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## Abstract

This paper deals with the Aw-Rascle-Zhang model for traffic flow on uni-directional road networks. For the conservation of the mass and the generalized momentum, we construct weak solutions for Riemann problems at the junctions. We particularly focus on a novel approximation to the homogenized pressure by introducing an additional equation for the propagation of a reference pressure. The resulting system of coupled conservation laws is then solved using an appropriate numerical scheme of Godunov type. Numerical simulations show that the proposed approximation is able to approximate the homogenized pressure sufficiently well. The difference of the new approach compared with the Lighthill-Whitham-Richards model is also illustrated.
arXiv:2005.12060v1 [math.NA] 25 May 2020
Second-order traﬃc ﬂow models on networks
Simone G¨ottlichMichael HertySalissou MoutariJennifer Weißen
May 26, 2020
Abstract
This paper deals with the Aw-Rascle-Zhang model for traﬃc ﬂow on uni-directional
road networks. For the conservation of the mass and the generalized momentum, we
construct weak solutions for Riemann problems at the junctions. We particularly focus on
a novel approximation to the homogenized pressure by introducing an additional equation
for the propagation of a reference pressure. The resulting system of coupled conservation
laws is then solved using an appropriate numerical scheme of Godunov type. Numerical
simulations show that the proposed approximation is able to approximate the homogenized
pressure suﬃciently well. The diﬀerence of the new approach compared with the Lighthill-
Whitham-Richards model is also illustrated.
AMS Classiﬁcation. 35L65, 90B20, 65M08
Keywords. Conservation laws on networks, Aw-Rascle-Zhang model, homogenized pressure
1 Introduction
Macroscopic modeling of traﬃc ﬂow has been of research interest over the past decades, since
the ﬁrst order Lighthill-Whitham-Richards (LWR) model [22,23]
tρ+x(ρV (ρ)) = 0 (1)
was introduced in the middle of the 20th century. Aw-Rascle-Zhang [2,26] proposed a second
order traﬃc ﬂow model (ARZ model), which accounts for a more detailed description of traﬃc
phenomena. The ARZ model is given by
tρ+x(ρv) = 0
t(ρw) + x(ρvw) = 0.(2)
It combines a conservation law for the density ρwith an additional conservation law for the
mean traﬃc speed v. The Lagrangian marker wis given by
w=v+p(ρ),(3)
where pis a pressure function. The LWR model can be viewed as a special case of the second
order model, where the speed of traﬃc is always at the level of the equilibrium speed V(ρ) and
University of Mannheim, Department of Mathematics, 68131 Mannheim, Germany (goettlich@uni-
mannheim.de, jennifer.weissen@uni-mannheim.de)
RWTH Aachen University, IGPM, 52064 Aachen, Germany (herty@igpm.rwth-aachen.de)
Queen’s University Belfast, BT7 1NN Belfast, United Kingdom (s.moutari@qub.ac.uk)
1
all the drivers possess the same Lagrangian marker w=V(ρ) + p(ρ) = w0. In comparison to
the LWR model, which is in general not able to capture traﬃc instabilities, the ARZ model
enables to portray traﬃc phenomena such as growing traﬃc waves and the capacity drop
eﬀect at junctions .
Coupling or boundary conditions are required to extend equations 1or (2) to models for traﬃc
ﬂow on road networks. The appropriate junction modelling is of utmost importance to model
traﬃc dynamics and it has been the focus of recent research in the ﬁeld, see e.g. [5,10,12]. The
proposed models for traﬃc are generally based on the conservation of mass at the junctions.
At the same time, further conditions are imposed to obtain a unique boundary condition. As
an example, the non–negative ﬂux ρv at the junction is distributed according to given ratios
and the total ﬂux through the junction is maximized according to the speciﬁed ratios. While
many studies, including [9,10,12,13,15,19], use the LWR model 1to investigate traﬃc
modelling on road networks, we are interested in models based on the ARZ model (2). Further,
we are interested in a description in terms of the so-called macroscopic quantities (ρ, ρv).Since
macroscopic models are known to be coarse in the description of traﬃc dynamics, especially
at junctions, the maximization of the ﬂux at junctions may sometimes lead to unrealistically
high values of ﬂux ρv. To address these shortcomings, hybrid models, providing a more
accurate description of traﬃc dynamics at the junctions, have been introduced [4,16,25].
These models are computationally more expensive and we refer to the cited literature for a
more detailed discussion.
Here, we are interested in junction models for the ARZ model. Conditions have been con-
sidered for example in [7,10,12,14,17,18,20,21]. A detailed review would be beyond the
scope of the paper, however, note that the Riemann solver at the junction proposed in 
does not conserve the pseudo momentum and therefore does not yield a weak solution to
the ARZ model (2). Except for [17,18], all proposed coupling conditions [14,20,21] use
a ﬁxed pressure function p=p(ρ), see equation (3) on each road of the network. The cou-
pling conditions therein conserve both the mass and a (predeﬁned) mixture of the Lagrangian
attribute w. Speciﬁcally, the case of a 2-1 merging junction, where the Riemann solver is
based on ﬁxed assigned coeﬃcients, was presented . The ﬂux through the junction was
maximized through a multi-objective maximization of the incoming ﬂuxes. Other examples
 also propose conversation of mass and a mixture of the Lagrangian marker. However,
as outlined in , those solutions are not consistent with the Lagrangian form of equation
(2). Without entering the discussion in detail, the crucial point is that any change in the
value of winduces also a change in the corresponding pressure paccording to equation (3).
This leads to coupling conditions at junctions that not only prescribe coupling or boundary
conditions in terms of (ρ, ρv) but also in the form of the pressure p, see e.g. [17,18]. This
interpretation is also consistent with a discretization in Lagrangian coordinates leading to the
so–called follow-the-leader model . In order to pass from a microscopic to a macroscopic
limit, only low regularity is required on the Lagrangian marker w. The homogenization limit
of the system (2) was investigated in , where the homogenized relationship between the
traﬃc density, velocity and Lagrangian marker through the pressure function was established.
In [17,18], coupling conditions for the pressure using the homogenized pressure have been
established. Therein, the derivation of the homogenized pressure for the 2-1 merge junction
and the situation where the initial pressure p(ρ) is given by p(ρ) = ρhas been illustrated.
It should also be noted that whenever the incoming w-values or the mixture rule changes,
the homogenized pressure changes. This leads to computational challenges and to the best
of our knowledge, an explicit formula to determine the adapted pressure and a numerical
2
scheme complying with these changes are yet to be established. In this paper we propose a
numerical approximation to the homogenized pressure in order to obtain a computable but
still consistent traﬃc ﬂow model based on the ARZ equations (2).
