authors showed how solutions to
the Follow the Leader model, with appropriate rescaling, in the limit tend to the
solutions of the ARZ model. In Ref. 20 a hybrid model is built based on results from
Ref. 2. However, due to manifest di±culties in the interface connections between
micro and macro models, coupling at boundaries has been investigated only under a
numerical point of view (see, for instance, Refs. 16 and 17). Our intent is to give an
analytical notion of solution, taking into account the well-known connections illustrated above, among Follow the Leader and ARZ models.
Let us recall the precise expressions for the microscopic Follow the Leader
model12,21:
8
>
<x_ i ¼ vi ;
viþ1 vi
:
>
;
:vi ¼ C
ðxiþ1 xi Þ þ1
and the macroscopic AwRascleZhang model3,27:
(
t þ ðvÞx ¼ 0;
ðwÞt þ ðwvÞx ¼ 0:
The coupling is obtained by solving two initial boundary value problems for the two
models, with boundary data mutually in°uenced from each side.
As a prerequisite, we need to de¯ne the dynamic of Follow the Leader model in
each of the two quadrants t > 0; x 7 0. In particular, we shall investigate how a car
can be inserted in the model in x > 0 according to the boundary data at x ¼ 0, which we
refer to as emission procedure. Again, in the quadrant t > 0; x < 0, we shall interpret
the boundary data at x ¼ 0 as \position" and \velocity" of a ¯ctitious car ahead
Math. Models Methods Appl. Sci. 2010.20:2349-2370. Downloaded from www.worldscientific.com
by NATIONAL AUTONOMOUS UNIVERSITY OF MEXICO (UNAM) on 04/02/20. Re-use and distribution is strictly not permitted, except for Open Access articles.
Coupling of Microscopic and Macroscopic Tra±c Models at Boundaries
2351
(in this case, we refer to a forcing e®ect of the boundary). Moreover, it is clear that, for
these initial boundary value problems, we shall obtain a \continuous" dynamics until a
car is inserted into the model (resp. a car exits from the model) for the case x > 0 (resp.
x < 0). The aforementioned discrete events de¯ne an appropriate restarting procedure,
which shall then play a crucial role in the interplay with the continuous model.
As a by-product of our analysis toward the comprehension of the possible connections between the two models, we obtain interesting new °uidodynamical insights
of the microscopic Follow the Leader model as the behavior of its dynamics with
respect to the invariant region of the AwRascleZhang model. Moreover, we show
how the characteristic interaction time of the former model is proportional to the
interaction time of two approaching waves of the latter.
The remaining part of this paper is organized as follows.
After this Introduction, we review the main properties of the macroscopic model
(Sec. 2) and of the microscopic one (Sec. 3), together with the crucial de¯nitions of
the two initial boundary value problems for the latter.
In Sec. 4, we investigate the aforementioned °uidodynamical interpretation of the
Follow the Leader model and ¯nally in Sec. 5 we state our de¯nition of solutions of
the coupled models, starting from the prescription of the dynamics for Riemann
problems, and prove the existence for the related Cauchy problems.
{
{
2. Description of the Aw Rascle Zhang Model
As a macroscopic model for the tra±c °ow on highways, we focus on the following
second-order model, known as the AwRascleZhang model (from now on, ARZ) in
Eulerian coordinates3,27:
t þ ðvÞx ¼ 0;
ð2:1Þ
ðwÞt þ ðwvÞx ¼ 0;
where w ¼ v þ pðÞ and the \pressure" p veri¯es the usual monotonicity for the gas
dynamic models, namely p 0 ðÞ > 0. In (2.1), ðx; tÞ > 0 stands for the density of the
cars at point x and time t and vðx; tÞ for their average velocity. Once we have ¯xed the
maximal density of cars along the street and the reference velocity both equal to one,
the common choices for the function p are
logðÞ; formally for ¼ 0;
pðÞ ¼
ð2:2Þ
;
for > 0:
Let us brie°y describe reformulations of the ARZ model in conserved variables, in
Riemann invariants and in Lagrangian coordinates. To this end, de¯ne the \momentum"
variable by y ¼ w ¼ ðv þ pðÞÞ. Then (2.1) can be written in ð; yÞ coordinates as
8
þ ðy pðÞÞx ¼ 0;
>
< t
2
ð2:3Þ
y
>
:yt þ
ypðÞ ¼ 0;
x
Math. Models Methods Appl. Sci. 2010.20:2349-2370. Downloaded from www.worldscientific.com
by NATIONAL AUTONOMOUS UNIVERSITY OF MEXICO (UNAM) on 04/02/20. Re-use and distribution is strictly not permitted, except for Open Access articles.
2352
C. Lattanzio & B. Piccoli
and its characteristic speeds are given by 1 ð; yÞ ¼ y ðpðÞÞ 0 ; 2 ð; yÞ ¼ y pðÞ ¼ 1 ð; yÞ þ p 0 ðÞ > 1 ð; yÞ whenever > 0. Note that the second eigenvalue is
indeed the average velocity v of the cars along the road. The (right) eigenvectors are
given by
0 1
0
1
1
1
0
A
r1 ð; yÞ ¼ @ y A; r2 ð; yÞ ¼ @ y
ð;
yÞ
þ
¼
r
1
þ p 0 ðÞ
p 0 ðÞ
and the corresponding Riemann invariants are w ¼
Hence, the quasilinear diagonal version of (2.3) is
(
wt þ 2 ð; yÞwx ¼ 0;
y
and v ¼ y pðÞ ¼ 2 ð; yÞ.
vt þ 1 ð; yÞvx ¼ 0:
ð2:4Þ
The ¯rst characteristic ¯eld is genuinely nonlinear if r1 ð; yÞ r1 ð; yÞ never vanishes, that is if
2p 0 ðÞ p 00 ðÞ 6¼ 0;
ð2:5Þ
for any . Conversely, the second characteristic ¯eld is linearly degenerate since
r2 ð; yÞ r2 ð; yÞ ¼ 0, for any ð; yÞ under consideration.
Finally, as pointed out in Ref. 2, the ARZ model has a very natural connection
with the microscopic Follow the Leader model, when it is written in Lagrangian
coordinates (still denoted with t and x), as follows:
t vx ¼ 0;
ð2:6Þ
wt ¼ 0;
where ¼
1
> 0 is the \speci¯c volume" and therefore v ¼ w pð 1 Þ, i.e.
