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ESAIM: M2AN 56 (2022) 213–235 ESAIM: Mathematical Modelling and Numerical Analysis

https://doi.org/10.1051/m2an/2022002 www.esaim-m2an.org

NETWORK MODELS FOR NONLOCAL TRAFFIC FLOW

Jan Friedrich , Simone G¨

ottlich*and Maximilian Osztfalk

Abstract. We present a network formulation for a traﬃc ﬂow model with nonlocal velocity in the

ﬂux function. The modeling framework includes suitable coupling conditions at intersections to either

ensure maximum ﬂux or distribution parameters. In particular, we focus on 1-to-1, 2-to-1 and 1-to-

2 junctions. Based on an upwind type numerical scheme, we prove the maximum principle and the

existence of weak solutions on networks. We also investigate the limiting behavior of the proposed

models when the nonlocal inﬂuence tends to inﬁnity. Numerical examples show the diﬀerence between

the proposed coupling conditions and a comparison to the Lighthill-Whitham-Richards network model.

Mathematics Subject Classiﬁcation. 35L65, 65M12, 90B20.

Received April 30, 2021. Accepted January 3, 2022.

1. Introduction

Macroscopic traﬃc ﬂow models have been studied by researchers for several decades. All started with the

famous Lighthill-Whitham-Richards (LWR) [41,44] model, which has been introduced in the 1950’s. Since then

the approach of modeling traﬃc ﬂow by conservation law has been extended in many directions. A second

equation has been introduced to describe the evolution of the speed and include acceleration [2,48]. The LWR

model has been further adapted to multilane traﬃc ﬂow [18,30,32,33,35,42] and complex road networks

[16,25,26,33,34]. From an application point of view, macroscopic models can be used to investigate the so-

called capacity drop eﬀect and to optimize traﬃc ﬂow [19,21,29,31,33,40].

Nowadays, with the progress in autonomous driving, new challenges in road traﬃc arise. In order to manage

these challenges mathematically, nonlocal traﬃc ﬂow models have been introduced [4,9,14,24,27]. They include

more information in a certain nonlocal range about the traﬃc of the road. This nonlocal range can stand for

the connection radius of autonomous cars or for the sight of a driver. Nonlocal models for traﬃc ﬂow are widely

studied in current research concerning existence of solutions [4,9,24,27,36], numerical schemes [4,8,22,24,27] or

convergence to local conservation laws [5,6,17,20,38] - even, in general, this question is still an open problem.

Modeling approaches include microscopic models [12,13,28,45], second order models [12], multiclass models

[10], multilane models [3,23] and also time delay models [37]. But to the best of our knowledge only a few works

deal with network models.

Some ﬁrst attempts can be found in [7,11,13,39]. The work [7] considers measure valued solutions for nonlocal

transport equations and [39] deals with nonlocal conservation laws on bounded domains while [11,13] includes

Keywords and phrases. Nonlocal scalar conservation laws, traﬃc ﬂow networks, coupling conditions, upwind scheme.

University of Mannheim, Department of Mathematics, 68131 Mannheim, Germany.

*Corresponding author: goettlich@uni-mannheim.de

c

○The authors. Published by EDP Sciences, SMAI 2022

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0),

which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

214 J. FRIEDRICH ET AL.

1-to-1 junctions. In [11], the existence and well-posedness of solutions at a 1-to-1 junction is shown, where the

roads are allowed to diﬀer in the velocity and maximum road capacities. To the best of our knowledge, network

models for other types of junctions have not been studied in literature and a general framework to deal with

nonlocal traﬃc ﬂow models on a network, similar to the LWR model, is missing in current research.

Hence, in this work we propose a network formulation for a class of nonlocal traﬃc ﬂow models. The work is

thereby structured as following: in the next section we introduce the network setting. We concentrate on single

junctions and deﬁne a weak solution. Additionally, we introduce rather general assumptions on the coupling

conditions necessary to obtain a well-posed model. It turns out that the 1-to-1 junction model from [11] fulﬁlls

these assumptions. Furthermore, we also present two diﬀerent explicit choices of the coupling conditions for

1-to-2 and 2-to-1 junctions which are inspired by well-studied couplings of the LWR model on a network.

In Section 3, we present a numerical scheme to solve the nonlocal equations on a network and argue that

it converges to a weak solution. In particular, we also prove that the numerical scheme and hence the weak

solution inherit a suitable maximum principle. Furthermore, we investigate the limit behavior of the junction

models for a nonlocal range tending to inﬁnity in Section 4. This case is of special interest for nonlocal traﬃc

ﬂow models since they are motivated by autonomous cars and an inﬁnitely large range can be interpreted as

prefect information for each driver. We close this work in Section 5by numerically comparing the proposed

coupling conditions for the nonlocal model. On the one hand, we demonstrate diﬀerences between the nonlocal

approaches and on the other hand, we give a comparison to the LWR network model.

2. Network modeling

Following the ideas presented in [26,29] we describe a traﬃc ﬂow network as a directed graph 𝐺= (𝑉, 𝐸 ),

where the arcs 𝐸correspond to roads and the vertices 𝑉to junctions or intersections.

On each road 𝑒∈𝐸the density of cars is given by 𝜌𝑒(𝑡, 𝑥) and for a given initial state 𝜌𝑒(0, 𝑥)∀𝑒∈𝐸the

dynamics are governed by conservation laws of the form

𝜕𝑡𝜌𝑒(𝑡, 𝑥) + 𝜕𝑥𝑓𝑒(𝑡, 𝑥, {𝜌𝑘(𝑡, ·)}𝑘∈𝐸)=0 ∀𝑒∈𝐸, 𝑥 ∈(𝑎𝑒, 𝑏𝑒), 𝑡 > 0.(2.1)

Equation (2.1) allows for diﬀerent choices of the ﬂux functions 𝑓𝑒as for example the well-studied Lighthill-

Whitham-Richards (LWR) model of local type

𝑓𝑒(𝑡, 𝑥, {𝜌𝑘(𝑡, ·)}𝑘∈𝐸) = 𝜌𝑒(𝑡, 𝑥)𝑣𝑒(𝜌𝑒(𝑡, 𝑥)),(2.2)

where 𝑣𝑒are suitable velocity functions. On the other hand, the ﬂux function can be of nonlocal type and hence

depend on the whole density of the road 𝑒at time 𝑡. In this work, we focus on the nonlocal case but ﬁrst shortly

give some remarks about the modeling of the local setting.

Generally, for solving hyperbolic partial diﬀerential equations (PDEs) on a network the boundary treatment

is essential. Therefore, the boundary (or coupling) conditions at junctions 𝑣∈𝑉have to be deﬁned to ensure

the conservation of mass. The modeling of a junction in the case of the LWR model is not unique and there

exists a wide literature, see e.g., [15,26,29]. All these approaches describe the ﬂow which passes directly through

the junction depending on the purpose as for instance maximizing the ﬂux through the junction [29], satisfying

certain distribution rates [26] or considering the vanishing viscosity approach [15]. Most of the models also share

the similarity that they treat the junction only by determining the ﬂows at the intersection point. In contrast

to that, for nonlocal ﬂuxes in (2.1) these boundary (or coupling) conditions are already present in a transition

area in front of the junction point, coming from the nonlocal range. This makes the network coupling of the

nonlocal conservation laws more involved.

The model we are interested in this work is a nonlocal version of the LWR model. In the case of a single road

the ﬂux reads

𝑓1(𝑡, 𝑥, 𝜌1(𝑡, ·)) = 𝜌1(𝑡, 𝑥)𝑉1(𝑡, 𝑥) with 𝑉1(𝑡, 𝑥) = 𝑥+𝜂

𝑥

𝑣1(𝜌(𝑡, 𝑦))𝜔𝜂(𝑦−𝑥)𝑑𝑦 (2.3)

NETWORK MODELS FOR NONLOCAL TRAFFIC FLOW 215

for any 𝜂 > 0.The model includes that drivers adapt their speed based on a weighted mean of downstream

velocities, where 𝜂represents the nonlocal interaction range and 𝜔𝜂is a kernel function. In order to have a

well-posed network model, we need the following assumptions.

Assumption 2.1. We impose the following assumptions on the velocity function 𝑣𝑒,𝑒∈𝐸, and the kernel

function 𝜔𝜂:

𝑣𝑒∈𝐶2([0, 𝜌max

𝑒]; R+): 𝑣′

𝑒≤0, 𝑣𝑒(𝜌max

𝑒) = 0,

𝜔𝜂∈𝐶1([0, 𝜂]; R+) : 𝜔′

𝜂≤0,𝜂

0

𝜔𝜂(𝑥)𝑑𝑥 = 1,lim

𝜂→∞ 𝜔𝜂(0) = 0 ∀𝜂 > 0.(2.4)

The well-posedness under these assumptions on a single arc with (𝑎1, 𝑏1)=(−∞,∞) has been shown in [24].

Note that the considered kernel function 𝜔𝜂is assumed to be equal for all roads 𝑒∈𝐸as it is rather a property

of the driver than of the road. In addition to the Assumptions 2.1, we assume the following:

Assumption 2.2. We assume 𝜂 < 𝑏𝑒−𝑎𝑒∀𝑒∈𝐸.

This assumption ensures that the nonlocal range is restricted by the length 𝑏𝑒−𝑎𝑒of road 𝑒. Hence, we only

consider one junction in the nonlocal downstream term.