This paper focuses on the following three important aspects. First and in contrast with ,
the solution proposed in this study is a weak solution of the network problem and therefore
satisﬁes the conservation of mass and generalized momentum. Thus, the quantities ρv and
ρwv are conserved through the junction. Second, compared to [14,20,21], this study
considers the homogenization of the pressure at the junction and, hence it is consistent with
the formulation of the model in Lagrangian coordinates and to the microscopic follow-the-
leader model. Finally, in contrast [17,18], we present an approach for approximating the
homogenized pressure. Such an approach allows for numerical simulation of traﬃc in an entire
road network and the corresponding computational results are presented.
After the presentation of the approximation of the homogenized pressure, we discuss the prop-
erties of the homogenized system and construct the demand and supply functions. These are
used to determine the ﬂux at the junction and allow for an appropriate description of the
boundary conditions. Using the approximation of the homogenized pressure in the supply
function, admissible states at the junction are deﬁned and the network solution is constructed.
We provide a suitable numerical scheme for simulating traﬃc on road networks, which is easy
to implement and based on a Godunov discretization. It is well known, that the classical
Godunov scheme produces nonphysical oscillations near contact discontinuities . These os-
cillations then lead to numerical solutions that do not precisely capture the Riemann invariants
of the system (2). Since an accurate description of the Riemann invariants is of paramount
importance for our network model, we leverage the non-conservative scheme from , which
was speciﬁcally designed for the system (2), and adapt it to our network model. The numerical
examples show that the solution is suﬃciently close to the homogenized solution.
The outline of the paper is as follows: In Section 2, we exemplify some coupling conditions
for the LWR and the AR models, and we deﬁne weak entropy solutions for the corresponding
Riemann problem. As in , the need for the homogenization of the pressure function after
merge type junctions is shown. In Section 3, we provide a simple and suitable approxima-
tion for the adapted pressure function, and we show that with our approach, we can indeed
approximate the homogenized pressure and the correct homogenized solution. The approx-
imation is then generalized for the general n-m-junction and the quality of the solution is
assessed through a comparison of the ﬂux with true and approximated homogenized pressure
for the 2-1-junction. A numerical scheme, which accounts for the variation of the pressure, is
introduced in Section 4. Finally, we compare our network solution against the solution with
the homogenized pressure in Section 5. Furthermore, the numerical results highlight a more
accurate description of traﬃc dynamics using the proposed approach compared to the LWR
network model.
2 Coupling conditions for traﬃc ﬂow networks
A road network is modeled as a directed graph G= (V, E ). Each edge iEcorresponds to
a road, which is modeled as an interval Ii= [ai, bi] with length Li=biai. The junctions
are represented by the nodes kV. For a given junction k, let δ
k(resp. δ+
k) denote the
set of indices representing incoming (resp. outgoing) roads to (resp. from) the junction k.
On each road iEof the network, a traﬃc network model given by 1or (2) is required to
3
hold. Furthermore, some initial data are assumed to be described on each road. Coupling
conditions at the junctions are imposed to deﬁne suitable boundary conditions for the traﬃc
model at hand. Those conditions will be discussed in detail in the following section.
2.1 The Lighthill-Whitham-Richards model
On each road iEof the network, we require the following equation to hold:
tρi+x(ρiVi(ρi)) = 0.(4)
The traﬃc density and velocity on road iare denoted ρi=ρi(x, t) and vi=V(ρi(x, t)),
respectively. The velocity is a function of the density and for a maximum density ρmax
iit holds
that Vi(ρmax
i) = 0. Moreover, the ﬂux ρVi(ρ) is strictly concave and has a unique maximum
σi. For a given junction k, let {Φi}i(δ
kδ+
k)denote a family of smooth test functions, where
Φi:Ii×[0,+]R2has a compact support in Iiand is also smooth across the junction,
i.e.,
Φi(bi,·) = Φj(aj,·)iδ
k,jδ+
k.
A set of functions ρi, i (δ
kδ+
k), is called a weak solution of (4) at the junction kif, for all
families of test functions smooth across the junction, the following equation holds:
X
i(δ
kδ+
k)Z
0Zbi
ai
[ρi·tΦi+ρivi·xΦi] dxdtZbi
ai
ρi,0·Φi(0, x)dx= 0.(5)
In the above equation, ρi,0denotes the initial data. First, we provide a discussion of the
Riemann problem at a single junction k, located at x= 0. For each road i(δ+
kδ
k), we
consider the following (half-)Riemann problem :
tρi+x(ρivi) = 0
ρi(x, 0) = ρ+
ifor x > 0
ρ
ifor x0.!.(6)
Depending on whether the road is incoming or outgoing, only one of the Riemann data is
deﬁned for t= 0. If iδ
k, then ρ
i=ρi,0, bi= 0 and if iδ+
k, then ρ+
i=ρi,0,which will be
the case ai= 0. The other datum is deﬁned by the solution through some suitable coupling
conditions be discussed below.
2.1.1 Nodal conditions for the LWR model
In this section, we discuss the construction of weak entropy solutions for the Riemann prob-
lem (6) for traﬃc networks. On each road, we consider the LWR model. To deﬁne the solution
at the junction, we consider a Riemann solver giving an admissible, weak entropy solution,
see also [9,10,12,19]. The entropy criterion is expressed by a demand and supply formu-
lation for admissible ﬂux values. We impose additional conditions on the ﬂux distribution
from incoming to outgoing roads in the network. Using a ﬂux maximization with respect
to the additional conditions, we obtain the network solution. In the following, we specify
4
the constraints on the network solution. From (5), a weak solution satisﬁes the Kirchhoﬀ
condition
X
iδ
k
(ρivi)(0, t)
|{z }
:=qi
=X
jδ+
k
(ρjvj)(0+, t)
|{z }
:=qj
.(7)
Let qji R0for jδ+
k, i δ
kdenote the initially unknown ﬂux coming from road iand
entering road jand denote by qiand qjthe total ﬂuxes at the junction:
qi=X
jδ+
k
qji, qj=X
iδ
k
qji.
The following constraint (H1) speciﬁes the admissible ﬂuxes for entropy solutions. Conditions
for the ﬂux distributions between diﬀerent roads of the network will be given in (H2)-(H3)
below.
(H1) The ﬂuxes at the junction are bounded by demand diand supply sj
0qidi(pi)iδ
k,0qjsj(ρj)jδ+
k,(8)
where the demand, di(ρ), and the supply, si(ρ), on road iare deﬁned as follows:
di(ρ) = (ρVi(ρ) if ρσi
σiVi(σi) if ρ > σi
, si(ρ) = (σiVi(σi) if ρσi
ρVi(ρ) if ρ > σi
(9)
(H2) Consider a junction with nincoming and moutgoing roads. As in [9,10,12,18,19],
a traﬃc distribution matrix A= (αji)iδ
k,jδ+
kis assumed to be known. It describes the
distribution of traﬃc at the junction, where 0 αj i 1 denotes the percentage of cars on
kαji = 1. The ﬂuxes at the junction must satisfy the
following equality qji =αj iqi.