8
1
>
< t w p
¼ 0;
x
>
:
wt ¼ 0:
ð2:7Þ
The charactristic speeds are given by 1 ð; wÞ ¼ p 0 ð 1 Þ 12 < 0; 2 ð; wÞ ¼ 0, with
(right) eigenvectors r1 ð; wÞ ¼ ð1; 0Þ; r2 ð; wÞ ¼ ð1; p 0 ð 1 Þ 12 Þ and Riemann invariants v and w as above. In other words, the quasilinear diagonal version of (2.7) is
given by
8
1 1
>
<vt p 0
v ¼ 0;
2 x
>
:
wt ¼ 0:
At this point, we brie°y recall the main properties of such model (2.3) (see also
Ref. 3).
Math. Models Methods Appl. Sci. 2010.20:2349-2370. Downloaded from www.worldscientific.com
by NATIONAL AUTONOMOUS UNIVERSITY OF MEXICO (UNAM) on 04/02/20. Re-use and distribution is strictly not permitted, except for Open Access articles.
Coupling of Microscopic and Macroscopic Tra±c Models at Boundaries
2353
2.1. Hugoniot curves and invariant region
From now on, let us consider the ARZ model with pðÞ given by (2.2). Then, we
assign a maximal velocity for cars along the street depending on their density so that
0 v vmax ðÞ:
An obvious restriction for vmax ðÞ is that it must be zero for ¼ 1, namely, when the
cars are bumper-to-bumper (recall that 0 < 1Þ; the common choice is
logðÞ; formally for ¼ 0;
vmax ðÞ ¼
ð2:8Þ
for > 0:
1 ;
Hence, in view of (2.8), we have the following bounds for the variable y ¼ w ¼
ðv þ pðÞÞ:
logðÞ y 0; formally for ¼ 0;
þ1 y ; for > 0:
ð2:9Þ
In this description, we shall focus in the case > 0, the case ¼ 0 being similar.
For this choice of the pressure p, condition (2.5) becomes
ð þ 1Þ 1 6¼ 0;
which is obviously true for any > 0. Hence the simple waves of the ¯rst family are
either shock or rarefaction waves, while the simple waves of the second family are
contact discontinuities.
2.1.1. 2-contact discontinuities
A state Y ¼ ð ; y Þ is connected to a state Y ¼ ð; yÞ to the right by a 2-contact
discontinuities if vðY Þ ¼ vðY Þ, i.e.
y ¼ vðY Þ þ þ1 ;
for arbitrary :
2.1.2. 1-rarefaction waves
A state Y ¼ ð ; y Þ is connected to a state Y ¼ ð; yÞ to the right by a 1-rarefaction
wave if wðY Þ ¼ wðY Þ and 1 ðY Þ < 1 ðY Þ, i.e.
y ¼ wðY Þ;
with < :
2.1.3. 1-shock waves
A state Y ¼ ð ; y Þ is connected to a state Y ¼ ð; yÞ to the right by a 1-shock wave
of speed s if
sð Þ ¼ y þ1 y þ þ1
;
y2
y2
y þ y ;
1 ðY Þ < s < 1 ðY Þ;
s < 2 ðY Þ:
sðy y Þ ¼
Math. Models Methods Appl. Sci. 2010.20:2349-2370. Downloaded from www.worldscientific.com
by NATIONAL AUTONOMOUS UNIVERSITY OF MEXICO (UNAM) on 04/02/20. Re-use and distribution is strictly not permitted, except for Open Access articles.
2354
C. Lattanzio & B. Piccoli
Hence, we obtain
y
y
s ¼ s ¼ c;
y y
¼ 0;
c
1 ðY Þ < s < 1 ðY Þ;
s < 2 ðY Þ
and, excluding the case c ¼ 0 which yields to the 2-contact discontinuities already
described, we end up with
y ¼ wðY Þ;
with > :
2.1.4. Invariant region
Let us consider the region R, depicted in Fig. 1(a), in the quadrant 0; y 0
de¯ned by (2.9):
R ¼ fð; yÞ 2 R þ R þ : 0 1; þ1 y g:
ð2:10Þ
By an accurate analysis of Riemann problems, it is possible to show that such region
is invariant for (weak) admissible solutions of (2.3).3 Moreover (see Sec. 3.3), it is
useful to recast this invariant region also in Riemann invariants coordinates ðw; vÞ:
R ¼ fðw; vÞ 2 R þ R þ : 0 w 1; 0 v wg;
ð2:11Þ
this region is depicted in Fig. 1(b).
2.2. Boundary value problems
We conclude this brief review by describing the initial boundary value problems for
the ARZ model in the two cases t > 0; x 7 0.
As it is well known (see, for instance, Refs. 23, 1 and 24), such a problem is wellposed according to the choices of boundary data assigned at x ¼ 0. This is related to
the signs of characteristic speeds, which de¯ne the incoming and outgoing waves with
respect to the domain (see (2.4)). Indeed, while the second eigenvalue 2 ð; yÞ ¼ v is
always non-negative, the ¯rst one can be either positive or negative and this will
a®ect the de¯nition of the boundary data at x ¼ 0.
Remark 2.1. In spite of the clear advantages of the Lagrangian formulation given
by (2.6), both for the connections established in Ref. 2 and for the trivial nature of its
characteristic speeds, we are forced to consider its Eulerian version in the de¯nition of
initialboundary value problems to make use of \physical variables" x and t and, in
particular, to work with a \physical" boundary in x ¼ 0. This requirement is needed
only to merge these problems with the corresponding initialboundary value problems
for the microscopic Follow the Leader model through x ¼ 0 (see Secs. 3 and 5 below).
Indeed, a boundary value problem for the Lagrangian system (2.6), without a further
coupling with another model at the Eulerian boundary x ¼ 0, would clearly be justi¯ed
Math. Models Methods Appl. Sci. 2010.20:2349-2370. Downloaded from www.worldscientific.com
by NATIONAL AUTONOMOUS UNIVERSITY OF MEXICO (UNAM) on 04/02/20. Re-use and distribution is strictly not permitted, except for Open Access articles.
Coupling of Microscopic and Macroscopic Tra±c Models at Boundaries
2355
(a)
(b)
Fig. 1. Invariant region in ð; yÞ and ðw; vÞ variables.
under the modeling point of view: we assign at x ¼ 0 information for drivers so that
they adjust their velocity forward along the °ux.