2.1. Modeling of a junction

In the following, we consider a set of incoming arcs ℐ={1, . . . , 𝑀}and a set of outgoing arcs 𝒪={𝑀+

1, . . . , 𝑀 +𝑁}at a ﬁxed junction. In addition to simplify notation, we set 𝑎𝑒=−∞, 𝑏𝑒= 0 for 𝑒= 1, . . . , 𝑀

and 𝑎𝑒= 0, 𝑏𝑒=∞for 𝑒=𝑀+ 1, . . . , 𝑀 +𝑁. As already noticed, coupling conditions are needed in order

to be able to deﬁne a solution at the junction. The coupling is induced by a function 𝑔𝑒and plays a crucial

role in the junction modeling of network models governed by nonlocal conservation laws. We will give concrete

examples of 𝑔𝑒for speciﬁc junction types in Section 2.2. Inspired by equation (2.3) we set

𝑓𝑒(𝑡, 𝑥, {𝜌𝑘}𝑘∈𝐸) = 𝜌𝑒(𝑡, 𝑥)𝑉𝑒(𝑡, 𝑥) + 𝑔𝑒{𝜌𝑖(𝑡, ·)}𝑖∈ℐ ,{𝑉𝑜(𝑡, 𝑥)}𝑜∈𝒪, 𝑒 ∈ ℐ,

𝜌𝑒(𝑡, 𝑥)𝑉𝑒(𝑡, 𝑥), 𝑒 ∈ 𝒪,(2.5)

with

𝑉𝑒(𝑡, 𝑥) :=

min{𝑥+𝜂,0}

min{𝑥,0}𝑣𝑒(𝜌𝑒(𝑡, 𝑦))𝜔𝜂(𝑦−𝑥)𝑑𝑦, 𝑒 ∈ ℐ,

max{𝑥+𝜂,0}

max{𝑥,0}𝑣𝑒(𝜌𝑒(𝑡, 𝑦))𝜔𝜂(𝑦−𝑥)𝑑𝑦, 𝑒 ∈ 𝒪.(2.6)

In addition, we couple equations (2.1), (2.5) with the initial conditions

𝜌𝑒,0∈𝐿1∩BV((𝑎𝑒, 𝑏𝑒); [0, 𝜌max

𝑒]), 𝑒 ∈ {1, . . . , 𝑀 +𝑁}.(2.7)

This setting allows for diﬀerent velocity functions and road capacities, respectively. As can be seen the diﬀerences

in the velocities are included by computing the mean velocity with the respective weights on the incoming or

outgoing roads. By having a closer look at the deﬁnition of 𝑉𝑒(𝑡, 𝑥) for 𝑒∈ 𝒪 in (2.6) we recognize that the

nonlocal velocities of the outgoing roads can become positive at 𝑥=−𝜂as soon as drivers notice the junction

and the properties of the next road. The functional relationship on the coupling 𝑔𝑒do not only depend on the

velocities of the outgoing roads but also on the densities of all incoming roads. In particular, the changes in

the maximum capacity 𝜌max

𝑒are captured by the function 𝑔𝑒such that no non-physical densities might occur.

Furthermore, we see that the velocities of the outgoing roads are important for all junction models due to the

nonlocality of the problem, while the densities on the incoming roads play an important role at the junction

for 𝑀 > 1. Naturally, it can be reasonable to also derive models which include the velocities of all incoming

roads or the densities of all outgoing roads. Note that the latter are indirectly included through the nonlocal

velocities.

Following ([26], Def. 4.2.4) we deﬁne a weak solution for (2.1) and (2.5) at a single junction as:

216 J. FRIEDRICH ET AL.

Deﬁnition 2.3 (Weak solution).A collection of functions 𝜌𝑒∈𝐶(R+;𝐿1

loc((𝑎𝑒, 𝑏𝑒))) and 𝑒∈ {1,...𝑀 +𝑁}

with 𝑎𝑒=−∞, 𝑏𝑒= 0 for 𝑒= 1, . . . , 𝑀 and 𝑎𝑒= 0, 𝑏𝑒=∞for 𝑒=𝑀+ 1, . . . , 𝑀 +𝑁, is called a weak

solution of (2.1) and (2.5) with (2.6) if

(1) for every 𝑒∈ {1, . . . , 𝑀 }𝜌𝑒is a weak solution on (−∞,−𝜂) to

𝜕𝑡𝜌𝑒(𝑡, 𝑥) + 𝜕𝑥𝜌𝑒(𝑡, 𝑥)𝑥+𝜂

𝑥

𝑣𝑒(𝜌𝑒(𝑡, 𝑦))𝜔𝜂(𝑦−𝑥)dy = 0,(2.8)

and for every 𝑒∈ {𝑀+ 1, . . . , 𝑀 +𝑁}on (0,∞) to (2.8);

(2) for every 𝑒∈ {1, . . . , 𝑀 +𝑁}and for a.e. 𝑡 > 0, the function 𝑥↦→ 𝜌𝑒(𝑡, 𝑥) has a bounded total variation;

(3) the following integral equality holds

𝑀

𝑖=1 R+0

−∞

𝜌𝑖(𝑡, 𝑥)𝜕𝑡𝜑𝑖(𝑡, 𝑥) + 𝑓𝑖(𝑡, 𝑥, {𝜌𝑘(𝑡, ·)}𝑘∈𝐸)𝜕𝑥𝜑𝑖(𝑡, 𝑥)𝑑𝑥𝑑𝑡

+

𝑀+𝑁

𝑜=𝑀+1 R++∞

0

𝜌𝑜(𝑡, 𝑥)𝜕𝑡𝜑𝑜(𝑡, 𝑥) + 𝑓𝑜(𝑡, 𝑥, {𝜌𝑘(𝑡, ·)}𝑘∈𝐸)𝜕𝑥𝜑𝑜(𝑡, 𝑥)𝑑𝑥𝑑𝑡= 0

for every collection of test function 𝜑𝑖∈𝐶1

0((0,∞)×(−∞,0]; R), 𝑖 = 1, . . . , 𝑀 and 𝜑𝑜∈𝐶1

0((0,∞)×

[0,∞); R), 𝑜 =𝑀+ 1, . . . , 𝑀 +𝑁satisfying:

𝜑𝑖(·,0) = 𝜑𝑜(·,0), 𝜕𝑥𝜑𝑖(·,0) = 𝜕𝑥𝜑𝑜(·,0)

for 𝑖∈ {1, . . . , 𝑀 }and 𝑜∈ {𝑀+ 1,...𝑀 +𝑁}.

We note that in contrast to ([26], Def. 4.2.4), the functions 𝜌𝑒have to be weak solutions outside the transition

area and for 𝑒∈ {𝑀+ 1, . . . , 𝑀 +𝑁}.This is ensured in the network setting by equation (2.6). In addition to

([26], Def. 4.2.4), we introduce the last condition in Deﬁnition 2.3 to also include the transition area into the

deﬁnition of a weak solution. Moreover, from the last condition follows the ﬂux conservation at the junction,

i.e., for a.e. 𝑡 > 0, it holds

𝑀

𝑖=1

𝑓𝑖(𝑡, 0−,{𝜌𝑘(𝑡, ·)}𝑘∈𝐸) =

𝑀+𝑁

𝑜=𝑀+1

𝑓𝑜(𝑡, 0+,{𝜌𝑘(𝑡, ·)}𝑘∈𝐸).(2.9)

Let us now deﬁne admissible conditions that weak solutions should satisfy. Therefore, as in [26], we introduce

the distribution matrix 𝐴= (𝛼𝑖,𝑜)𝑖=1,...,𝑀,𝑜=𝑀+1,...,𝑀+𝑁with 𝑁+𝑀

𝑜=𝑀+1 𝛼𝑖,𝑜 = 1. Here, 𝛼𝑖,𝑜 determines the

proportion of traﬃc coming from road 𝑖and going to road 𝑜.

Deﬁnition 2.4 (Admissible weak solution).A collection of functions 𝜌𝑒∈𝐶(R+;𝐿1

𝑙𝑜𝑐((𝑎𝑒, 𝑏𝑒))) and 𝑒∈

{1,...𝑀 +𝑁}, with 𝑎𝑒=−∞, 𝑏𝑒= 0 for 𝑒= 1, . . . , 𝑀 and 𝑎𝑒= 0, 𝑏𝑒=∞for 𝑒=𝑀+ 1, . . . , 𝑀 +𝑁, is

called an admissible weak solution to (2.1) and (2.5) if it is a weak solution in the sense of Deﬁnition 2.3 and

additionally satisﬁes at least one of the following conditions:

(i) If 𝜌𝑖(𝑡, 𝑥)≤𝜌max

𝑜∀𝑖∈ {1, . . . , 𝑀 }, 𝑜 ∈ {𝑀+ 1, . . . , 𝑀 +𝑁}a.e. 𝑡 > 0 and 𝑥∈[−𝜂, 0), then for every

𝑖∈ {1, . . . , 𝑀 }𝜌𝑖is an (entropy) weak solution on (−∞,0) to

𝜕𝑡𝜌𝑖(𝑡, 𝑥) + 𝜕𝑥𝜌𝑖(𝑡, 𝑥)(𝑉𝑖(𝑡, 𝑥) +

𝑁+𝑀

𝑜=𝑀+1

𝛼𝑖,𝑜𝑉𝑜(𝑡, 𝑥))= 0.

NETWORK MODELS FOR NONLOCAL TRAFFIC FLOW 217

(ii) For all 𝑜∈ {𝑀+ 1, . . . , 𝑀 +𝑁}we have

𝑓𝑜(𝑡, 0+,{𝜌𝑘(𝑡, ·)}𝑘∈𝐸) =

𝑀

𝑖=1

𝛼𝑖,𝑜𝑓𝑖(𝑡, 0−,{𝜌𝑘(𝑡, ·)}𝑘∈𝐸).

The ﬁrst admissible condition introduced in Deﬁnition 2.4 means the following: If no capacity restrictions of

the outgoing roads aﬀect the incoming road, the nonlocal model should produce the natural idea of the model

(2.3). This means that the drivers should adapt their velocity according to the mean velocity. Here, the mean

velocity is given by a part of the velocity on the current road and by a part of all outgoing roads weighted

with the distribution parameters. The second condition represents the distribution parameters which should be

fulﬁlled at the junction. As we will see, there exist couplings that do not always satisfy both conditions at the

same time. Hence, we have a kind of a trade oﬀ between satisfying the distribution parameters and the natural

behavior of the model.