For junctions with more than one incoming road, conditions (H1)-(H2) are not suﬃcient to
determine unique ﬂux values at the junction. A further constraint has to be introduced to
single out a solution. Here, we impose the following mixture rule, see also [12,18].
(H3) On an outgoing road j, the ﬂuxes ~qi= (qi)iδ
kare proportional to a given priority
vector ~
β= (βij )iδ
k. Hence, we impose the following constraint:
~qi=z~
β, z R.(10)
We require the network solution to be the ﬂux maximizing weak solution, subject to con-
straints (H1)-(H3). The existence of the network solution is shown in the references [12,19].
2.2 The Aw–Rascle–Zhang model
On each road of the network, we require the following system to hold:
tρi
ρiwi+xρivi
ρiwivi= 0,(11)
where ρiand viare the density and the velocity on road i, respectively. The Lagrangian
marker, wi, is deﬁned by wi=vi+pi(ρi).The form of the pressure function pidepends on
5
the initial data and the type of the junction. For each i,pi,0(ρ) is a pressure function, which
is initially given and for which the ﬂux ρ(wpi,0(ρ)) has a unique maximum σi,0. A set of
functions Ui= (ρi, wi), i (δ
kδ+
k) is called a weak solution of (2) at the junction kif, for
all families of test functions smooth across the junction, the following equation holds:
X
i(δ
kδ+
k) Z
0Zbi
ai ρi
ρiwi·tΦi+ρivi
ρiviwi·xΦidxdt
Zbi
aiρi,0
ρi,0wi,0·Φi(0, x)dx!= 0.
(12)
We refer to  for an analysis of the model (11). As before, for each i(δ+
kδ
k), we consider
the following (half-)Riemann problem:
t ρi
ρiwi!+x ρivi
ρiwivi!= 0,
(ρi, wi) = (ρ+
i, w+
i) for x > 0
(ρ
i, w
i) for x0.!.
(13)
As in the case of the LWR model, depending on whether the road is incoming or outgoing, only
one of the Riemann data is deﬁned for t= 0. In the following, we denote with U= (ρ, ρw)
the traﬃc state deﬁned by the density ρand the Lagrangian marker w.
2.2.1 Nodal conditions for the ARZ model
In this section we state conditions for a network solution to the ARZ model. To deﬁne a
unique network solution for the Riemann problem (13) of the 2 ×2 system (2), additional
conditions compared to the Riemann problem of the scalar conservation law (6), are necessary,
see also [14,17,18,20]. From (12), a weak solution for the ARZ model is required to satisfy
the Kirchhoﬀ condition for the conservation of mass (7) as well as momentum:
X
iδ
k
(ρiviwi)(0, t) = X
jδ+
k
(ρjvjwj)(0+, t).(14)
Additionally, the homogenization of the Lagrangian marker given by the rule (H4) below has
to be considered [17,18].
(H4) The homogenized w-value on an outgoing road jis motivated by the underlying mi-
croscopic model, see section 6 in . Cars passing trough the junction have the average
property wjassociated with the Young measure µxdescribing the mixture of cars. Once the
proportions βij =qji/qjare known, the homogenized Lagrangian marker wjis given by
wj:= Zwdµx(w) = X
iδ
v
qji
qj
wi(Ui,0) = ~
β ~wTjδ+
k.(15)
The unique weak entropy solution of (11) is characterized by the relation
τ=X
iδ
k
qji
qj
P1
j(wi,0v) =: (P
j)1(wjv),(16)
6
see . The pressure P
j(τ) is, in fact, redeﬁned such that for each τ, the velocity vis the
velocity of the weak entropy solution of the Lagrangian formulation of the ARZ equations.
This solution is then equivalent to the solution of the ARZ equations (11). The modiﬁed
pressure function and the Lagrangian marker are deﬁned as follows:
p
j(ρ) = P
j(1),(17)
wj(U) = wj=v+p
j(ρ).(18)
The new pressure function corresponds to the homogenized Lagrangian marker wj, and for
diﬀerent values w6=wj, a diﬀerent pressure pis obtained. The construction of the new
pressure function is well-deﬁned once the proportions qj i
qjof the incoming ﬂuxes are known.
The total ﬂuxes qiand qjfulﬁll similar bounds in terms of the demand and supply as in the
case of the LWR model. More precisely, we replace (H1) by
(H1*) The ﬂuxes at the junction are bounded by the demand diand the supply sj
0qidi(pi;ρi,0, wi,0)iδ
k0qjsj(p
j; ˜ρj,wj)jδ+
k.(19)
Here, the density ˜ρjare deﬁned below in Section 4. The value wjis the homogenized La-
grangian marker given by equation (15). Furthermore, the pressure p
jis the homogenized
pressure given by equation (17).
The demand and supply functions for the ARZ model are given by
di(p;ρ, w) = (ρ(wp(ρ)) if ρσi(w)
σi(w)(wp(σi(w))) if ρ > σi(w),
si(p;ρ, v) = (σi(w)(wp(σi(w))) if ρσi(w)
ρ(wp(ρ)) if ρ > σi(w),
where σi(w) is the unique maximum of the ﬂux ρ(wpi(ρ)). For a given state U= (ρ, ρw),
the demand on the incoming road is evaluated using w(U) = v+pi(ρ) whereas the supply on
the outgoing road uses w(U) = v+p
j(ρ).
We refer to the reference  (section 7, Theorem 7.1) for the proof of existence and uniqueness
of the network solution.
3 Approximation of homogenized pressure
Suitable conditions for ﬁnding the network solution have been proposed in , where a full
discussion of the Riemann problem was presented. It is assumed that the proportions qji/qj
of the incoming ﬂuxes at the junction are known and ﬁxed. The homogenized ﬂow on parts
of the outgoing roads is derived. However, even in the case where initially Riemann data are
available on all roads, the pressure law on the outgoing roads changes over time. Further, the
adapted pressure function deﬁned by (17) is not obvious to ﬁnd, due to the nonlinear and
implicit relationship between (15) and (16). The coupling conditions suggested by [17,18]
require the adapted pressure function to be recalculated for any change in the homogenized
Lagrangian marker w=~
β ~wT. Due to this complexity, the adaption of the pressure law in
numerical simulations for traﬃc networks has not been considered so far. In this work, we
suggest a practical approach for the simulation of traﬃc networks, which accounts for the
changes in the pressure law on outgoing roads. We present an approximation for computing
7
the adapted pressure, whenever the homogenized Lagrangian marker at the junctions changes.