2.2.1. Boundary on the left
Let us start by considering the ARZ model in the quadrant x > 0 and t > 0. Then,
besides BV Cauchy data 0 and y0 assigned at t ¼ 0 and x > 0, to de¯ne a well-posed
Math. Models Methods Appl. Sci. 2010.20:2349-2370. Downloaded from www.worldscientific.com
by NATIONAL AUTONOMOUS UNIVERSITY OF MEXICO (UNAM) on 04/02/20. Re-use and distribution is strictly not permitted, except for Open Access articles.
2356
C. Lattanzio & B. Piccoli
initialboundary value problem, at x ¼ 0 we must prescribe:
(i) only w, if the ¯rst characteristic speed is non-positive;
(ii) both w and v, if the ¯rst characteristic speed is positive.
2.2.2. Boundary on the right
In that case, if the ¯rst characteristic speed is positive, then both waves of the ARZ
model leave the quadrant x < 0 and t > 0 and therefore we must prescribe only BV
Cauchy data 0 and y0 at t ¼ 0 and x < 0. Finally, if the ¯rst characteristic speed is
non-positive, besides the aforementioned Cauchy data, we must prescribe the value of
the velocity v at x ¼ 0 to de¯ne a well-posed initialboundary value problem.
3. Description of the Follow the Leader Model
The microscopic Follow the Leader model without relaxation (from now on, FL) is
given by the following system of ODEs:
8
>
<x_ i ¼ vi ;
ð3:1Þ
viþ1 vi
:
>
;
:vi ¼ C
þ1
ðxiþ1 xi Þ
where xi and vi are respectively the position and the velocity of the i th car and 0.
We normalize the maximal density of cars along the street and the reference velocity
both equal to one and choose the constant C as C ¼ ð4XÞ , where 4X is the
length of a single car. Then, introducing the variables \density" i , \speci¯c volume"
i and \modi¯ed pressure" wi for the ith car:
i ¼
4X
;
xiþ1 xi
1
x xi
;
¼ iþ1
i
4X
1
wi ¼ vi þ p
;
i
i ¼
with p given by (2.2), Eq. (3.1) can be rewritten as:
8
<: i ¼ viþ1 vi ;
4X
:
w_ i ¼ 0:
ð3:2Þ
As already pointed out before,2 the form (3.2) is the one used to connect (via the
scaling limit) to the ARZ model considered in Lagrangian coordinates (2.6). Since our
coupling between the ARZ °uidodynamic system and the microscopic FL model will
be done in terms of boundary conditions, we investigate how to interpret an initial
boundary value problem for the latter. To this end, we introduce pointwise de¯ned
Math. Models Methods Appl. Sci. 2010.20:2349-2370. Downloaded from www.worldscientific.com
by NATIONAL AUTONOMOUS UNIVERSITY OF MEXICO (UNAM) on 04/02/20. Re-use and distribution is strictly not permitted, except for Open Access articles.
Coupling of Microscopic and Macroscopic Tra±c Models at Boundaries
2357
density and velocity (up to the car length 4X):
I ðx; Þ ¼
4X
;
xIþ1 ðÞ xI ðÞ
vI ðx; Þ ¼ vIþ1 ðÞ;
x 2 ½xI ; xIþ1 4X:
ð3:3Þ
We now want to treat the case of domains with boundaries. Unavoidably there
will be some discrete events separating time intervals where the evolution is continuous, namely when a car starts from the boundary (in the case of boundary on the
left, see Sec. 3.1) or when a car reaches the boundary (in the case of boundary on the
right, see Sec. 3.2). A restarting procedure will be necessary after such discrete events.
In the description below, we shall ¯rst focus our attention on the \continuous"
dynamics, namely the one de¯ned between restarting procedures.
Remark 3.1. We cannot expect to have in general equivalence between the free
dynamic of FL and de¯ned dynamic in the presence of boundaries: the presence of the
boundary a®ects the dynamic by inserting a new car in the model, hence it acts as an
initial datum and therefore it is equivalent to a change made in a space of ¯nite
dimension. On the other hand, boundary values can be assigned by spaces of in¯nite
dimension. Necessarily there are singularities of the map from boundary data to FL
data. However, we shall check the aforementioned equivalence in the case of cars with
constant velocities for both cases in the corresponding sections below.
To de¯ne a solution for given boundary data for the FL model, we ¯rst consider
the model on the whole R and then restrict it to a quadrant ft 0; x 7 0g. Then we
\read" or \impose" the values of the ¯ctitious density and of the velocity at the
boundary placed on the vertical t axis.
3.1. Boundary on the left : Emission procedure
Assume the boundary is at x ¼ 0 and let I be the index such that xI < 0 and
xIþ1 > 0. Then, according to (3.3), for the full FL model (de¯ned on the whole R), the
density and the velocity detected at x ¼ 0 are:
I ð0; Þ ¼
4X
;
xIþ1 ðÞ xI ðÞ
vI ð0; Þ ¼ vIþ1 ðÞ:
ð3:4Þ
If the FL model is taken only at the right of the boundary x ¼ 0, then xi > 0 for any i
and the ¯rst car is indeed xIþ1 .
Fix the initial time 0 as the time for which xIþ1 ð0Þ ¼ 0 (the ðI þ 1Þth car passes
through the boundary x ¼ 0) and assign boundary values b and vb . Then we de¯ne
an emission procedure of a car xI at x ¼ 0 (and thus xI ¼ 0 in the ¯rst equality
of (3.3)) at a time t de¯ned by the following relation:
Z t
1
4X
ð3:5Þ
t 0 b ðtÞdt ¼ xIþ1 ðt Þ :
Remark that for b 1, there is no emission of cars if xIþ1 4X (there is not enough
room between the boundary and the ¯rst car on the right). Moreover, we assign a
Math. Models Methods Appl. Sci. 2010.20:2349-2370. Downloaded from www.worldscientific.com
by NATIONAL AUTONOMOUS UNIVERSITY OF MEXICO (UNAM) on 04/02/20. Re-use and distribution is strictly not permitted, except for Open Access articles.
2358
C. Lattanzio & B. Piccoli
velocity vI to that car given by
Z
vI ¼
t
0
b ðtÞvb ðtÞdt
:
Z t
b ðtÞdt
ð3:6Þ
0
The meaning of Eq. (3.5) is the following:
the I th car is inserted in the model at the origin when the mean
value of the assigned density at the boundary is such that the ¯rst of
equality in (3.3) is veri¯ed for xI ¼ 0 (see also (3.4)).