The goal is now to ﬁnd appropriate coupling functions 𝑔𝑒to close the model proposed in (2.5). We restrict to

𝑀-to-1 and 1-to-𝑁junctions since the coupling of 𝑀-to-𝑁junctions is more involved in the nonlocal setting.

Assumption 2.5. We impose the following restrictions on the function 𝑔𝑒, 𝑒 ∈ ℐ and the inﬂow on the outgoing

roads 𝑓𝑒(𝑡, 0), 𝑒 ∈ 𝒪:

(1) The inﬂow has to be positive and smaller than the maximum possible ﬂow on the outgoing road, i.e.,

0≤𝑓𝑒(𝑡, 0) ≤𝜌max

𝑒𝑉𝑒(𝑡, 0) ∀𝑒∈ 𝒪.(2.10)

(2) The coupling function 𝑔𝑒has to be positive and smaller than the desired ﬂow, i.e., the current density times

the mean velocity of the outgoing roads:

0≤𝑔𝑒{𝜌𝑖(𝑡, ·)}𝑖∈ℐ ,{𝑉𝑜(𝑡, 𝑥)}𝑜∈𝒪≤𝜌𝑒(𝑡, 𝑥)

𝑁+𝑀

𝑜=𝑀+1

𝛼𝑖,𝑜𝑉𝑜(𝑡, 𝑥).(2.11)

(3) The coupling function 𝑔𝑒does not decrease in the velocities of the outgoing roads and concerning the density

of the road 𝑒it is always smaller or equal than the corresponding maximum density, i.e.,

𝑔𝑒(𝜌𝑒(𝑡, 𝑥), . . . )≤𝑔𝑒(𝜌max

𝑒, . . . )

𝑔𝑒(...,𝑉𝑘(𝑡, 𝑥), . . . )≤𝑔𝑒(...,𝐶,...)∀𝐶≥𝑉𝑘(𝑡, 𝑥)and 𝑘∈ 𝒪.(2.12)

(4) The function 𝑔𝑒is Lipschitz continuous in 𝜌𝑒with Lipschitz constant 𝐿≤max𝑒∈𝐸‖𝑣𝑒‖∞.

The ﬁrst two assumptions are limitations on the ﬂows while the third assumption states upper bounds on 𝑔𝑒.

Note that the second assumption together with the deﬁnition of the velocity (2.6) implies that the ﬁrst item of

Deﬁnition 2.3 is satisﬁed for 𝑒∈ {1, . . . , 𝑀 }.

2.2. Junction types

In the following, we consider 1-to-1, 1-to-2 and 2-to-1 junctions, specify the assumptions from above and give

concrete examples for the junction models. The extensions to 1-to-N and M-to-1 junctions are then straightfor-

ward. We note that by construction the proposed models satisfy the Rankine-Hugoniot condition (2.9).

2.2.1. 1-to-1 junctions

The 1-to-1 junction has been already studied extensively in [11]. It has a kind of special role as it can

be interpreted as a model on a single arc with the velocity function changing at the intersection point. This

consideration is not possible for the other types of junctions. The coupling in [11] is given by

𝑔1(𝜌1(𝑡, 𝑥), 𝑉2(𝑡, 𝑥)) := min{𝜌1(𝑡, 𝑥), 𝜌max

2}𝑉2(𝑡, 𝑥).(2.13)

Under the Assumptions 2.1 the well-posedness and uniqueness of weak entropy solutions has been shown in [11].

As we have just one incoming and one outgoing road, both items in the Deﬁnition 2.4 are fulﬁlled.

218 J. FRIEDRICH ET AL.

2.2.2. 1-to-2 junctions

For 1-to-2 junctions we have 𝑀= 1 and 𝑁= 2. So 𝑔1only depends on 𝜌1(𝑡, 𝑥), 𝑉2(𝑡, 𝑥) and 𝑉3(𝑡, 𝑥). We

need to prescribe distribution parameters 𝛼1,2and 𝛼1,3with 𝛼1,2+𝛼1,3= 1 which give us the desired ﬂow

from the incoming road to the outgoing roads. These distribution rates can be for example determined through

historical data. Using the distribution rates we can specify the second condition of the Assumption 2.5,i.e.,

0≤𝑔1(𝜌1(𝑡, 𝑥), 𝑉2(𝑡, 𝑥), 𝑉3(𝑡, 𝑥)) ≤𝜌1(𝑡, 𝑥) (𝛼1,2𝑉2(𝑡, 𝑥) + 𝛼1,3𝑉3(𝑡, 𝑥)) .(2.14)

The modeling choice of the function 𝑔1is of course not unique. The function should preserve the densities in

the given intervals (which is achieved by fulﬁlling Assumption 2.5) and follow the purpose of modeling. Here

we present two approaches: the ﬁrst approach allows for a maximum ﬂow (satisfying only (i) in Def. 2.4) and

the second approach satisﬁes the distribution parameters at all costs (satisfying only (ii) in Def. 2.4).

Example 2.6. The approach for the maximum ﬂow is very similar to the 1-to-1 junction and is inspired by

the model presented in [29]. Due to the distribution parameters the ﬂow, from the incoming to one outgoing

road is either given by the distribution rate times the ﬂow or restricted by the maximum ﬂow/capacity on the

corresponding outgoing road. So we get

𝑔1(𝜌1, 𝑉2, 𝑉3) = min{𝛼1,2𝜌1(𝑡, 𝑥), 𝜌max

2}𝑉2(𝑡, 𝑥) + min{𝛼1,3𝜌1(𝑡, 𝑥), 𝜌max

3}𝑉3(𝑡, 𝑥).(2.15)

In the case that the capacity restrictions are not active we would simply have the ﬂow deﬁned by the density

times mean velocity weighted by the distribution rates. The corresponding inﬂows on the outgoing roads are

then given by:

𝑓𝑒(𝑡, 0+, 𝜌1) = min{𝛼1,𝑒𝜌1(𝑡, 0−), 𝜌max

𝑒}𝑉𝑒(𝑡, 𝑥), 𝑒 ∈ {2,3}.

In addition, it is obvious that condition (ii) in Deﬁnition 2.4 cannot be satisﬁed in all cases.

Example 2.7. In order to always satisfy the distribution parameters, we adapt the idea introduced in [26].

The ﬂow of the incoming road in the transition area is, if possible, the density times the mean (in terms

of distribution rates) nonlocal velocity or the maximum feasible ﬂows of the outgoing roads divided by the

corresponding distribution, i.e.,

𝑔1(𝜌1, 𝑉2, 𝑉3) = min 𝜌1(𝑡, 𝑥)(𝛼1,2𝑉2(𝑡, 𝑥) + 𝛼1,3𝑉3(𝑡, 𝑥)),𝜌max

2𝑉2(𝑡, 𝑥)

𝛼1,2

,𝜌max

3𝑉3(𝑡, 𝑥)

𝛼1,3(2.16)

with the inﬂows

𝑓𝑒(𝑡, 0+, 𝜌𝑒) = 𝛼1,𝑒𝑓1(𝑡, 0−, 𝜌1), 𝑒 ∈ {2,3}.

If the desired ﬂow is not restricted by the outgoing roads, also condition (i) in Deﬁnition 2.4 is satisﬁed.

2.2.3. 2-to-1 junctions

Similar to the discussion above we intend to present again two models satisfying the Assumption 2.5: one

approach allows for the maximum possible ﬂux and the other one satisﬁes the priority rules at all costs.

Therefore, priority rules have to be prescribed in the sense that the percentage of cars going from the incom-

ing roads to the outgoing road is 𝑞1,3+𝑞2,3= 1. In a ﬁrst attempt, we assume the functional relationship

𝑔𝑒(𝜌𝑒(𝑡, 𝑥), 𝜌𝑒−(𝑡, ·)), 𝑉3(𝑡, 𝑥)) with 𝑒−being the other incoming road. So far, we did not specify the exact den-

sity to be taken into account from the other incoming road. However, to keep a maximum principle, it turns

out that we need to consider the density at the junction point itself. Hence, we set

𝑔𝑒(𝜌𝑒(𝑡, 𝑥), 𝜌𝑒−(𝑡, ·)), 𝑉3(𝑡, 𝑥)) := 𝑔𝑒(𝜌𝑒(𝑡, 𝑥), 𝜌𝑒−(𝑡, 0−)), 𝑉3(𝑡, 𝑥)).(2.17)

NETWORK MODELS FOR NONLOCAL TRAFFIC FLOW 219

The inﬂow on the outgoing road is simply given by the sum of the two outﬂows at 𝑥= 0, i.e.,

𝑓3(𝑡, 0+, 𝜌1, 𝜌2) = 𝑔1(𝜌1, 𝜌2, 𝑉3) + 𝑔2(𝜌2, 𝜌1, 𝑉3).

Therefore, both presented approaches satisfy (ii) in Deﬁnition 2.4.

Example 2.8. To maximize the ﬂux through the junction, several possibilities can be considered. First, the

ﬂux can pass on to the outgoing road without violating the ﬂux restrictions coming from the maximum possible

ﬂux and the priority parameter. Second, the ﬂux restriction can become active. However, if there is not enough

mass coming from the other incoming road we allow the ﬂow to be higher allowing the maximum possible ﬂow.

As in [29], this results in the following coupling function in the transition area:

𝑔𝑒(𝜌𝑒, 𝜌𝑒−, 𝑉3) = min{𝜌𝑒(𝑡, 𝑥),max{𝑞𝑒,3𝜌max

3, 𝜌max

3−𝜌𝑒−(𝑡, 0−)}}𝑉3(𝑡, 𝑥).(2.18)

Example 2.9. To always satisfy the priority rules, we have to assume that 𝜌𝑒(𝑡, 0) >0 for all 𝑡 > 0 and

𝑒∈ {1,2}. Otherwise, the model does not give meaningful results since the solution is always to let no ﬂow

through the junction. Inspired by [26], we want to maximize the ﬂux through the junction but at the same time

always satisfy the priority parameters. Applying this idea to the transition area leads to

𝑔𝑒(𝜌𝑒, 𝜌𝑒−, 𝑉3) = min{𝜌𝑒(𝑡, 𝑥), 𝑞𝑒,3𝜌max

3,(𝑞𝑒,3/𝑞𝑒−,3)𝜌𝑒−(𝑡, 0−)}𝑉3(𝑡, 𝑥).(2.19)

3. Numerical scheme

So far we have presented junction models which seem to be a reasonable choice concerning the Deﬁnitions

2.3 and 2.4. In this section, we now deal with the question of existence for weak solutions on networks. To do

so, we present a numerical discretization scheme of upwind type and consider its convergence properties. We

follow the ideas presented in [11] and [24] to derive a scheme for (2.1) and (2.5).