To achieve this, we replace (H4) by the following condition on the pressure:
(H4*) Assume that the pressure on an outgoing road is initially given by p(ρ), the La-
grangian markers on the roads entering the junction are given by ~w and the priority vector is
~
β. The homogenized Lagrangian marker is given by (15) and we approximate the homogenized
pressure (17) as follows:
p(ρ)p∗∗(ρ) = c(~
β, ~w)p(ρ).(20)
The value of c(~
β, ~w) will be deﬁned below. Its value will be transported with the outgoing
velocity. Approximation properties and further details will be discussed in the following
sections.
3.1 Properties of the homogenized system
Assuming the pressure law is of the type c(~
β, ~w)p(ρ) with a value c(~
β, ~w) independent of ρ,
the pressure propagates with the velocity vof the cars, according to equation (11). However,
the value of cmight change over time due to the coupling at the junction. This will change
the value of ccorresponding to the value of the homogenized Lagrangian marker w. Once
this marker enters the outgoing road it is transported with the ﬂow. The idea is to propose
an additional advection equation to the original system (2) to propagate the value of c(~
β, ~w)
together with the marker w. The value of c(~
β, ~w) is then used as correction to the pressure
p(ρ).We will show below that this correction c(~
β, ~w)p(ρ) is in fact an approximation to the
homogenized pressure in the sense of equation (20).
If cis transported similarly to w, then in conservative form, the system reads:
t
ρ
ρw
ρc
+x
ρv
ρwv
ρcv
= 0.(21)
This system (with a diﬀerent motivation) has been studied in . In contrast with , here we
do not work with the Lagrangian formulation, but instead consider the Eulerian formulation.
The meaning of cis also diﬀerent compared with . Nevertheless the mathematical properties
are the same and will be recalled here for convenience. The system (21) is hyperbolic with
eigenvalues λ1=vp(ρ)ρ < v =λ2=λ3.
The ﬁrst ﬁeld is genuinely nonlinear and the second and third ﬁeld are linearly degenerate.
The Riemann invariants associated with their corresponding eigenvalue are (w, c), (v, c) and
(v, ρ), respectively. We study the solution to the Riemann problem, i.e. the initial value
problem with constant initial data (ρ0, w0, c0), for the system (21). The solution, for some
given constant initial data, consists of a shock or rarefaction wave associated with the ﬁrst
eigenvalue, followed by a contact discontinuity associated with the second and third eigenvalue
(2-3-contact discontinuity). The following proposition summarizes the Riemann problem in
Eulerian coordinates (see  for the Lagrangian formulation):
Proposition 1. Consider the Riemann problem
8
t
ρ
ρw
ρc
+x
ρv
ρwv
ρcv
= 0
(ρ, w, c)(x, 0) = (ρ+, w+, c+)for x > 0
(ρ, w, c)for x0.!.
(22)
The solution U(x, t) = (ρ, w, c)(x, t), to the system (22), is as follows:
(i) We connect Uto an intermediate state ˜
U= (˜ρ, ˜w, ˜c)such that ˜w=w,˜c=c,˜v=v+
holds. The wave connecting Uand ˜
Uis either a 1-shock if v+< vand a 1-rarefaction
if v+> v. The states ˜
Uand U+are then connected by a 2-3 contact discontinuity of
velocity v+.
(ii) Additionally, wand ctake only the values (w, c)and (w+, c+). The velocity vis a
monotone function of x/t with min{v, v+} ≤ v(x, t)max{v, v+}and for x > tv+,
we always have U(x, t) = U+.
(iii) U(x, t)and v(x, t)remain in an invariant region Raway from the vacuum
R:= {(ρ, w, c)|(v, w, c)[vmin , vmax]×[wmin, wmax ]×[cmin, cmax]}
Subsequently, we will refer to this solution, with our pressure approximation, as the ”Adapted
Pressure ARZ model” (AP). We now deﬁne the network solution.
Deﬁnition 1 (Network solution AP).Consider a junction kwith nincoming and moutgoing
roads, with constant initial data Ui,0, i (δ
kδ+
k). We say that the family {Ui(x, t)}i(δ
kδ+
k)
is an admissible solution to the Riemann problem (13)with approximated homogenized pres-
sure if and only if
i(δ
kδ+
k),Ui(x, t)is a weak solution of the network problem (12)where the
pressure is piiδ
k. On an outgoing road jδ+
k, the solution in the triangle
{(x, t)|0< x < t vj,0}is the approximated homogenized solution with pressure p∗∗
j. For
{(x, t)|t vj,0< x}, the pressure is pj,0;
the constraint (20)for the pressure is satisﬁed (H4*), and the homogenized w-value is
given by wj:= Piδ
k
qji
qjwi(Ui,0)jδ+
k;
the sum of the incoming ﬂuxes is maximal subject to (H1*)-(H2) and they satisfy the
mixture rule (H3).
We remark that in comparison to the ARZ model, we only replace the true homogenized
pressure pby the approximation p∗∗.
Note that due to (H4*), the pressure law is always of the form cp(ρ). Thus, the demand and
supply function in (H1*) can be simpliﬁed to the compact form (23)-(24)
DAP(ρ, w, c) = ((wcp(ρ))ρif ρσ(w, c),
(wcp(σ(w, c)))σ(w, c) otherwise, (23)
SAP(ρ, w, c) = ((wcp(σ(w, c)))σ(w, c) if ρσ(w, c),
(wcp(ρ))ρotherwise. (24)
9
3.2 The case of the n-1-junction
To exemplify, we consider the case iδ
k,|δ
k|=nand jδ+
k,|δ+
k|= 1.We describe the
approximation of the pressure p∗∗
j. Assume that initially, the pressure is given by pj,0(ρ) =
cj,0ργ, j δ+
k, which is the prototype pressure function used in the literature [2,11]. Assume
that the mixture of incoming ﬂows complies with the mixture rule ~
β= (βi)i=1,...,n with
Pn
i=1 βi= 1. Together with the vector of incoming Lagrangian markers ~w = (wi)i=1,...,n, the
homogenized value w(15) is given by
wj=
n
X
i=1
βiwi, j δ+
k.(25)
We approximate the homogenized pressure p
jby p∗∗
jas in equation (20). We deﬁne the
approximated pressure therefore by
p∗∗
j(ρ) = c(~
β, ~w)pj,0(ρ) = c(β1,...βn, w1,...wn)cj,0ργ:= γ,
where cis given by equation 28.