To check the consistency of the proposed solutions to boundary value problems
with the original FL model, we may proceed as follows:
Fix an evolution of the FL model on the whole R;
Determine the values of ¯ctitious continuous variables ð; vÞ as in (3.3) at the
boundary x ¼ 0;
. Solve the obtained Initial–Boundary Value problem (brie°y IBV) on the ¯rst
quadrant;
. Check if the solution to the IBV problem coincides with the initial solution to the
FL model, restricted to the ¯rst quadrant.
.
.
Due to Remark 3.1, we do not expect this consistency to hold in general. However, we
can check it for the simple case of cars with constant velocities. Therefore assume that
all velocities are equal for all cars and all time, say to v, let z be the ¯xed distance
among cars and assume also xIþ1 ð0Þ ¼ 0 for some I. Then, the continuous variables
de¯ned in (3.3) satisfy for any x; t and I :
X
vI v:
¼ ;
z
Now setting accordingly the boundary values, namely b ðtÞ and vb ðtÞ v, condition (3.5) reads
I X X
¼ z
vt
from which we get t ¼ z =
v , which is precisely the time such that xI ðt Þ ¼ 0. Finally,
consistency holds true.
3.2. Boundary on the right: Forcing e®ect
Consider now a FL model only at the left of the boundary x ¼ 0, that is xi < 0 for any
i and w.l.o.g. let xI be the leading car (i.e. the closest to the origin). If one ¯xes
boundary values b 1 and vb , then the last car xI \sees" a ðI þ 1Þth car at position
xIþ1 ¼
4X
4X > 0
b
ð3:7Þ
Math. Models Methods Appl. Sci. 2010.20:2349-2370. Downloaded from www.worldscientific.com
by NATIONAL AUTONOMOUS UNIVERSITY OF MEXICO (UNAM) on 04/02/20. Re-use and distribution is strictly not permitted, except for Open Access articles.
Coupling of Microscopic and Macroscopic Tra±c Models at Boundaries
2359
with velocity vb , according to (3.3) at x ¼ 0. In this case, the assigned boundary
values and relation (3.7) are interpreted as follows:
Since the assigned boundary value for the density b must be
independent from the dynamic of xI to obtain a well-posed initial
boundary value problem, the function b represents the portion of (de¯ned via (3.3)) in the positive real line, due to the presence of a
¯ctitious (I þ 1)th car ahead.
Thus, the equations for the I th car take the form:
8
>
<x_ I ¼ vI ;
vb vI
:
>
þ1
:vI ¼ C
ð4X xI b Þ þ1 b
ð3:8Þ
and the boundary acts as a source term in the equations for the last car xI .
Let us now consider consistency as de¯ned in the previous section. Again consider
a solution with constant velocities equal to v, constant distance among cars equal to z
and xIþ1 ð0Þ ¼ 0. The boundary values b and vb are as before. Then the equation for
the last car velocity vI is:
vb vI
:
vI ¼ C
þ1 ¼ 0;
ð4X xI b Þ þ1 b
thus the solution to the IBV problem has also constant velocities.
Let us prove consistency also in another manner. The IBV problem dynamic is
equivalent to the free dynamic of FL with a ðI þ 1Þth car with
vIþ1 ¼ vb ;
xIþ1 ¼
4X
;
b
again in the case of constant data. Indeed, vb ¼ vI ¼ vIþ1 is the only constant velocity
admissible for the dynamic of the car xI , because it must verify
vIþ1 vI
:
vI ¼ C
¼0
ðxIþ1 xI Þ þ1
(all cars have the same velocity in the constant case!) and the constant in time
density detected at x ¼ 0 is
I ¼
4X
4X
;
¼
xIþ1 xI
4X=b xI ð0Þ
that is, the contribution due to the boundary value and the presence of the I th car in
the negative real line.
Remark 3.2. Let us emphasize that the dynamic of the FL model, and thus of our
\equivalent" boundary value problems, is de¯ned so that collisions between cars are
avoided in the following sense. We cannot expect to keep the distance xiþ1 ðtÞ xi ðtÞ 4X, because, as pointed out before, the equations admit solutions with equal,
Math. Models Methods Appl. Sci. 2010.20:2349-2370. Downloaded from www.worldscientific.com
by NATIONAL AUTONOMOUS UNIVERSITY OF MEXICO (UNAM) on 04/02/20. Re-use and distribution is strictly not permitted, except for Open Access articles.
2360
C. Lattanzio & B. Piccoli
constant, positive velocities and equal, constant, non-negative and arbitrary mutual
distance between cars. However, we can prove that xiþ1 xi 0 for any t as follows,
provided v 1; 1 being the reference velocity of the street, as already mentioned.
To this aim, let us consider the worst possible case, that is a car xiþ1 sitting at
0 and
point 4X with zero velocity, and a moving car xi , starting for t ¼ 0 from x
with velocity 0 < v 1. Then the equations describing the dynamic of xi and vi are
given by
8
>
<x_ i ¼ vi ;
ð3:9Þ
vi
:
>
;
:vi ¼ C
þ1
ð4X xi Þ
which is indeed Eq. (3.8) for b ¼ 1 and vb ¼ 0, namely, again the worst possible case
for any t and recalling
of our IBV problem described in Sec. 3.2. Then, since xi ðtÞ x
C ¼ ð4XÞ , from (3.9) we obtain
:
vi vi ;
¼
ð4XÞ ;
Þ þ1
ð4X x
and therefore
vi ðtÞ ve t
for any t 0. Moreover, from the above bound we obtain
Z t
v
þ
þ
xi ðtÞ x
:
ve s ds x
0
Now, as pointed out before, condition xi 0 (that is, distance between the two cars
at least 4X) is not granted in general, but it can be forced provided initial velocity is
su±ciently small:
¼
v x
ð4XÞ ð
x Þ:
Þ þ1
ð4X x
As a matter of fact, if we perform the above analysis for the IBV problem (3.8), for
this velocity we do not have restarting procedures in the model, because the i th car
never reaches the boundary x ¼ 0 in ¯nite time, while in general this possibility,
being admissible, is included in the model, as mentioned above.