The numerical scheme uses the following ingredients: For 𝑗∈Z, 𝑛 ∈Nand 𝑒∈𝐸, let 𝑥𝑒,𝑗−1/2=𝑗𝛥𝑥 be the

cell interfaces, 𝑥𝑒,𝑗 = (𝑗+ 1/2)𝛥𝑥 the cells centers, corresponding to a space step 𝛥𝑥 such that 𝜂=𝑁𝜂𝛥𝑥 for

some 𝑁𝜂∈N, and let 𝑡𝑛=𝑛𝛥𝑡 be the time mesh. In particular, 𝑥=𝑥𝑒,−1/2= 0 is a cell interface. Note that

we assume the same step sizes for each road and that we have 𝑗≥0 iﬀ 𝑒∈ 𝒪 and 𝑗 < 0 iﬀ 𝑒∈ ℐ.

The ﬁnite volume approximate solution is given by 𝜌𝛥𝑥

𝑒such that 𝜌𝛥𝑥

𝑒(𝑡, 𝑥) = 𝜌𝑛

𝑒,𝑗 for (𝑡, 𝑥)∈[𝑡𝑛, 𝑡𝑛+1)×

[𝑥𝑒,𝑗−1/2, 𝑥𝑒,𝑗+1/2) and 𝑒∈𝐸. The initial datum 𝜌𝑒,0in (2.7) is approximated by the cell averages

𝜌0

𝑒,𝑗 =1

𝛥𝑥 𝑥𝑒,𝑗+1/2

𝑥𝑒,𝑗−1/2

𝜌𝑒,0(𝑥)𝑑𝑥, 𝑗 < 0, 𝑒 ∈ ℐ,

𝑗≥0, 𝑒 ∈ 𝒪.

Following [11,24], we consider the numerical ﬂux function

𝐹𝑛

𝑒,𝑗 := 𝜌𝑛

𝑒,𝑗 𝑉𝑛

𝑒,𝑗 +𝑔𝑒𝜌𝑛

𝑒,𝑗 ,{𝜌𝑛

𝑖,−1}𝑖∈ℐ ,{𝑉𝑛

𝑜,𝑗 }𝑜∈𝒪, 𝑒 ∈ ℐ

𝜌𝑛

𝑒,𝑗 𝑉𝑛

𝑒,𝑗 , 𝑒 ∈ 𝒪,(3.1a)

with

𝑉𝑛

𝑒,𝑗 =min{−𝑗−2,𝑁𝜂−1}

𝑘=0 𝛾𝑘𝑣𝑒(𝜌𝑛

𝑒,𝑗+𝑘+1 ), 𝑒 ∈ ℐ,

𝑁𝜂−1

𝑘=max{−𝑗−1,0}𝛾𝑘𝑣𝑒(𝜌𝑛

𝑒,𝑗+𝑘+1 ), 𝑒 ∈ 𝒪,(3.1b)

𝛾𝑘=(𝑘+1)𝛥𝑥

𝑘𝛥𝑥

𝜔𝜂(𝑥)𝑑𝑥, 𝑘 = 0, . . . , 𝑁𝜂−1,(3.1c)

where we set, with some abuse of notation, 𝑏

𝑘=𝑎= 0 whenever 𝑏 < 𝑎. The inﬂuxes 𝐹𝑛

𝑒,−1for 𝑒∈ 𝒪 are deﬁned

by discrete versions of the ﬂuxes given by the modeling approach.

220 J. FRIEDRICH ET AL.

In this way, we can deﬁne the following ﬁnite volume numerical scheme

𝜌𝑛+1

𝑒,𝑗 =𝜌𝑛

𝑒,𝑗 −𝜆𝐹𝑛

𝑒,𝑗 −𝐹𝑛

𝑒,𝑗−1with 𝜆:= 𝛥𝑡

𝛥𝑥, 𝑛 ∈N,𝑗 < 0, 𝑒 ∈ ℐ,

𝑗≥0, 𝑒 ∈ 𝒪.(3.1d)

Note that, due to the accurate calculation of the integral in (3.1c) and the deﬁnition of the convoluted velocities

in (3.1b), there holds

0≤𝑉𝑛

𝑒,𝑗 ≤𝑣max

𝑒∀𝑗∈Z, 𝑛 ∈N, 𝑒 ∈𝐸.

We set

‖𝑣‖:= max

𝑒∈𝐸‖𝑣𝑒‖∞,‖𝑣′‖:= max

𝑒∈𝐸‖𝑣′

𝑒‖∞,‖𝜌‖:= max

𝑒∈𝐸𝜌max

𝑒

and consider the following CFL condition:

𝜆≤1

𝛾0‖𝑣′‖‖𝜌‖+ 2‖𝑣‖·(3.2)

The proposed discretization for appropriate choices of 𝑔𝑒and the inﬂuxes are the basis to prove a maximum

principle and the existence of weak solutions.

3.1. Existence of weak solutions

We prove that the scheme (3.1) satisﬁes a maximum principle under the Assumption 2.5. We start by deriving

some elementary inequalities concerning the diﬀerences of the velocities.

Lemma 3.1. Consider the velocities computed in (3.1b)and let the conditions (2.4)hold, then we have the

following estimates

𝑉𝑛

𝑒,𝑗−1−𝑉𝑛

𝑒,𝑗 ≤

𝛾0‖𝑣′‖(𝜌max

𝑒−𝜌𝑛

𝑒,𝑗 ), 𝑗 ≤ −1,

0, 𝑗 ≥0,𝑒∈ ℐ,

0, 𝑗 ≤ −1,

𝛾0‖𝑣′‖(𝜌max

𝑒−𝜌𝑛

𝑒,𝑗 ), 𝑗 ≥0,𝑒∈ 𝒪.

(3.3)

In addition, we have

𝑉𝑛

𝑒,𝑗−1𝜌max

𝑒−𝑉𝑛

𝑒,𝑗 𝜌𝑛

𝑒,𝑗 ≤

(𝛾0‖𝑣′‖‖𝜌‖+𝑉𝑛

𝑒,𝑗 )(𝜌max

𝑒−𝜌𝑛

𝑒,𝑗 ), 𝑗 ≤ −1,

𝑉𝑛

𝑒,𝑗 (𝜌max

𝑒−𝜌𝑛

𝑒,𝑗 ), 𝑗 ≥0,𝑒∈ ℐ,

𝑉𝑛

𝑒,𝑗 (𝜌max

𝑒−𝜌𝑛

𝑒,𝑗 ), 𝑗 ≤ −1,

(𝛾0‖𝑣′‖‖𝜌‖+𝑉𝑛

𝑒,𝑗 )(𝜌max

𝑒−𝜌𝑛

𝑒,𝑗 ), 𝑗 ≥0,𝑒∈ 𝒪.

(3.4)

Proof. Let us consider the case 𝑒∈ ℐ:

𝑉𝑛

𝑒,𝑗−1−𝑉𝑛

𝑒,𝑗 =

𝑁𝜂−1

𝑘=1

(𝛾𝑘−𝛾𝑘−1)𝑣𝑒(𝜌𝑛

𝑒,𝑗+𝑘)−𝛾𝑁𝜂−1𝑣𝑒(𝜌𝑛

𝑒,𝑗+𝑁𝜂)

+𝛾0𝑣𝑒(𝜌𝑛

𝑒,𝑗 ),

𝑗≤ −𝑁𝜂−1,

−𝑗−1

𝑘=1 (𝛾𝑘−𝛾𝑘−1)𝑣𝑒(𝜌𝑛

𝑒,𝑗+𝑘) + 𝛾0𝑣𝑒(𝜌𝑛

𝑒,𝑗 ),−𝑁𝜂≤𝑗≤ −2,

𝛾0𝑣𝑒(𝜌𝑛

𝑒,−1), 𝑗 =−1,

0, 𝑗 ≥0,

≤𝛾0𝑣𝑒(𝜌𝑛

𝑒,𝑗 ), 𝑗 ≤ −1,

0, 𝑗 ≥0.

NETWORK MODELS FOR NONLOCAL TRAFFIC FLOW 221

Using 𝑣𝑒(𝜌max

𝑒) = 0 and the mean value theorem yields inequality (3.3) for 𝑒∈ ℐ. Then, the inequality (3.4)

follows from multiplying the ﬁrst equation by 𝜌max

𝑒and adding and subtracting 𝑉𝑛

𝑒,𝑗 𝜌𝑛

𝑒,𝑗 .

The inequalities for 𝑒∈ 𝒪 can be obtained analogously.

Next, we give the details of the proof in the particular case of 1-to-2 situations.

Proposition 3.2. Under hypothesis (2.4), initial conditions as in (2.7)and the CFL condition (3.2), the

sequence generated by the numerical scheme (3.1)for a 1-to-2 junction with distribution parameters 𝛼1,2+𝛼1,3=

1satisﬁes the following maximum principle:

0≤𝜌𝑛

1,𝑗 ≤𝜌max

1for 𝑗≤ −1and 0≤𝜌𝑛

𝑒,𝑗 ≤𝜌max

𝑒for 𝑗≥0, 𝑒 ∈ {2,3} ∀𝑛∈N.