Lemma 1. Consider a junction kwith nincoming roads and a single outgoing road, with
constant initial data Ui= (ρi,0, wi,0), i (δ
kδ+
k), initial pressure p(ρ) = cj,0ργ, γ 1on
the outgoing road j=n+ 1 and the rules (H1*),(H2),(H3),(H4*). The approximation of the
homogenized pressure (20)on the outgoing road is given by
p∗∗
j(ρ) = cj,0 n
X
i=1
βiwi n
X
l=1
βl
w1
l!γ!ργ.(26)
Then there exists a unique network solution in the sense of Deﬁnition 1.
Proof. For γ1, we have Pj(τ) = pj(1) = pj(ρ) = cj,0ργand τfollowing (16) is deﬁned
by the relationship
τ=ZP1
j(wiv) dµx=
n
X
i=1
βicj,0
wiv1
γ
,
where xdescribes the mixture with priorities βi, i = 1,...,n. Due to (18), the homogenized
Lagrangian marker wis the sum of the velocity vand the homogenized pressure p
j,i.e.,
wj=v+p(ρ).According to assumption (H4*) we have
wj=v+p∗∗(ρ)(20)
=v+c(~
β, ~w)cj,01
τγ
=: v+c1
τγ
,(27)
with a function cdepending only on ~
βand ~w. Those are quantities known due to the initial
data. Comparing now the value of wjand cwe propose the following choice for c:
c=cj,0 n
X
i=1
βiwi n
X
l=1
βl
w1
l!γ!.(28)
Once the pressure p∗∗ for the outgoing road with the priority rule ~
βgiven by (H3) is deter-
mined, we can determine the ﬂux through the junction, which is maximal subject to (H1*),
10
with the demand and supply functions (23)-(24). Since there there is a unique outgoing road,
the ﬂuxes leaving the incoming roads have to exit the junction at the outgoing road by rule
(H2). On the incoming roads, the pressure law is unchanged and the solution Ui(x, t) is the
solution of a (half-)Riemann problem (13). On the outgoing road, the pressure function p∗∗
j(ρ)
possesses the same properties as the prototype pressure function pj,0(ρ). In particular, the
ﬂux function ρ(wjp∗∗
j(ρ)) is concave and has a unique maximum σ(wj, c). The state U
j
corresponding to the ﬂux value exiting the junction is therefore uniquely determined. Away
from the junction, the solution is the unique entropy solution of the Riemann problem (22)
given in Proposition 1with U
jon the left and U+
j=Uj,0on the right.
The crucial point in the previous proof is the deﬁnition of cgiven by equation (28). This choice
is motivated by the following consideration. According to the result in  the homogenized
pressure p
jfor a ﬁxed value wjthe relation between v, w and ρon road jcan be expressed as
w=v+p
j(ρ).(29)
We may use the general Ansatz p(ρ) = C(ρ, v)ργto deﬁne p
j.Hence, the choice (28) implicitly
assumes that Cis independent of ρ. Then we may proceed as in the previous and obtain the
following functional dependence C=C(v)
C(v) = (wjv) n
X
i=1
βicj,0
wiv1
γ!γ
.(30)
Comparing now (30) and (28) we observe that the proposed choice is simply c=C(0),
i.e., setting v= 0 in equation (30). Obviously, other choices are possible. However, for the
presented approach it is crucial that the ﬁnal approximation p∗∗
jcontains a value cindependent
of the dynamic quantities(ρ, v, w).Only this fact allows to propagate the value cwith velocity
vand only this fact justiﬁes the additional equation in Proposition 1.
Example: The 2-1-junction, γ= 1:We state the explicit formula for p∗∗ for a junction
merging of two incoming roads i= 1,2 into a single outgoing road j= 3 Assume that ﬁxed
ratios β1=βand β2= 1 βare assigned to the incoming roads such that the homogenized
the Lagrangian marker is given by (25) and the pressure according to Lemma 1 by (32).
w3=β1w1,0+β2w2,0,(31)
p∗∗
3(ρ) = c3,0 2
X
i=1
2
X
l=1
βiβlwl,0
wi,0!ρ=c3,0 1 + β(1 β)(w1,0w2,0)2
w1,0w2,0!ρ. (32)
For the choice β= 0 or β= 1 the homogenized pressure p
3can be computed exactly and we
compare it with the approximation p∗∗
3. In this case p
3p∗∗
3.Furthermore, for w1,0=w2,0
and any value of βwe have p∗∗
3(ρ) = p
3(ρ) = p3,0(ρ). Figure 1illustrates numerically
the diﬀerences in pressure approximations for β= 0.5, w1,0= 9/2, w2,0= 7/2 and w3= 4.
Initially, the pressure is set to p3,0(ρ) = ρ. The homogenized pressure, p
3(ρ) is numerically
computed by the homogenization formula (17) and the pressure p∗∗
3(ρ) is obtained from (32).
3.3 The case of a n-m-junction
Consider a junction with i= 1,...n incoming roads and j= 1,...m outgoing roads. Using
distribution rates αji and priority coeﬃcients βij complying with the rules (H2)-(H3), we
determine the solution on each outgoing road using a construction similar to the 1-njunction.
11
01234
0
0.2
0.4
0.6
ρ
pressure diﬀerence
|p(ρ)p0(ρ)|
|p(ρ)p∗∗(ρ)|
(a) Approximation error
01234
0
1
2
3
4
ρ
ﬂux
for p0(ρ)
for p∗∗(ρ)
for p(ρ)
(b) Flux functions with diﬀerent pressures
Figure 1: Homogenized pressure pand its approximation p∗∗ as well as the initial pressure
p0.
Lemma 2. Consider a junction with nincoming roads and moutgoing roads, with constant
initial data Ui= (ρi,0, wi,0), i (δ
kδ+
k), initial pressures pj(ρ) = cj,0ργ, γ 1on the
outgoing roads j=n+ 1,...n+mand the rules (H1*), (H2), (H3), (H4*). Then, there exists
a unique network solution according to Deﬁnition 1and the approximation of the homogenized
pressure (20)on each outgoing road, setting v= 0, is given by
p∗∗
j(ρ) = cj,0 n
X
i=1
βij wi n
X
l=1
βlj
w1
l!γ!ργ.(33)
Proof. Once the ratios of the incoming ﬂuxes (βij )iδ
vfor an outgoing road jare ﬁxed, the
approximation of the homogenized pressure is determined using Lemma 1. On each outgoing
road j, the pressure is then given by (33).