On the other hand, the prescribed dynamic implies the restriction xi ðtÞ 4X for
¼ 0. Indeed, in that case we have
any t, even if x
¼
1
4X
Math. Models Methods Appl. Sci. 2010.20:2349-2370. Downloaded from www.worldscientific.com
by NATIONAL AUTONOMOUS UNIVERSITY OF MEXICO (UNAM) on 04/02/20. Re-use and distribution is strictly not permitted, except for Open Access articles.
Coupling of Microscopic and Macroscopic Tra±c Models at Boundaries
2361
and
xi ðtÞ v
¼ 4X
v 4X;
because v 1.
In conclusion, collisions of cars, that is xi ðtÞ ¼ xiþ1 ðtÞ for some i and for some t, are
always excluded by the model, provided xiþ1 ð0Þ xi ð0Þ 4X. On the other hand,
we can keep distances between cars greater than the physical dimension of cars 4X,
provided we ¯x compatible initial data for the FL model, namely, cars too close to
each other must not have too di®erent initial velocities.
We conclude the section by analyzing the behavior of the invariant region
(Eqs. (2.10) and (2.11)) in the dynamics de¯ned by the FL model.
3.3. FL dynamics and ARZ invariant region
In the next section we shall put into evidence a new connection between FL and ARZ
models. To this aim, we need to emphasize that there exist physically meaningful
admissible solutions for the FL model which are not admissible for the ARZ model,
because they are not bounded to the invariant regions (2.10) or (2.11). Indeed, the FL
model admits solutions with maximal density and positive velocity, that is a bunch of
cars bumper-to-bumper which travel with the same positive velocity. This situation
has no counterpart in the macroscopic ARZ model: the only point in the invariant
region with maximal density ¼ 1 veri¯es also v ¼ 0 by construction (see Sec. 2.1
and Fig. 1). In other words, as we shall see later, this invariant region is not invariant
for the dynamics de¯ned by the FL model for 6¼ 0; 1 and thus the latter admits
states which are indeed excluded by the ARZ model. The above remark is crucial for
the discussion in the next section, because our °uidodynamical interpretation of the
FL model shall involve the above states.
Let us start by showing that the invariant region (2.11) is not invariant for the
dynamics de¯ned by (3.1). To this aim, we recall the de¯nition of \density" and
\modi¯ed pressure" for the ith car:
i ¼
4X
;
xiþ1 xi
wi ¼ vi þ pði Þ:
Then, the dynamic for wi is given by
w_ i ¼ ði Þ viþ1 vi
v vi
v vi
4Xði Þ 1 iþ1
¼ ð1 Þði Þ iþ1
:
2
xiþ1 xi
ðxiþ1 xi Þ
xiþ1 xi
Thus, we can choose states in FL at the boundary w ¼ 1 of Fig. 1 such that w_ i > 0 at
that boundary, which implies the aforementioned region is not invariant for FL (for
7 1, it su±ces to require viþ1 ? vi ). Hence the dynamics can exit from that region
and it can reach the line v ¼ w 1, i.e. ¼ 1, even with a positive velocity v.
Math. Models Methods Appl. Sci. 2010.20:2349-2370. Downloaded from www.worldscientific.com
by NATIONAL AUTONOMOUS UNIVERSITY OF MEXICO (UNAM) on 04/02/20. Re-use and distribution is strictly not permitted, except for Open Access articles.
2362
C. Lattanzio & B. Piccoli
4. A New Fluidodynamical Interpretation of the
Microscopic FL Model
In this section we present a heuristic connection between the microscopic FL and the
macroscopic ARZ models, besides the classical connection already established in Ref. 2.
More precisely, we shall see that the characteristic interaction time of the FL model
ðxiþ1 xi Þ þ1 xiþ1 xi
¼
C
i
can be seen as interaction time of two waves of the °uidodynamic ARZ model.
In view of our pointwise reconstruction of density and velocity in (3.3), we obtain
in particular, dropping for simplicity the t dependence:
wI ðxÞ ¼ vI ðxÞ þ pðI ðxÞÞ ¼ vIþ1 þ pðI Þ;
for x 2 ½xI ; xIþ1 4X . For what concerns the w variable, the above functions can
be considered as representation of a weak solution to the ARZ model, with waves
which are 2-contact discontinuities.
Then we focus on the Riemann's problem at points xI and xIþ1 4X to determine the time-interaction of waves emanating from them. We depict this situation in
Fig. 2, with the notations
wþ
I ¼ limþ wI ðxÞ ¼
x!x I
lim
x!ðxIþ1 4XÞ wI ðxÞ ¼ w I :
As one can see from Sec. 2.1, the 2-wave emanating from the point xI is a contact
discontinuity, with speed þ
2 ¼ vIþ1 . Moreover, the 1-wave emanating from the point
xIþ1 4X has speed of propagation at least 1 , namely, its value from the left.
Therefore the time interaction of these two °uidodynamical waves is given by
xIþ1 xI 4X
xIþ1 xI 4X
xIþ1 xI 4X
¼
¼
v
ðw ðð1
þ
Þv
w
Þ
þ
Iþ1
Iþ1
1
I
I vIþ1 Þ
2
¼
xIþ1 xI 4X
;
I
Fig. 2. Interaction of °uidodynamical waves in microscopic FL model.
Math. Models Methods Appl. Sci. 2010.20:2349-2370. Downloaded from www.worldscientific.com
by NATIONAL AUTONOMOUS UNIVERSITY OF MEXICO (UNAM) on 04/02/20. Re-use and distribution is strictly not permitted, except for Open Access articles.
Coupling of Microscopic and Macroscopic Tra±c Models at Boundaries
2363
which is of the same order of the one obtained for the FL model, up to an error of the
order of 4X.
5. The Coupled Models
Our coupling of models (from now on, ARZFL or FLARZ, according to the way
the two models are coupled) is de¯ned by considering the \merging" of two Initial
Boundary Value problems (brie°y IBV), with initial data given in the two half-lines
x > 0 and x < 0. To avoid confusion, where necessary we use superscript FL for the
Follow the Leader model and the superscript ARZ for the AwRascleZhang model.
The connection between the two models shall be done through the boundary in x ¼ 0,
according to the previous discussions. More precisely we will assign boundary values
at x ¼ 0 for the ARZ model, indicated by ð ARZ
; v ARZ
Þ, and for the FL model,
b
b
FL
FL
indicated by ð b ; v b Þ. The evolution of the two IBV problems will be considered for
a (in general) ¯nite time interval ½0; 4t; 4t > 0, bounded by:
.
the ¯rst emission time (if ¯nite) of a new car in the FL model side, for the ARZ–FL
coupled model (see (3.5));
. the ¯rst time for which a car in the FL model side reaches the boundary x ¼ 0,
according to its dynamics (see (3.8)), for the FL–ARZ coupled model.