Proof. We start with the incoming road and the lower bound which can be obtained by applying (2.14):

𝜌𝑛+1

1,𝑗 =𝜌𝑛

1,𝑗 +𝜆𝐹𝑛

1,𝑗−1−𝐹𝑛

1,𝑗

≥𝜌𝑛

1,𝑗 −𝜆𝜌𝑛

1,𝑗 𝑉𝑛

1,𝑗 −𝜆𝑔1(𝜌𝑛

1,𝑗 , 𝑉 𝑛

2,𝑗 , 𝑉 𝑛

3,𝑗 )

≥𝜌𝑛

1,𝑗 (1 −𝜆𝑉𝑛

1,𝑗 +𝛼1,2𝑉𝑛

2,𝑗 +𝛼1,3𝑉𝑛

3,𝑗 )

≥0.

To obtain the upper bound we use the third and fourth property on 𝑔𝑒of Assumption 2.5 and the estimates

from Lemma 3.1:

𝜌𝑛+1

1,𝑗

≤𝜌𝑛

1,𝑗 +𝜆𝑉𝑛

1,𝑗−1𝜌max

1−𝑉𝑛

1,𝑗 𝜌𝑛

1,𝑗 +𝜆𝑔1(𝜌max

1, 𝑉 𝑛

2,𝑗−1, 𝑉 𝑛

3,𝑗−1)−𝑔1(𝜌𝑛

1,𝑗 , 𝑉 𝑛

2,𝑗 , 𝑉 𝑛

3,𝑗 )

≤𝜌𝑛

1,𝑗 +𝜆𝛾0‖𝑣′‖‖𝜌‖+𝑉𝑛

1,𝑗 (𝜌max

1−𝜌𝑛

1,𝑗 ) + 𝜆𝑔1(𝜌max

1, 𝑉 𝑛

2,𝑗 , 𝑉 𝑛

3,𝑗 )−𝑔1(𝜌𝑛

1,𝑗 , 𝑉 𝑛

2,𝑗 , 𝑉 𝑛

3,𝑗 )(3.5)

≤𝜌𝑛

1,𝑗 +𝜆𝛾0‖𝑣′‖‖𝜌‖+𝑉𝑛

1,𝑗 +𝐿(𝜌max

1−𝜌𝑛

1,𝑗 ) (3.6)

≤𝜌𝑛

1,𝑗 +𝜆(𝛾0‖𝑣′‖‖𝜌‖+ 2‖𝑣‖) (𝜌max

1−𝜌𝑛

1,𝑗 )

≤𝜌max

1.

For the outgoing roads, we have to consider the cell 𝑗= 0 since for 𝑗 > 0 the maximum principle is given in [24].

Here, we use the conditions on the inﬂow which is positive and obtain with 𝑒∈ {2,3}:

𝜌𝑛+1

𝑒,0≥𝜌𝑛

𝑒,0−𝜆𝐹𝑒,0≥𝜌𝑛

𝑒,0(1 −𝜆‖𝑣‖)≥0.

Similarly with Lemma 3.1,

𝜌𝑛+1

𝑒,0=𝜌𝑛

𝑒,0+𝜆(𝐹𝑛

𝑒,−1−𝐹𝑛

𝑒,0)

≤𝜌𝑛

𝑒,0+𝜆(𝜌max

𝑒𝑉𝑛

𝑒,−1−𝜌𝑛

𝑒,0𝑉𝑛

𝑒,0)

≤𝜌𝑛

𝑒,0+𝜆(𝛾0‖𝑣′‖‖𝜌‖+‖𝑣‖)(𝜌max

𝑒−𝜌𝑛

𝑒,0)

≤𝜌max

𝑒.

Remark 3.3. The proof for the 2-to-1 junction is completely analogous except that we additionally need to

satisfy condition (2.17). This is necessary when proving the upper bound on the incoming roads. Instead of

(3.5) we have, e.g., for road 1:

𝜌𝑛+1

1,𝑗 ≤𝜌𝑛

1,𝑗 +𝜆𝛾0‖𝑣′‖‖𝜌‖+𝑉𝑛

1,𝑗 (𝜌max

1−𝜌𝑛

1,𝑗 ) + 𝜆𝑔1(𝜌max

1, 𝜌𝑛

2,−1, 𝑉 𝑛

3,𝑗 )−𝑔1(𝜌𝑛

1,𝑗 , 𝜌𝑛

2,−1, 𝑉 𝑛

3,𝑗 ).

As we assume (2.17) we can proceed as in (3.6) and apply the Lipschitz continuity of 𝑔𝑒. The rest of the proof

remains the same.

222 J. FRIEDRICH ET AL.

Remark 3.4. Note that the CFL condition (3.2) can be further relaxed. This is possible due to the properties

of the kernel function 𝜔𝜂and the deﬁnition of the velocities in (2.6) and (3.1b), respectively. Exemplary, for the

1-to-2 junction we can estimate 𝑉𝑛

1,𝑗 +𝐿in (3.6) by ‖𝑣‖. Hence, the relaxed CFL condition reads

𝜆≤1

𝛾0‖𝑣′‖‖𝜌‖+‖𝑣‖·

In order to prove a BV bound in space we consider the following total variation:

𝑖∈ℐ

𝑗<−1𝜌𝑛

𝑖,𝑗+1 −𝜌𝑛

𝑖,𝑗 +

𝑜∈𝒪

𝑖∈ℐ 𝜌𝑛

𝑜,0−𝜌𝑛

𝑖,−1+

𝑜∈𝒪

𝑗≥0𝜌𝑛

𝑜,𝑗+1 −𝜌𝑛

𝑜,𝑗 .(3.7)

For the derivation of a bounded variation estimate, we need to carefully follow the lines of ([11], Lem. 2)1. For

the outgoing roads we can proceed as outlined there for the case 𝑥 > 0 and for the incoming roads, we need to

use a regularization of the limiter as all proposed couplings 𝑔𝑒are only weakly diﬀerentiable in 𝜌𝑒. A suitable

regularization of the minimum function is for instance given in equation (4.9) of [11]. Then, we are able to follow

again the lines of ([11], Lem. 2) for 𝑥 < 0. Finally, we need to calculate a direct estimate on the middle sum in

(3.7). Putting all three estimates together by carefully collecting all the corresponding terms we end up with

an estimate on the total variation of the form

𝑇 𝑉 (𝜌)≤exp(𝐶(𝑇))𝑇 𝑉 (𝜌0) + const.

Applying ([24], Thm. 3.3) and similar steps as already described above we get a BV estimate in space and time

such that with Helly’s Theorem the convergence of a sub-sequence can be concluded. Without mentioning all

the details we aim to demonstrate that the collection of limiting functions 𝜌*

𝑒are weak solutions in the sense of

the Deﬁnition 2.3 and 2.4. The ﬁrst item of Deﬁnition 2.3 can be shown by using Lax-Wendroﬀ types arguments,

the calculations done in [11,24] as well as equations (2.6) and (3.1b), respectively. The calculation can be even

simpliﬁed since we deal with weak solutions. The second condition in Deﬁnition 2.3 holds by the BV estimates

derived above and Helly’s Theorem. The third condition can be also obtained by using Lax-Wendroﬀ type

arguments. Furthermore, the ﬂux conservation through the junction (2.9) becomes obvious by the deﬁnition of

the coupling conditions.

Now we turn to the conditions (i) and (ii) in Deﬁnition 2.4. Starting with (i), we see that the Examples

2.6–2.9 satisfy

𝑔𝑒({𝜌𝑖(𝑡, 𝑥)}𝑖∈ℐ ,{𝑉𝑜(𝑡, 𝑥)}𝑜∈𝒪)≤𝜌𝑒(𝑡, 𝑥)

𝑁+𝑀

𝑜=𝑀+1

𝛼𝑖,𝑜𝑉𝑜(𝑡, 𝑥)

if 𝜌𝑖(𝑡, 𝑥)≤𝜌max

𝑜,∀𝑜∈ 𝒪, 𝑖 ∈ ℐ. Equality is only obtained in the Example 2.6, while in the other models

further conditions have to be satisﬁed. In contrast, the condition (ii) is satisﬁed for all models by construction,

except Example 2.6.

Therefore, we can conclude that weak solutions in the sense of the Deﬁnitions 2.3 and 2.4 exist. Note that

we do not consider the uniqueness of those solutions, since the standard techniques are not applicable in a

straightforward way.

4. Limit 𝜂→ ∞

As mentioned in the introduction nonlocal traﬃc ﬂow models have been introduced to incorporate the chal-

lenges occurring in nowadays traﬃc such as autonomous cars. The nonlocal range 𝜂can be therefore interpreted

as a connection radius between cars, where the latter only need information about the downstream traﬃc.

Apparently, in case of non-autonomous cars, the nonlocal range can be seen as the sight of a human driver. The

1We note that the precise calculations are more involved than in ([11], Lem. 2), in particular with regard to Example 2.7 and 2.9.

NETWORK MODELS FOR NONLOCAL TRAFFIC FLOW 223

question arises what happens if autonomous cars would have perfect information about the downstream traﬃc.

To treat this question from a theoretical point of view, the nonlocal range should tend to inﬁnity in the network

setting (2.1) and (2.5). For a similar traﬃc model on a single road, this feature has been already analyzed in

Corollary 1.2 of [9]. Therein, the model tends to a linear transport equation with maximum velocity. From the

modeling perspective this result is kind of intuitive: All drivers know exactly what happens in front of them on

the whole road such that they can react in advance. So they are able to keep the speed at the maximum velocity

regardless of the current traﬃc situation. However, a network model includes diﬀerent types of junctions and

diﬀerent maximum densities or speed functions, respectively, leading to a non-intuitive behavior of drivers.

For our considerations, we start with some general estimates on the nonlocal velocities and discuss the 1-to-1

junction model in more detail (since all necessary estimates have been already established in [11]). Note that

we now use the notation 𝑉𝜂

𝑒(𝑡, 𝑥) for the velocities to emphasize their dependence on the nonlocal range 𝜂.