4 Numerical solution procedure for the traﬃc network
We introduce the following notations used for the numerical solution procedure. Let Yi(x, t) =
(ρi, ρiwi, ρici)(x, t) denote the traﬃc state in conservative variables on road iat position x
and time t, whereas f(Yi) = (ρivi, ρiwivi, ρicivi) denotes the ﬂux. We introduce a grid in
time and space with step sizes ∆tand ∆xto discretize the system (21). Each road is divided
into Nxi=Li/xcells of equal size and we consider a ﬁnite number of time discretizations
Nt=T/t. Let Ii,j denote the open interval (xi,j0.5, xi,j +0.5) for j= 1,...Nxi, and let Ys
i,j
denote the average value of the function Yi(x, t) on the interval Ii,j at time ts=st, i.e.,
Ys
i,j =1
xZIi,j
Yi,j(x, ts) dx,
such that the following CFL condition is satisﬁed:
t
xmax
imax
j{|λl(Ys
i,j)|, l = 1,2,3} ≤ 1
2s= 1,...,Nt.(34)
12
4.1 The Transport-Equilibrium scheme
We discretize (21) using a transport-equilibrium scheme based on a Godunov discretization.
The analysis in 3highlighted the importance of the description of the Lagrangian marker
since its variation results in the recalculation of the pressure. The Godunov scheme is known
to have unphysical oscillations, at least for realistic grid sizes, which reduce the quality of
the numerical solution. In , it was shown, that the scheme does not comply with the
maximum principle on the Riemann invariants for the system (2). To better treat the 2-3-
contact discontinuity and ensure the correct depiction of the Riemann invariant w, we use the
transport equilibrium (TE) scheme  and expand it to the AP system. The TE scheme is a
wave splitting strategy where the contact discontinuities and the 1-waves evolve separately.
The ﬁrst step only accounts for the contact discontinuity and the second step focuses on the
1-wave. Let ˜
Y(Y, Y+) denote the intermediate state in conservative variables in the solution
to the Riemann problem in Section 3.1 with initial states Yon the left and Y+on the right.
The ﬁrst step is based on Glimm’s random sampling strategy. The intermediate value Ys+1/2
i,j
is determined by means of a well distributed random sequence (αs) within (0,1) and is set to
Ys+1/2
i,j =(˜
Y(Ys
i,j1, Y s
i,j) if αs+1 (0,t
xvs
i,j),
Ys
i,j if αs+1 [t
xvs
i,j,1).
Here, we use the van der Corput random sequence deﬁned by αs=Pz
l=0 il2(l+1),computed
using the binary expansion of the integer s=Pz
l=0 il2l,il∈ {0,1}. In other words, the random
sampling decides whether the intermediate value is set to the initial state or to the state of
a possibly present contact discontinuity. In either case, the correct ﬂuxes through the cell
interfaces have to be determined in the second step. In the absence of a contact discontinuity,
the scheme is equivalent to the Godunov scheme. Assuming the CFL condition (34), the
complete scheme is given by
Ys+1
i,j =Ys
i,j t
xFs+1/2,L
i,j+1/2Fs+1/2,R
i,j1/2,(35)
where the left and right numerical ﬂux functions are deﬁned as follows
Fs+1/2,L
i,j+1/2=Gs
i,j+1/2(Ys+1/2
i,j , Y s
i,j+1),
Fs+1/2,R
i,j1/2=(Gs
i,j1/2(Ys
i,j1, Y s+1/2
i,j ) if ˜
Y(Ys
i,j1, Y s+1/2
i,j ) = Ys+1/2
i,j ,
f(Ys+1/2
i,j ) otherwise.
The ﬂux terms Gare the usual Godunov ﬂuxes given by
Gs
i,j+1/2=Gs
i,j+1/2(Ys
i,j, Y s
i,j+1 ) =
qs
i,j+1
/2
ws
i,jqs
i,j+1
/2
cs
i,j qs
i,j+1
/2
,
where qs
i,j+1
/2= min{DAP (ρs
i,j, ws
i,j , cs
i,j ), SAP (˜ρs
i,j+1 , ws
i,j , cs
i,j )},
and ˜ρs
i,j+1 is either given by the intersection of the curves {wi(Y) = ws
i,j, ci(Y) = cs
i,j }and
{vi(Y) = vs
i,j+1}, if this intersection exists; otherwise ˜ρs
i,j+1 is set to zero.
13
4.1.1 The boundary conditions
The ﬂux terms Fs+1/2,R
i,j1/2for j= 1 and Fs+1/2,L
i,j+1/2for j=Nxe, in the TE scheme (35), are obtained
by coupling the boundary conditions. The ﬂux at the junction is determined by the coupling
condition, which depends upon the type of the junction. We illustrate the computation of the
boundary ﬂuxes at the 2-1-junction and the Riemann data Yi,0= (ρi,0, ρi,0wi,0, ρi,0ci,0), i =
1,2,3. Assume that the priorities are βand 1βand that the pressure on the outgoing road is
p3,0(ρ) = c3,0ρ. To determine the solution to the Riemann problem according to Deﬁnition 1,
we proceed as follows: We compute the incoming w-value w3given by (31) and the new
constant for the pressure function c3=c3,0c((β, 1β),(w1,0, w2,0)) deﬁned by Lemma 1once
at t= 0. The demand and the supply at the junction are
D1=DAP(ρ1,0, w1,0, c1,0)D2=DAP(ρ2,0, w2,0, c2,0)S3=SAP (˜ρ, w3, c3),(36)
where ˜ρis either given by the intersection of the curves {w(Y) = v+p∗∗(ρ) = w3, c3(Y) = c3}
and {v(Y) = v3,0}or ˜ρ= 0. The ﬂow into the outgoing road is given by
q3
w3q3
c3q3
where q3= min{D1/β, D2/(1 β), S3},
and the ﬂows at the end of the two incoming roads are given by
q1
w1q1
c1q1
where q1=βq3
q2
w2q2
c2q2
where q2= (1 β)q3.
The new pressure law on the outgoing road is p∗∗ (ρ) = c3ρ. We illustrate now, how to
determine the boundary states for s= 0. For the computation with the Godunov scheme,
the ﬂuxes q1, q2and q3would be suﬃcient, since the scheme is based purely on ﬂux terms. In
contrast, the computation of the ﬂux terms F1/2,R
i,1/2and F1/2,L
i,Nxi+1/2, in the TE scheme, requires
the density values of the states Y0
3,0and Y0
1,Nx1, Y 0
2,Nx2. We begin with the computation of
Y0
3,0. Once the ﬂux q3is determined, we can compute two states for which
f1(Y) = q3.(37)
Since waves on the outgoing road must have positive velocities, the state Ys
i,0is uniquely
determined as the state such that
λ1(Y)0.(38)
We can obtain the boundary state Y0
3,0= (ρ, ρw, ρc) deﬁned by (37)-(38) as follows.
ρ=w3
2c3
sw3
2c32
q3
c3
, w =w3, c =c3.