Then a restarting procedure will be considered, and we shall solve again the two IBV
problems with updated initial conditions. We start by describing the dynamics we
assign for the Riemann problems for the two couplings.
5.1. Riemann problem for FLARZ coupled model
In this section we shall describe how to solve a Riemann problem for the FLARZ
model. To this end, let us ¯x initial data xi ð0Þ; vi ð0Þ for FL with i I; xi ð0Þ < 0 (cars
at the left of the boundary x ¼ 0 and the last car is xI ) and constant initial data
0 < ARZ
1; y ARZ
for ARZ in x > 0.
0
0
Then, according to Sec. 3.2, the in°uence of ARZ to the dynamics of the last car xI
is taken into account introducing a forcing e®ect with
ARZ
þ
v FL
;
b ¼ v0 ð0 Þ ¼ v 0
ARZ
þ
FL
b ¼ 0 ð0 Þ ¼ 0
(see Fig. 3). These values are obtained using the following relation:
y ARZ
¼ ARZ
w ARZ
¼ ARZ
ðv ARZ
þ pð ARZ
ÞÞ:
0
0
0
0
0
0
ð5:1Þ
The in°uence of the FL model on the ARZ model will happen only when a car
reached the boundary x ¼ 0. Then at the beginning we set:
ð ARZ
; v ARZ
Þ ¼ ð ARZ
; v ARZ
Þ:
0
0
b
b
Then let t be such that xI ðt Þ ¼ 0, namely, the last car of FL reaches the boundary.
Then a restarting procedure is needed and FL shall in°uence ARZ through the
Math. Models Methods Appl. Sci. 2010.20:2349-2370. Downloaded from www.worldscientific.com
by NATIONAL AUTONOMOUS UNIVERSITY OF MEXICO (UNAM) on 04/02/20. Re-use and distribution is strictly not permitted, except for Open Access articles.
2364
C. Lattanzio & B. Piccoli
Fig. 3. Riemann problem for FLARZ coupled model before the restarting procedure: during
\continuous" dynamics, ARZ in°uences FL.
Fig. 4. Riemann problem for FLARZ coupled model at the restarting procedure: In discrete events, FL
in°uences ARZ.
boundary. More precisely we will set:
ð ARZ
; v ARZ
ÞðsÞ ¼ ðI1 ðX; t Þ; vI1 ðX; t ÞÞ ¼
b
b
X
; v ðt Þ
xI ðt Þ xI1 ðt Þ I
for all s t and up to next time in which a car reached the boundary x ¼ 0. The
corresponding IBV problem, for s t , is equivalent to the following Riemann
problem (Fig. 4):
8
X
>
>
;
y
ð
t
Þ
; x < 0;
<
xI ðt Þ xI1 ðt Þ I1
>
>
: ARZ ARZ
ð 0 ; y 0 Þ;
x > 0;
using again (5.1) to reconstruct the value yI1 ðt Þ from the density I1 ðt Þ ¼ X xI ðt ÞxI1 ðt Þ
and the velocity vI1 ðX; t Þ ¼ vI ðt Þ. As usual for boundary value problems for
Math. Models Methods Appl. Sci. 2010.20:2349-2370. Downloaded from www.worldscientific.com
by NATIONAL AUTONOMOUS UNIVERSITY OF MEXICO (UNAM) on 04/02/20. Re-use and distribution is strictly not permitted, except for Open Access articles.
Coupling of Microscopic and Macroscopic Tra±c Models at Boundaries
2365
hyperbolic equations,23,1,24 it is useful to rewrite the above Riemann problem in terms
of Riemann invariant coordinates:
(
ðwI1 ðt Þ; vI ðt ÞÞ; x < 0
ð5:2Þ
; v ARZ
Þ;
x > 0;
ðw ARZ
0
0
where wI1 ðt Þ ¼ vI1 ðt Þ þ pðI1 ðt ÞÞ. Finally, according to the waves of ARZ (see
Sec. 2), at x ¼ 0 we obtain
Þ, if the ¯rst characteristic speed is non-positive;
(i) ðwI1 ðt Þ; v ARZ
0
(ii) ðwI1 ðt Þ; vI ðt ÞÞ, if the ¯rst characteristic speed is positive.
FL
Then, according to case (1) or case (2) above, we de¯ne boundary values v FL
b and b
in FL in x < 0 for the last car xI1 , together with new initial data xi ðt Þ; vi ðt Þ; i I 1,
which gives the dynamics in the negative real line after the restarting procedure.
Finally, the dynamics in x > 0 is given by the Riemann problem (5.2) described above.
5.2. Riemann problem for ARZFL coupled model
Let us now analyze the Riemann problem for the ARZFL coupled model. Hence, let
us ¯x initial data xi ð0Þ; vi ð0Þ for FL with i I þ 1; xi ð0Þ > 0 (cars at the right of the
boundary x ¼ 0 and the ¯rst car is xIþ1 ). Moreover, we ¯x constant initial data 0 <
ARZ
1; y ARZ
for ARZ in x < 0 and we solve the latter in the quadrant x < 0; t > 0.
0
0
Let us ¯rst discuss the boundary values ð ARZ
; v ARZ
Þ for the ARZ model on the left.
b
b
It is equivalent to discuss the data in Riemann invariant coordinates, thus assigning
ðw ARZ
; v ARZ
Þ. Since the characteristic speed of w is always positive, we need not assign
b
b
the boundary value w ARZ
, thus we simply set:
b
v ARZ
ðtÞ ¼ vIþ1 ðtÞ
b
up to the ¯rst emission time. Regarding the boundary data for the FL model, we set:
FL
ARZ ð0 ; tÞ; v ARZ ð0 ; tÞÞ;
ð FL
b ; v b Þ ¼ ð
again reconstructed by (5.1) (see Fig. 5). In the above formula, we denote with
ð ARZ ðx; tÞ; v ARZ ðx; tÞÞ the solution of ARZ model in the quadrant t > 0; x < 0 with
the above-mentioned Cauchy and boundary data.