Lemma 4.1. Let Ωℐ= (0,∞)×(−∞,0) and Ω𝒪= (0,∞)×(0,∞). Deﬁne Ω𝑒= Ωℐif 𝑒∈ ℐ and Ω𝑒= Ω𝒪if

𝑒∈ 𝒪. In addition, let 𝜌𝑒∈𝐿∞(Ω𝑒)be a collection of functions which are either of compact support on Ω𝑒or

identical to zero and let 𝐾𝑒be a compact subset of Ω𝑒. Then, we obtain for the velocities (2.6)the following:

lim

𝜂→∞ 𝐾𝑒

|𝑉𝜂

𝑒(𝑡, 𝑥)|𝑑𝑡𝑑𝑥 = 0, 𝑒 ∈ ℐ,lim

𝜂→∞ 𝐾𝑒

|𝑉𝜂

𝑒(𝑡, 𝑥)−𝑣𝑒(0)|𝑑𝑡𝑑𝑥 = 0, 𝑒 ∈ 𝒪,

lim

𝜂→∞ 𝐾𝑖

|𝑉𝜂

𝑒(𝑡, 𝑥)−𝑣𝑒(0)|𝑑𝑡𝑑𝑥 = 0, 𝑒 ∈ 𝒪, 𝑖 ∈ ℐ.

(4.1)

Proof. We start with 𝑒∈ ℐ. If 𝜌𝑒(𝑡, ·)≡0 we directly obtain

𝑉𝜂

𝑒(𝑡, 𝑥) = min{𝑥+𝜂,0}

𝑥

𝜔𝜂(𝑦−𝑥)𝑣𝑒(0)𝑑𝑦 ≤𝜔𝜂(0)𝑣𝑒(0)|min{𝑥+𝜂, 0} − 𝑥}|

which goes to zero for 𝜂→ ∞ due to (2.4). Now let us consider the case 𝜌𝑒(𝑡, ·) being of compact support.

Without loss of generality, we assume that the support is on [𝑎𝑒(𝑡), 𝑏𝑒(𝑡)] with 𝑎𝑒(𝑡)< 𝑏𝑒(𝑡)≤0. Here, we have

𝑉𝜂

𝑒(𝑡, 𝑥) = min{𝑏𝑒(𝑡),𝑥+𝜂}

max{𝑥,𝑎𝑒(𝑡)}

𝜔𝜂(𝑦−𝑥)𝑣𝑒(𝜌𝑒(𝑡, 𝑦))𝑑𝑦 +min{𝑎𝑒(𝑡),𝑥+𝜂}

min{𝑥,𝑎𝑒(𝑡)}

𝜔𝜂(𝑦−𝑥)𝑣𝑒(0)𝑑𝑦

+min{max{𝑥+𝜂,𝑏𝑒(𝑡)},0}

max{𝑥,𝑏𝑒(𝑡)}

𝜔𝜂(𝑦−𝑥)𝑣𝑒(0)𝑑𝑦

𝜂→∞

→𝑏𝑒(𝑡)

max{𝑥,𝑎𝑒(𝑡)}

𝜔𝜂(𝑦−𝑥)𝑣𝑒(𝜌𝑒(𝑡, 𝑦))𝑑𝑦 +𝑎𝑒(𝑡)

min{𝑥,𝑎𝑒(𝑡)}

𝜔𝜂(𝑦−𝑥)𝑣𝑒(0)𝑑𝑦

+0

max{𝑥,𝑏𝑒(𝑡)}

𝜔𝜂(𝑦−𝑥)𝑣𝑒(0)𝑑𝑦

≤𝜔𝜂(0)𝑣𝑒(0) (|𝑏𝑒(𝑡)−max{𝑥, 𝑎𝑒(𝑡)}| +|𝑎𝑒(𝑡)−min{𝑥, 𝑎𝑒(𝑡)}| +|max{𝑥, 𝑏𝑒(𝑡)}|)

<∞

𝜂→∞

→0.

As we consider the pointwise limit, 𝑡and 𝑥are ﬁxed and so are 𝑎𝑒(𝑡) and 𝑏𝑒(𝑡). Hence, the intervals in the last

estimate are all ﬁnite and with lim𝜂→∞ 𝜔𝜂(0) = 0, cf. (2.4), we obtain that the upper bound on 𝑉𝜂

𝑒(𝑡, 𝑥) goes

to zero. As we also have 𝑉𝜂

𝑒(𝑡, 𝑥)≥0, we can conclude that 𝑉𝜂

𝑒(𝑡, 𝑥)→0 pointwise for 𝜂→ ∞ on Ωℐ.

224 J. FRIEDRICH ET AL.

If we consider 𝑒∈ 𝒪,𝑥 > 0 and 0 ≤𝑎𝑒(𝑡)< 𝑏𝑒(𝑡), we get

𝑉𝜂

𝑒(𝑡, 𝑥) = min{𝑏𝑒(𝑡),𝑥+𝜂}

max{𝑥,𝑎𝑒(𝑡)}

𝜔𝜂(𝑦−𝑥)𝑣𝑒(𝜌𝑒(𝑡, 𝑦))𝑑𝑦 +min{𝑎𝑒(𝑡),𝑥+𝜂}

min{𝑥,𝑎𝑒(𝑡)}

𝜔𝜂(𝑦−𝑥)𝑣𝑒(0)𝑑𝑦

+max{𝑥+𝜂,𝑏𝑒(𝑡)}

max{𝑥,𝑏𝑒(𝑡)}

𝜔𝜂(𝑦−𝑥)𝑣𝑒(0)𝑑𝑦

𝜂→∞

=𝑏𝑒(𝑡)

max{𝑥,𝑎𝑒(𝑡)}

𝜔𝜂(𝑦−𝑥)𝑣𝑒(𝜌𝑒(𝑡, 𝑦))𝑑𝑦 +𝑎𝑒(𝑡)

min{𝑥,𝑎𝑒(𝑡)}

𝜔𝜂(𝑦−𝑥)𝑣𝑒(0)𝑑𝑦

+∞

max{𝑥,𝑏𝑒(𝑡)}

𝜔𝜂(𝑦−𝑥)𝑣𝑒(0)𝑑𝑦

→𝑣𝑒(0).

Since the ﬁrst two terms can be again estimated from above by

𝜔𝜂(0)𝑣𝑒(0) (|max{𝑥, 𝑎𝑒(𝑡)} − 𝑏𝑒(𝑡)|+|min{𝑥, 𝑎𝑒(𝑡)} − 𝑎𝑒(𝑡)|)→0

and for the last term we can use that ∞

0𝜔𝜂(𝑦)𝑑𝑦 = 1 due to (2.4), we end up with

1≥∞

max{𝑥,𝑏(𝑡)}

𝜔𝜂(𝑦−𝑥)𝑑𝑦 = 1 −max{𝑥,𝑏(𝑡)}

𝑥

𝜔𝜂(𝑦−𝑥)𝑑𝑦 ≥1−𝜔𝜂(0)|𝑏(𝑡)−𝑥| → 1

for 𝜂→ ∞. If 𝜌𝑒(𝑡, ·)≡0, we proceed as for the last term. Hence, we have 𝑉𝜂

𝑒(𝑡, 𝑥)→𝑣𝑒(0) pointwise for 𝑒∈ 𝒪

on Ω𝒪. In addition, if we consider 𝑥∈(−𝜂, 0) the above calculations for 𝑉𝜂

𝑒(𝑡, 𝑥) and 𝑒∈ 𝒪 are completely

similar by setting 𝑥= 0 in the lower bounds of the integrals. Note that the case 𝑥≤ −𝜂disappears in the

pointwise limit by choosing 𝜂large enough for a ﬁxed 𝑥. So we also obtain 𝑉𝜂

𝑒(𝑡, 𝑥)→𝑣𝑒(0) pointwise for 𝑒∈ 𝒪

on Ωℐ.

In addition, we have 𝑉𝜂

𝑒(𝑡, 𝑥)≤𝑣𝑒(0) ∀(𝑡, 𝑥)∈Ω𝑒, 𝑒 ∈𝐸with 𝑣𝑒(0) being an integrable function on each

compact subset of Ω𝑒. Hence, Lebesgue’s dominated convergence theorem yields the assertion.

Next, we prove the convergence for 𝜂→ ∞ for the 1-to-1 junction model. Note that for simplicity we write

𝜌instead of 𝜌1and 𝜌2, since in the case of a 1-to-1 junction the solution is uniquely given on whole R.

Proposition 4.2. Let the hypotheses (2.4)hold and let 𝜌0∈BV(R, 𝐼 ). Then, the solution 𝜌𝜂of the 1-to-1

junction model given by (2.1),(2.5)with (2.13)converges for 𝜂→ ∞ to the unique entropy solution of the local

problem

𝜕𝑡𝜌+𝜕𝑥(min{𝜌𝑣2(0), 𝜌max

2𝑣2(0)})=0

𝜌(0, 𝑥) = 𝜌0(𝑥).(4.2)

Proof. We ﬁrst note that the 1-to-1 model can be rewritten by deﬁning 𝜌(𝑡, 𝑥) = 𝜌1(𝑡, 𝑥) if 𝑥 < 0 and 𝜌(𝑡, 𝑥) =

𝜌2(𝑡, 𝑥) if 𝑥 > 0 as

𝜕𝜌(𝑡, 𝑥) + 𝜕𝑥(𝜌(𝑡, 𝑥)𝑉1(𝑡, 𝑥) + 𝑔(𝜌(𝑡, 𝑥), 𝑉2(𝑡, 𝑥))) = 0,

which simpliﬁes the notation in the rest of the proof, see also [11].

The existence of solutions for 𝜂→ ∞ is given since the BV estimates and the maximum principle in [11] are

uniform as 𝜂→ ∞ such that Helly’s Theorem yields up to a subsequence the convergence of the solution in the

𝐿1

𝑙𝑜𝑐 norm.