The ﬂux term F1/2,R
3,1/2can then be computed using random sampling and the states Y3,0, Y3,1,
see (35). Analogously, we proceed for the end of road i= 1,2, where we have to choose the
state Yi,Nxi+1 with negative ﬁrst eigenvalue
λ1(Y)0.(39)
14
Thus, for road 1, the boundary state Y0
1,Nx1is given by
ρ=w1
2c1
+sw1
2c12
q1
c1
, w =w1, c =c1,
and one can proceed similarly for the second road. The procedure for obtaining the boundary
states for other junctions is identical. Once the ﬂux qifor road iat a junction is determined,
we can compute the two states Y, such that f1(Y) = qiand choose the correct state Y
depending on whether the boundary state is evaluated at the beginning or the end of a road
i.
Algorithm 1 Numerical simulation of a network
Require: Roads i= 1, .., Ne, nodes k= 1, .., Nv. Initial data ρi(x, 0), wi(x, 0), ci(x, 0)
Ensure: Densities ρi(x, t), Lagrangian marker wi(x, t) and pressure coeﬃcients ci(x, t) for
all times ts=stwith s∈ {1,...,Nt}
1: for s= 0,1,...,Ntdo
2: for k= 1,...,Nvdo
3: if kis a merging junction and s= 0 or ws16=wsthen
4: Update coeﬃcient for the pressure according to Lemma 1.
5: else
6: Use pressure coeﬃcient of ts1.
7: end if
8: Compute the maximum ﬂux at the junction at time tswith demand and supply (23)-
(24) respecting the coupling condition (36).
9: end for
10: for i= 1,...Nedo
11: Determine as+1 and the boundary states Ys
i,0, Y s
i,Nxi+1 according to (37)-(39) from the
ﬂuxes at the junction. Compute the solution at time ts+1 with the TE scheme (35).
12: end for
13: end for
5 Computational results
In this section, we provide numerical results to compare our solution with the homogenized
solution using the TE scheme. We also provide numerical results in the case of time-dependent
boundary data and a comparison to the LWR solution.
5.1 The Riemann problem at a 2-1 junction
We investigate the 2-1-junction and compare our network solution to the solution with true
homogenized pressure to show that we can indeed approximate the truly homogenized solu-
tion.Figure 2a depicts the solution to a Riemann problem with ~
β= (0.5,0.5). The initial data
is (ρ1, w1, c1) = (3,2,1),(ρ2, w2, c2) = (2,1,1),(ρ3, w3, c3) = (3,2,1). As shown in Figure 2a,
the solution on the outgoing road with the approximated pressure p∗∗ is close to the solution
using pressure p, determined by the homgenization process described in . Moreover, the
15
exact solution of the Riemann problem with the approximated pressure is compared with the
solution given by the TE scheme in Figure 2b. Numerically, the pressure coeﬃcient is changed
once at t= 0 according to (32) for the Riemann problem since after the initial interaction
all waves emerge from the junction. The ﬂuxes at the junction are computed using (36).
The simulation procedure to obtain the solution on the whole network is summarized in Al-
gorithm 1. For decreasing ∆xand ∆t= ∆x/10 fulﬁlling the CFL condition (34), we see
convergence towards the exact solution of the Riemann problem as Figure 2b illustrates for
the 1-wave at the junction and the contact discontinuity in the numerical solution at t= 0.12.
0 0.1 0.2 0.3 0.4 0.5
1
2
3
4
x
ρ(x, t)
solution for p∗∗
solution for p
(a) True homogenized and AP solution
0 0.1 0.2 0.3 0.4 0.5
2
4
6
8
x
ρ(x, t)
sol. for p∗∗ x= 1/100
x= 1/200 ∆x= 1/400
(b) Grid solutions and AP solution
Figure 2: Solution on the outgoing road for β= (0.5,0.5).
5.2 Network of merging junctions
We consider a network, which consists of a sequence of 2-1-junctions (in total Nmerging
junctions), see Figure 3. Moreover, we consider Riemann data (ρi, wi, ci) on each road i. The
homogenized Lagrangian marker on each road iis denoted by wi. The priorities at merges
are set to βand (1 β) with 0 < β < 1. We do not consider β∈ {0,1}here since the network
would reduce to a network with consecutive 1-1-junctions for which there is no need to adapt
the pressures (compared to 3.2).
B C D
A M1 M2 MN E
(1 β)
(1 β)
(1 β)
Figure 3: Network of merging junctions.
The network in Figure 3is used to investigate how a pertubation in the Lagrangian marker
propagates through the network. Initially, the Lagrangian marker is identical on all roads,
except for one single incoming road. We study the wave structure and the propagation speed
of the perturbation.
16
Lemma 3. Consider the network shown in Figure 3with priorities βand (1 β), initial
data Ui,0= (ρi,0, wi,0, ci,0)with ρi, wi, ci>0, i = 0,...2N. We set initial conditions w0,0=b
and wi,0=a, i = 1,...,2N , a, b > 0and pressures pi(ρ) = ci,0ρ. At a merging junction
l∈ {1,...N}, the time t
l, where p∗∗
l6=pl,0depends on the initial conditions and may be
inﬁnite. The approximated pressure on road lis given by
p∗∗
l(ρ) = cl,0 1 + β(1 β)(awl1)2
awl1!ρ=: cl,0dlρ, (40)
where (dl)lis monotonically decreasing sequence and liml→∞ dl= 1.
Proof. The propagation of the contact discontinuity through the network determines the in-
teraction time t
lof the contact discontinuity with the junction l. Without loss of generality,
let us consider the dynamics on road 1 after the merging junction. At the merging junctions
M1 and M2, there are two interactions to be considered (see Figure 4). If the contact discon-
tinuity arising from M1 with positive speed v1and a shock with negative speed arising from
the junction M2 interact, the speed of the contact discontinuity decreases after the interaction
and is still non-negative, but possibly zero. The new speed is dependent on the speed of the
state on the right of the shock front. In addition to a 1-shock, a 1-rarefaction can arise from
the merging junction M2 with positive speed. The contact discontinuity arising from the
junction M1 propagates to the junction M2. It changes its speed in the presence of a 1-shock
on road 1. In the absence of a 1-shock, its velocity stays unchanged (see Figure 4b).
M1M2
ts
ts+1
w0w1
¯w1
w2
(a) Interaction with a shock wave
M1M2
ts
ts+1
w0
w1
¯w1
¯w2
w2
(b) Interaction with rarefaction wave
Figure 4: Interaction of a contact discontinuity with 1-waves.
The contact discontinuity changes its speed at the junction M2 and afterwards interacts with
the 1-rarefaction. During the interaction with the rarefaction fan, the contact discontinuity
accelerates. Therefore, the times t
l, l = 1,...N, when the changes in the pressure are
triggered at the junctions, depend on the initial conditions. The homogenized w-value on
road l, after an interaction is triggered, is given by
wl=βlb+ (1 β)a
l1
X
i=0
βli1.