Now let us assume an emission of the I th car, according to the aforementioned
FL
boundary data ð FL
b ; v b Þ, takes place at time t (see Sec. 3.1). Then, at this time, we
solve the following Riemann problem for ARZ in Riemann invariant coordinates
(Fig. 6):
8
<ðw ARZ ð0 ; t Þ; v ARZ ð0 ; t ÞÞ; x < 0;
:ðw ðt Þ; v ðt ÞÞ;
I
I
x > 0;
Math. Models Methods Appl. Sci. 2010.20:2349-2370. Downloaded from www.worldscientific.com
by NATIONAL AUTONOMOUS UNIVERSITY OF MEXICO (UNAM) on 04/02/20. Re-use and distribution is strictly not permitted, except for Open Access articles.
2366
C. Lattanzio & B. Piccoli
Fig. 5. Riemann problem for ARZFL coupled model before the restarting procedure: during
\continuous" dynamics, FL in°uences ARZ.
Fig. 6. Riemann problem for ARZFL coupled model at the restarting procedure: In discrete events,
ARZ in°uences FL through the emission procedure, and FL in°uences ARZ through the corresponding
Riemann problem.
where
4X
I ðt Þ ¼
xIþ1 ðt Þ
and wI ðt Þ is deduced from I and vI by using (5.1). Finally, we obtain in this way the
ARZ ð0 ; tÞ and FL ¼ ARZ ð0 ; tÞ, t new values v FL
t to be used for the next
b ¼v
b
emission and the dynamics is completely de¯ned.
5.3. The Cauchy problems
In this section we prove the existence for the Cauchy problem of the coupled models
previously de¯ned. To this aim, let us ¯x initial data as follows:
(ID) for the coupled model ARZ–FL (resp. FL–ARZ), consider initial data for ARZ
model 0 ; y0 2 BV and in the invariant region (2.10), de¯ned in x < 0 (resp. x > 0),
Math. Models Methods Appl. Sci. 2010.20:2349-2370. Downloaded from www.worldscientific.com
by NATIONAL AUTONOMOUS UNIVERSITY OF MEXICO (UNAM) on 04/02/20. Re-use and distribution is strictly not permitted, except for Open Access articles.
Coupling of Microscopic and Macroscopic Tra±c Models at Boundaries
2367
i ; vi with x
i > 0 (resp. x
i < 0), and admissible,
and initial data for FL model x
kþ1 x
k 4X for any k (see Remark 3.2).
namely x
As pointed out before, our notion of solution de¯ned below will satisfy the
equations for almost all t > 0, namely, excluding the discrete times in which a
restarting procedure will take place.
De¯nition 5.1. A solution of ARZFL model (resp. FLARZ model) with initial
data as in (ID) is given by bounded measurable functions ððx; tÞ; yðx; tÞÞ, satisfying
ARZ model in x < 0 (resp. x > 0) and for almost every time t > 0, and absolutely
continuous functions xi ðtÞ; vi ðtÞ satisfying FL model for almost every time t > 0 and
with xi ðtÞ > 0 (resp. ðxi ðtÞ < 0).
The existence of such solutions is stated in the next theorem.
Theorem 5.1. For any initial data as in (ID), there exist solutions of the previously
de¯ned coupled models
Proof. Case 1. The FL ARZ model. The boundary data ð ARZ
; v ARZ
Þ for the
b
b
ARZ model is piecewise constant, because it changes only when a car reaches the
boundary x ¼ 0 and therefore the existence of IBV problem follows from standard
results (see, for instance, Ref. 1). Moreover, since the solution ð ARZ ; y ARZ Þ is BV and
con¯ned in the invariant region (2.10), the traces needed to get the boundary data
FL
1 for any t > 0. Therefore the
ð FL
b ; v b Þ for the FL model are well-de¯ned and in L
existence of absolutely continuous solutions of the FL model is classical, because in
particular the ODE for the last car involves forcing term in L 1 .
{
{
Case 2. The ARZ FL model. For the ARZ model, we have piecewise continuous
boundary data, namely, the velocities of cars between di®erent emissions, and thus
again existence of IBV problem follows from standard results. Moreover, for the FL
model, we only have to check that the emission procedure will de¯ne admissible initial
conditions, according to Remark 3.2 (see also (ID)). To this aim, recall de¯nition (3.5),
for which we conclude that the emission occurs only when the car xIþ1 is at least at
distance X from x ¼ 0, being FL
b 1. In addition, from our de¯nition we also exclude
the case of emissions of moving cars xI at x ¼ 0 with distance 4X with respect to the
car xIþ1 , which would actually be admissible in FL (see again Remark 3.2). Indeed,
assume that emission happens exactly when the ¯rst car of FL is at distance X and it
has velocity zero. Then necessarily the density FL
b 1 in the interval after the previous
emission, and therefore we must have ARZ ð0 ; tÞ 1 in that interval. Moreover, since
the (invariant region of) ARZ does not admit states with maximal density and positive
velocity, we conclude that v FL
b 0 and thus the car is emitted with velocity zero.
6. Conclusions
The principal aim of the present discussion is the analytic de¯nition of a dynamic for a
coupled microscopicmacroscopic model in the study of tra±c °ows, besides previous
Math. Models Methods Appl. Sci. 2010.20:2349-2370. Downloaded from www.worldscientific.com
by NATIONAL AUTONOMOUS UNIVERSITY OF MEXICO (UNAM) on 04/02/20. Re-use and distribution is strictly not permitted, except for Open Access articles.
2368
C. Lattanzio & B. Piccoli
attempts in this direction, which has not been rigorously justi¯ed, due to manifest
di±culties at the interface between the two regimes. We conclude our investigations
with some observations on these issues.
(i) Looking at (3.5), we conclude that for the ARZ–FL coupling, the emission
procedure in FL happens when two quantities are equal, namely the mean of the
boundary value FL
b ðtÞ
Z
1 t FL
ðsÞds
t 0 b
and the \density" on the interval ½0; xIþ1 4X
:
xIþ1 ðtÞ
However, the latter is always decreasing, while the former is not monotone in
general. Thus we cannot expect to have continuous dependence of the solutions
w.r.t. initial conditions. Indeed, in speci¯c cases, a small change of the boundary
function FL
b could lead to a uniformly large shift in the emission time t , or even
avoid emission at all, violating continuous dependence.
(ii) The coupling between the two models presented here is clearly one possibility, well
justi¯ed in our opinion under the modeling point of view, but one can consider also
di®erent interface conditions between the microscopic and the macroscopic
regimes, together with di®erent choices for the models describing the two regimes.