NETWORK MODELS FOR NONLOCAL TRAFFIC FLOW 225

We start from the entropy inequality and add and subtract 𝑣2(0) at the right place, such that for a ﬁxed

𝜅∈Rand 𝜑∈𝐶1

0([0,∞)×R;R+), we obtain

0≤∞

0∞

−∞

(|𝜌−𝜅|𝜑𝑡+|𝜌−𝜅|𝑉1𝜑𝑥+ sgn(𝜌−𝜅)(˜𝑔(𝜌)−˜𝑔(𝜅))(𝑉2±𝑣2(0))𝜑𝑥

−sgn(𝜌−𝜅)𝜅𝜕𝑥𝑉1𝜑−sgn(𝜌−𝜅)˜𝑔(𝜅)𝜕𝑥𝑉2𝜑)(𝑡, 𝑥)𝑑𝑥𝑑𝑡 +∞

−∞

|𝜌0(𝑥)−𝜅|𝜑(𝑥, 0)𝑑𝑥,

where ˜𝑔is deﬁned as ˜𝑔(𝜌) = min{𝜌, 𝜌max

2}. First, note that due to the compact support of the initial condition

and the ﬁnite speed of the waves, which are bounded by ‖𝑣‖, the solution 𝜌(𝑡, 𝑥) is also of compact support

such that we can apply Lemma 4.1 in the following. Additionally, as the test functions are of compact support,

there exist 𝑇 > 0 and 𝑅 > 0 such that 𝜑(𝑡, 𝑥) = 0 for |𝑥|> 𝑅 or 𝑡>𝑇. Using the latter and Lemma 4.1, we

obtain for 𝜂→ ∞:

∞

0∞

−∞

|𝜌−𝜅|𝑉1𝜑𝑥𝑑𝑥𝑑𝑡 ≤(‖𝜌‖+|𝜅|)‖𝜑𝑥‖𝑇

0𝑅

−𝑅

|𝑉1|𝑑𝑥𝑑𝑡 →0

∞

0∞

−∞

sgn(𝜌−𝜅)(˜𝑔(𝜌)−˜𝑔(𝜅))(𝑉2−𝑣2(0))𝜑𝑥𝑑𝑥𝑑𝑡 ≤2‖𝜌‖‖𝜑𝑥‖𝑇

0𝑅

−𝑅

|𝑉2−𝑣2(0)|𝑑𝑥𝑑𝑡

→0.

Again, thanks to the compactness of the test function and condition (2.4) of the kernel, we have

∞

0∞

−∞

sgn(𝜌−𝜅)𝜅𝜕𝑥𝑉1𝜑𝑑𝑥𝑑𝑡 ≤16|𝜅|‖𝜑‖‖𝑣‖𝑇 𝑅𝜔𝜂(0) →0,

∞

0∞

−∞

sgn(𝜌−𝜅)˜𝑔(𝜅)𝜕𝑥𝑉2𝜑)𝑑𝑥𝑑𝑡 ≤16‖𝜌‖‖𝜑‖‖𝑣‖𝑇 𝑅𝜔𝜂(0) →0.

Here, we use that

|𝜕𝑥𝑉1| ≤ min{𝑥+𝜂,0}

min{𝑥,0}

𝑣1(𝜌(𝑡, 𝑦))𝜔′

𝜂(𝑦−𝑥)𝑑𝑦

+𝑣1(𝜌(𝑡, min{𝑥, 0}))𝜔𝜂(0)

+𝑣1(𝜌(𝑡, min{𝑥+𝜂, 0}))𝜔𝜂(0)

≤‖𝑣‖min{𝑥+𝜂,0}

min{𝑥,0}

𝜔′

𝜂(𝑦−𝑥)𝑑𝑦

+ 2‖𝑣‖𝜔𝜂(0)

≤4‖𝑣‖𝜔𝜂(0),

and analogously we get |𝜕𝑥𝑉2| ≤ 4‖𝑣‖𝜔𝜂(0). We are left with

0≤∞

0∞

−∞

(|𝜌−𝜅|𝜑𝑡+ sgn(𝜌−𝜅)(𝑔(𝜌)−𝑔(𝜅))(𝑣2(0))𝜑𝑥)(𝑡, 𝑥)𝑑𝑥𝑑𝑡

+∞

−∞

|𝜌0(𝑥)−𝜅|𝜑(𝑥, 0)𝑑𝑥,

which is the entropy inequality of the corresponding local model (4.2).

Some remarks are in order.

Remark 4.3. The proof can be easily adapted to the model without considering any junctions, i.e.,

𝜕𝑡𝜌+𝜕𝑥(𝑉𝜂(𝑡, 𝑥)𝜌)=0

𝜌(0, 𝑥) = 𝜌0(𝑥),

226 J. FRIEDRICH ET AL.

with 𝑉𝜂(𝑡, 𝑥) = 𝑥+𝜂

𝑥𝑣(𝜌(𝑡, 𝑦))𝜔𝜂(𝑦−𝑥)𝑑𝑦. As already mentioned for a similar model a convergence result for

𝜂→ ∞ has already been discovered in Corollary 1.2 of [9]. Using the proof above we obtain the convergence to

the unique weak entropy inequality of the linear transport equation:

𝜕𝑡𝜌+𝜕𝑥(𝑣(0)𝜌) = 0

𝜌(0, 𝑥) = 𝜌0(𝑥),

i.e.,

0≤∞

0∞

−∞

(|𝜌−𝜅|𝜑𝑡+|𝜌−𝜅|𝑣(0)𝜑𝑥𝑑𝑥𝑑𝑡 +∞

−∞

|𝜌0(𝑥)−𝜅|𝜑(𝑥, 0)𝑑𝑥.

The linear transport is a situation on the road where no traﬃc jams occur since even if the traﬃc is at maximum

density all cars drive at the fastest velocity possible. Even though this sounds more like an idealistic property,

the model captures the limiting behavior for an inﬁnite interaction range resulting in an “optimized” traﬃc.

Remark 4.4. The model (4.2) has been also recovered in [1] in the context of production with velocity 𝑣2(0)

and capacity 𝑣2(0)𝜌max

2. Transferring the idea of a production model to traﬃc ﬂow, we observe that all cars

want to move at a constant speed 𝑣2(0) as long as there is enough capacity, which is mainly determined by the

maximum density of the second road 𝜌max

2.

Let us explain the model dynamics with the help of Riemann initial data, i.e.,

𝜌0(𝑥) = 𝜌𝐿,if 𝑥 < 0,

𝜌𝑅,if 𝑥 > 0,

with 𝜌𝐿∈[0, 𝜌max

1] and 𝜌𝑅∈[0, 𝜌max

2]. We recall that the change in the velocity is located at 𝑥= 0. There are

only two possible solutions:

– if 𝜌𝐿≤𝜌max

2, the solution is given by a linear transport with velocity 𝑣2(0),

– if 𝜌𝐿> 𝜌max

2, we have a rarefaction wave with the density equal to 𝜌max

2as an intermediate state.

Both cases are illustrated in Figure 1.

In the ﬁrst case, i.e., the linear transport, no traﬃc jams occurs and traﬃc is transported at a constant

velocity. As we allow the drivers to accelerate as soon as they are aware of the junction, which they always

are in the limit 𝜂→ ∞, they do not respect the given maximum velocity on the ﬁrst road. They always drive

with the maximum velocity of the second road which might be higher or lower than the one of the ﬁrst road.

So the model “optimizes” the traﬃc but might not care about velocity restrictions. The Riemann problems

producing a rarefaction wave induce a congestion at the end of the ﬁrst road and beginning of the second road

due to the fact that the initial traﬃc on the ﬁrst road cannot pass completely onto the second road. Again, the

model solves this problem in a reasonable way in terms of traﬃc congestion, i.e., as much ﬂow as possible is

sent through the junction at a constant speed.

Let us now turn to the more general nonlocal network setting. Here, we need to satisfy the local version of

Deﬁnition 2.3,i.e., Deﬁnition 4.2.4 of [26]. As aforementioned, these are basically the items 1 and 2 of Deﬁnition

2.3, even now 𝜌𝑒for 𝑒∈ {1, . . . , 𝑀}has to be a weak solution on (−∞,0) instead of (−∞,−𝜂). Obviously, the

Rankine-Hugoniot condition (2.9) is also satisﬁed for 𝜂→ ∞. Since the BV estimates established in ([24],

Thm. 3.2) and ([11], Lem. 2) also hold for 𝜂→ ∞ thanks to lim𝜂→∞ 𝜔𝜂(0) = 0, the second item is clear. The

calculations are similar to above. In particular, for the outgoing roads in the examples 2.6–2.9 the dynamics in

the limit are described by

𝜕𝑡𝜌𝑒+𝜕𝑥(𝜌𝑒𝑣𝑒(0)) = 0.

For the incoming roads the convergence of the coupling 𝑔𝑒plays the most important role. By obtaining the

pointwise limit of 𝑔𝑒we can proceed similarly as in the proof of Proposition 4.2 also using Lemma 4.1. Therefore,

we obtain the following limit models for 𝑥 < 0:

NETWORK MODELS FOR NONLOCAL TRAFFIC FLOW 227

Figure 1. Solution of the two Riemann problems for (4.2).

– Example 2.6:

𝜕𝑡𝜌1+𝜕𝑥(min{𝛼1,2𝜌1(𝑡, 𝑥), 𝜌max

2}𝑣2(0) + min{𝛼1,3𝜌1(𝑡, 𝑥), 𝜌max

3}𝑣3(0)) = 0,

– Example 2.7:

𝜕𝑡𝜌1+𝜕𝑥min{𝜌1(𝑡, 𝑥)(𝛼1,2𝑣2(0) + 𝛼1,3𝑣3(0)),𝜌max

2𝑣2(0)

𝛼1,2

,𝜌max

3𝑣3(0)

𝛼1,3

}= 0,

– Example 2.8,𝑒∈ {1,2}:

𝜕𝑡𝜌𝑒+𝜕𝑥(min{𝜌𝑒(𝑡, 𝑥),max{𝑞𝑒,3𝜌max

3, 𝜌max

3−𝜌𝑒−(𝑡, 0−)}}𝑣3(0)) = 0,

– Example 2.9,𝑒∈ {1,2}:

𝜕𝑡𝜌𝑒+𝜕𝑥min{𝜌𝑒(𝑡, 𝑥), 𝑞𝑒,3𝜌max

3, 𝑞𝑒,3/𝑞𝑒−,3𝜌𝑒−(𝑡, 0−)}𝑣3(0)= 0.