The sequence (wl) is either monotonically increasing (a > b) or decreasing (b > a) and
liml→∞ wl=a. At each junction, when the contact discontinuity reaches the junction, the
17
new pressure is computed using Lemma 1. Then, the new pressure law is given by (40)
where w0=b. Moreover, (dl) is monotonically decreasing and liml→∞ dl= 1. Summarizing,
d1> d2>···> dNand in the limit N→ ∞, we have that the pressure on the outgoing road
after the last merging junction remains unchanged.
Tracking the contact discontinuity is analogous to car path tracking in road networks, see
also , with the only diﬀerence that each interaction of the contact discontinuity with a 1-
wave triggers an additional 1-wave in our case, which has been omitted in the ﬁgures for sake
of simplicity.Figure 5exempliﬁes on an x-t-plane the solution may look like. The contact
discontinuity starts at the junction M1 and propagates at positive speed. Whenever, the
contact discontinuity interacts with a 1-wave, the speed of the contact discontinuity changes.
Therefore, the new speed depends on the initial data on the roads that are located behind the
merges. After an interaction with a rarefaction wave, the speed increases. After an interaction
with a shock, the speed decreases. It is also possible that junctions in the network are not
reached by the contact discontinuity, if the speed of the contact discontinuity approaches zero
after an interaction with a shock wave. In Figure 5, the contact discontinuity interacts with
a shock wave on road 2, and its speed approaches zero. Therefore, the Lagrangian marker
beyond the junction M3 stays unchanged.
M1M2M3
0
t
b
a
¯w1
¯w2
a a
Figure 5: Perturbation of the Lagrangian marker at one incoming road.
5.2.1 Numerical evaluation
We set a= 2, b = 1, β =1
2, N = 10 and ci,0= 1. Each road has length Li=L= 0.5 and
the end of the time horizon is T= 12. We consider two scenarios. In the ﬁrst scenario, we
set initial densities on all roads to ρi,0= 0.3, i = 1,...,N (free ﬂow case). Note that we will
refer to this case as the ’free ﬂow case’ to indicate the corresponding initial data.
The second scenario considered is identical to the ﬁrst one, except for the density on the
last road, where we consider maximal congestion ρN,0= 1 (congested case). Since the last
road is congested, a shock wave moves backwards through the network. Simulation results
are obtained with the discretization ∆x= 1/100 and ∆t= ∆x/4, fulﬁlling the CFL condi-
tion (34). Table 1shows the adaption times t
lat the junctions l= 1,...,10. We see that the
pressure is recalculatedat each merging junction with t
1< t
2< . . . t
10 < T = 12 in the free
Numerically, we update the pressure coeﬃcient at a junction laccording to Algorithm 1whenever one of
the incoming Lagrangian markers changes its value from w=ato w=wl1.
18
ﬂow scenario. For the congested case, we observe less changes in the pressure constants. For
roads l= 1,...,5, the times t
lare identical to the free ﬂow scenario. On road 5, the contact
discontinuity interacts with the shock wave downstream. The traﬃc beyond the shock wave
has zero velocity and the propagation of the Lagrangian marker is stopped, see Figure 6a.
land constants dl
t
1t
2t
3t
4t
5t
6t
7t
8t
9t
10
free ﬂow 0 0.42 0.84 1.42 2.14 3.6 5.8 7.48 8.74 9.66
congested 0 0.42 0.84 1.42 2.14 ∞ ∞ ∞
d1d2d3d4d5d6d7d8d9d10
1.0800 1.0427 1.0241 1.0141 1.0084 1.0051 1.0032 1.0020 1.0012 1.0008
The constants dlof Lemma 3are also shown in Table 1. We observe that dlapproaches the
value one already for N= 10. In the congested case, the constants are identical, but the
change in the Lagrangian marker is not propagated across the junctions l= 6,...,10. Hence,
the pressure law is identical to the initial pressure law on these roads.
5.2.2 Comparison to the LWR model
We may re-write the LWR model as a special case of the ARZ model. In the LWR model,
the velocity function Vi(ρ) = vmax(1 ρimax
i) = (wi,0ipi,0(ρ)) is used as the equilibrium
velocity for a ﬁxed value wi,0and ﬁxed pressure pi,0(ρ) = wi,0max
iρ. Note that, the value
wi,0idoes not depend on time and space in contrast to the ARZ model. The network
structure induces diﬀerences in the solution of the LWR and the AP model, that are not only
connected to the modeling of the pressure function after merges, but are due to the coupling
conditions in general. Using the supply and demand formulation (23)-(24), we couple the 2-
1-junction in the LWR model using the demand and supply functions (9) and use the demand
and supply evaluations D1=d1(ρ1,0), D2=d2(ρ2,0), S3=s3(ρ3,0) instead of (36). Since
there are no contact discontinuities present in the solution to the LWR model, we compute
numerical results of the LWR model with a Godunov scheme using the same numerical grid
x= 1/100,t= ∆x/4. Figure 6a shows the Lagrangian marker on each road at t=T. For
the AP model, we obtain, as expected, the homogenized Lagrangian marker on each road in
the free ﬂow scenario. In the congested scenario, the Lagrangian markers are given by wlon
roads l= 1,...,4 and w5at the beginning of road 5. On the remaining part, the Lagrangian
marker is equal to a. In the LWR model, the perturbation of the Lagrangian marker on the
incoming road does not aﬀect the solution on the remaining part of the network.
6 Conclusion
We have presented a suitable approximation of the homogenized pressure appearing as nec-
essary coupling conditions for networked ARZ equations. The novel pressure approximation
has been analyzed in detail and allows for an eﬃcient numerical computation even for time–
dependent boundary data and complex networks. A numerical method has been proposed
to eﬃciently compute coupled second–order traﬃc ﬂow models on networks. The numerical
19
01234567891011
0.5
1
1.5
2
x
w
Initial data, t= 0
AP (free ﬂow), t=T
AP (congested), t=T
0 2 4 6 8 10
0
0.5
1
time
ρ
AP LWR
(b) Density at node A
Figure 6: Lagrangian marker and densities in the sequential network.
results demonstrate the performance of the proposed procedure and highlight diﬀerences to
the homogenized pressure and the LWR model predictions.
Acknowledgment
S. G¨ottlich and J. Weissen were supported by the DFG grants GO 1920/7,10 and the DAAD
projects 57444394 (USA), 57445223 (France) while M.Herty gratefully acknowledges support
through the DFG grants HE 5386/15,18,19, 320021702/GRK2326 and DFG EXC-2023 Inter-
net of Production-390621612. The research of S. G¨ottlich and M. Herty was also supported
by the joint BMBF grant ENets.
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