(iii) A further compatibility with the FL model can be analyzed in the following
sense:
(a) Start with a FL model on the whole R;
(b) Use °uid equivalence to put half-plane in ARZ setting, using the pointwise
reconstruction of density and velocity in (3.3);
(c) Solve the microscopic–macroscopic model as stated above;
(d) Compare the resulting solution with the evolution of the original complete
FL model.
We expect to have in this way a good compatibility, because the typical interaction times of waves in the ARZ part is compatible with FL model (see
Sec. 4), thus the dynamic of the two model should be su±cienlty close.
(iv) Besides the di±culties already mentioned in point (i), we expect to be able to
derive easily microscopic–macroscopic numeric schemes from this description,
because we have a natural candidate for the interface conditions between the
two regimes (regarding this issue, see also Refs. 16 and 17). However, stability of
schemes is not expected to hold in general and this issue has to be duly investigated. The obvious advantage of our situation is to have an \analytic background" for the choice of interface conditions, given by the de¯ned analytic
solutions of the coupled models.
Math. Models Methods Appl. Sci. 2010.20:2349-2370. Downloaded from www.worldscientific.com
by NATIONAL AUTONOMOUS UNIVERSITY OF MEXICO (UNAM) on 04/02/20. Re-use and distribution is strictly not permitted, except for Open Access articles.
Coupling of Microscopic and Macroscopic Tra±c Models at Boundaries
2369
References
1. D. Amadori and R. M. Colombo, Continuous dependence for 2 2 conservation laws with
boundary, J. Di®erential Equations 138 (1997) 229266.
2. A. Aw, A. Klar, T. Materne and M. Rascle, Derivation of continuum tra±c °ow models
from microscopic follow-the-leader models, SIAM J. Appl. Math. 63 (2002) 259278.
3. A. Aw and M. Rascle, Resurrection of \second-order" models of tra±c °ow, SIAM J.
Appl. Math. 60 (2000) 916938.
4. N. Bellomo, M. Delitala and V. Coscia, On the mathematical theory of vehicular tra±c
°ow. I. Fluid dynamic and kinetic modelling, Math. Models Methods Appl. Sci. 12 (2002)
18011843.
5. N. Bellomo, A. Marasco and A. Romano, From the modelling of driver's behavior to
hydrodynamic models and problems of tra±c °ow, Nonlinear Anal. Real World Appl. 3
(2002) 339363.
6. S. Benzoni-Gavage and R. M. Colombo, An n-populations model for tra±c °ow, Eur. J.
Appl. Math. 14 (2003) 587612.
7. F. Berthelin, P. Degond, M. Delitala and M. Rascle, A model for the formation and
evolution of tra±c jams, Arch. Rational Mech. Anal. 187 (2008) 185220.
8. R. M. Colombo, Hyperbolic phase transitions in tra±c °ow, SIAM J. Appl. Math. 63
(2002) 708721 (electronic).
9. C. Daganzo, Requiem for second-order °uid approximation to tra±c °ow, Transport. Res.
B 29 (1995) 277286.
10. M. Delitala and A. Tosin, Mathematical modeling of vehicular tra±c: A discrete kinetic
theory approach, Math. Models Methods Appl. Sci. 17 (2007) 901932.
11. M. Garavello and B. Piccoli, On °uido-dynamic models for urban tra±c, Netw. Heterog.
Media 4 (2009) 107126.
12. D. Gazis, R. Herman and R. Rothery, Nonlinear follow-the-leader models of tra±c °ow,
Oper. Res. 9 (1961) 545567.
13. P. Goatin, The AwRascle vehicular tra±c °ow model with phase transitions, Math.
Comput. Model. 44 (2006) 287303.
14. J. M. Greenberg, A. Klar and M. Rascle, Congestion on multilane highways, SIAM J.
Appl. Math. 63 (2003) 818833 (electronic).
15. D. Helbing, Improved °uid dynamic model for vehicular tra±c, Phys. Rev. E 51 (1995)
31643169.
16. D. Helbing, From microscopic to macroscopic tra±c models, in A Perspective Look at
Nonlinear Media. From Physics to Biology and Social Sciences, eds. J. Parisi et al.
(Springer-Verlag, 1998), pp. 122139.
17. D. Helbing, A. Hennecke, V. Shvetsov and M. Treiber, Micro- and macro-simulation of
freeway tra±c, Math. Comput. Model. 35 (2002) 517547.
18. A. Klar and R. Wegener, A hierarchy of models for multilane vehicular tra±c. I. Modeling,
SIAM J. Appl. Math. 59 (1999) 9831001 (electronic).
19. M. J. Lighthill and G. B. Whitham, On kinematic waves. II. A theory of tra±c °ow on
long crowded roads, Proc. Roy. Soc. London Ser. A 229 (1955) 317345.
20. S. Moutari and M. Rascle, A hybrid Lagrangian model based on the AwRascle tra±c
°ow model, SIAM J. Appl. Math. 68 (2007) 413436.
21. I. Prigogine and R. Herman, Kinetic Theory of Vehicular Tra±c (Elsevier, 1971).
22. P. I. Richards, Shock waves on the highway, Oper. Res. 4 (1956) 4251.
23. M. Sable-Tougeron, Methode de Glimm et probleme mixte, Ann. Inst. H. Poincare Anal.
Non Lineaire 10 (1993) 423443.
24. D. Serre, Systems of Conservation Laws 2. Geometric Structures, Oscillations, and
InitialBoundary Value Problems (Cambridge Univ. Press, 2000).
Math. Models Methods Appl. Sci. 2010.20:2349-2370. Downloaded from www.worldscientific.com
by NATIONAL AUTONOMOUS UNIVERSITY OF MEXICO (UNAM) on 04/02/20. Re-use and distribution is strictly not permitted, except for Open Access articles.
2370
C. Lattanzio & B. Piccoli
25. F. Siebel and W. Mauser, On the fundamental diagram of tra±c °ow, SIAM J. Appl.
Math. 66 (2006) 11501162 (electronic).
26. F. Siebel and W. Mauser, Synchronized °ow and wide moving jams from balanced
vehicular tra±c, Phys. Rev. E 73 (2006) 10.
27. H. M. Zhang, A non-equilibrium tra±c model devoid of gas-like behavior, Transport.
Res. B 36 (2002) 275290.