By construction all models are similar to a production type model and inherit the property that the ﬂow moving

at a constant speed is restricted by a capacity.

We note that the computations for the Example 2.7 are slightly more involved since the nonlocal velocities are

inside the minimum.

5. Numerical simulations

In the following we consider a more complex network to demonstrate the properties of the proposed models.

The network under consideration has a diamond structure and consists of nine roads and six vertices, see

Figure 2.

As we have not prescribed inﬂow and outﬂow conditions in this work the roads 0 and 8 are used as artiﬁcial

roads to avoid posing boundary conditions at intersections. In particular, due to the nonlocality of the dynamics,

we need to impose more than just a single value boundary condition at the end of the network. The length

228 J. FRIEDRICH ET AL.

Figure 2. Network structure.

Table 1. Parameters of the diamond network.

Road 𝑒012345678

¯𝜌𝑒0.4 0.4 0.4 0.4 0.8 0.4 0.8 0.2 0.2

𝑣max

𝑒0.5 0.5 2 2 0.5 2 0.5 1 1

of these artiﬁcial roads is set to (𝑎0, 𝑏0) = (−∞,0) and (𝑎8, 𝑏8) = (0,∞), respectively. For all other roads

𝑒= 1,...,7 we set (𝑎𝑒, 𝑏𝑒) = (0,1). We also remark that we do not consider the artiﬁcial roads when calculating

traﬃc measures, i.e., the outﬂow of the system is measured at the end of road 7. At the vertices 2 and 3 we

have to prescribe distribution parameters and priority parameters at the vertices 4 and 5. These are given by

𝛼1,2=𝛼1,3= 0.5, 𝛼2,4= 1/5, 𝛼2,5= 4/5 and 𝑞3,6= 4/5, 𝑞4,6= 1/5, 𝑞5,7= 4/5, 𝑞6,7= 1/5. The velocity

function on all roads 𝑒∈ {0,...,8}is described by

𝑣𝑒(𝜌) = 𝑣max

𝑒(1 −𝜌) (5.1)

and hence the maximum density on all roads is equal. In addition, the initial conditions are given by

𝜌𝑒,0(𝑥) = ¯𝜌𝑒∀𝑥∈(𝑎𝑒, 𝑏𝑒), 𝑒 ∈ {0,...,8}

and explicit parameters can be found in Table 1.

We note that the delicate point in this network is the choice of the maximum velocities and initial densities for

the roads 4 and 5. Due to the high velocity on road 5 and the shorter distance to the end of road 7, the way over

the more congested road 4 seems to be not favorable in case of measuring travel times. In all simulations, we

choose a linear decreasing kernel function 𝜔𝜂= 2(𝜂−𝑥)/𝜂2and use 𝛥𝑥 = 0.01 combined with an adaptive CFL

condition determined by (3.2) for all norms in each time step 𝑡𝑛. In the following, we denote by the nonlocal

maximum ﬂux model the network model using the coupling condition described in Example 2.6 for the 1-to-2

junctions at the vertices 2 and 3 and the one described in Example 2.8 for the 2-to-1 junctions at the vertices 4

and 5. Analogously, the nonlocal distribution model corresponds to the coupling conditions of Example 2.7 and

Example 2.9.

For the comparison of the diﬀerent network models we consider the following traﬃc measures, see e. g.

[29,43,47]:

– total travel time

𝑇𝑇𝑇 =

7

𝑒=1 𝑇

01

0

𝜌𝑒(𝑡, 𝑥)𝑑𝑥𝑑𝑡,

NETWORK MODELS FOR NONLOCAL TRAFFIC FLOW 229

Table 2. Traﬃc measures.

Model Outﬂow 𝑇 𝑇 𝑇 Congestion

Distribution 2.1531 62.9 48.744

Maximum ﬂux 4.6774 44.577 16.144

– outﬂow

𝑂=𝑇

0

𝑓7(𝑡, 1, 𝜌7(𝑡, 1))𝑑𝑡,

– congestion measure

𝐶𝑀 =

7

𝑒=1 𝑇

0

max 0,1

0

𝜌𝑒(𝑡, 𝑥)−𝑓𝑒(𝑡, 𝑥, 𝜌𝑒(𝑡, 𝑥))

𝑣𝑒,ref

𝑑𝑥𝑑𝑡.

With some abuse of notation the ﬂux can be either local or nonlocal depending on the models used. We also

remark that the reference velocity is chosen to be 𝑣𝑒,ref = 0.5𝑣max

𝑒.

5.1. Nonlocal models

We start by comparing the nonlocal maximum ﬂux with the nonlocal distribution model. Therefore, we set

𝜂= 0.5 and consider the ﬁnal time of 𝑇= 20. Figure 3displays the approximate solution at the ﬁnal time 𝑇.

Apparently, the nonlocal distribution model results in a congested network. Over time road 6 becomes even

further congested with the traﬃc jam moving backwards to road 4 and also to road 1 over road 3 while the

other roads are rather empty.

These eﬀects can be also recognized in Table 2where the outﬂow of the nonlocal maximum ﬂux model is

more than twice as high as in the congested nonlocal distribution model. Similar observations can be made

for the total travel time (𝑇𝑇𝑇) and the congestion measure. The main reason for this behavior is that in the

nonlocal distribution model the prescribed distribution rates must be exactly kept. On the contrary, the nonlocal

maximum ﬂux does not necessarily fulﬁll them exactly. In particular, the actual distribution2at the vertex 3

over time shows that the distribution onto road 5 is in the interval [0.93,0.98] instead of the prescribed value

of 0.8. As a consequence less vehicles enter road 4 and a traﬃc jam can be avoided.

Furthermore, in the nonlocal maximum ﬂux model the actual priority parameters at vertex 5 start to change

away from the prescribed values around approximately 𝑡= 4 such that for 𝑡∈[5,20] the ratio from road 6 to

7 is higher as from road 5 to 7. This helps to resolve the traﬃc jam at road 7. The nonlocal distribution model

model keeps again the prescribed parameters and is not able to resolve the traﬃc jam.

This example also demonstrates that both models are in line with the original modeling ideas: The nonlocal

distribution model model obeys the distribution rates (by construction) while the nonlocal maximum ﬂux model

achieves maximum ﬂuxes.

5.2. Nonlocal vs. local models

Next, we compare the nonlocal modeling approaches to local network models. These are described by (2.1)

with

𝑓𝑒(𝑡, 𝑥, {𝜌𝑘}𝑘∈𝐸) = 𝜌𝑒(𝑡, 𝑥)𝑣𝑒(𝜌𝑒(𝑡, 𝑥))

2Actual means that the inﬂows of the outgoing roads over time are divided by the outﬂow of the incoming road.

230 J. FRIEDRICH ET AL.

Figure 3. Approximate solutions of the nonlocal maximum ﬂux (nl. max. f. model ) and the

nonlocal distribution model (nl. dist. model) at 𝑇= 20 for diﬀerent roads and 𝜂= 0.5.

for 𝑒∈𝐸,𝑡 > 0 and 𝑥∈(𝑎𝑒, 𝑏𝑒). The local model is equipped with coupling conditions satisfying demand and

supply functions, cf. [29]:

𝐷𝑒(𝜌) = 𝜌𝑣𝑒(𝜌),if 𝜌≤𝜎𝑒,

𝜎𝑒𝑣𝑒(𝜎𝑒),if 𝜌 > 𝜎𝑒,𝑆𝑒(𝜌) = 𝜎𝑒𝑣𝑒(𝜎𝑒),if 𝜌≤𝜎𝑒,

𝜌𝑣𝑒(𝜌),if 𝜌>𝜎𝑒,(5.2)

NETWORK MODELS FOR NONLOCAL TRAFFIC FLOW 231

Figure 4. Approximate solution of the local model with the coupling conditions from [29] and

the nonlocal maximum ﬂux model (nl. max. f. model) at 𝑇= 20 for 𝜂= 0.05 on road 4.

where 𝜎𝑒is the maximum point of the ﬂux function 𝜌𝑣𝑒(𝜌)3. As already mentioned the coupling conditions for

the nonlocal maximum ﬂux model are also inspired by this approach. For completeness these are in the local

model for the 1-to-2 junction (using the notation from the previous sections)

𝑓𝑒(𝑡, 0+, 𝜌1, 𝜌𝑒) = min{𝛼1,𝑒𝐷1(𝜌1(𝑡, 0−)), 𝑆𝑒(𝜌𝑒(𝑡, 0+))}, 𝑒 ∈ {2,3}

𝑓1(𝑡, 0−, 𝜌1, 𝜌2, 𝜌3) = 𝑓2(𝑡, 0+, 𝜌1, 𝜌2) + 𝑓3(𝑡, 0+, 𝜌1, 𝜌3)

and for the 2-to-1 junction:

𝑓𝑒(𝑡, 0−, 𝜌𝑒, 𝜌𝑒−) = min{𝐷𝑒(𝜌𝑒(𝑡, 0−)),max{𝑞𝑒,3𝑆3(𝜌3(𝑡, 0+)), 𝑆3(𝜌3(𝑡, 0+)) −𝐷𝑒−(𝜌𝑒−(𝑡, 0−))}}

𝑓3(𝑡, 0+, 𝜌1, 𝜌2, 𝜌3) = 𝑓1(𝑡, 0−, 𝜌1, 𝜌2) + 𝑓2(𝑡, 0−, 𝜌2, 𝜌1).

In contrast the local coupling conditions inspiring the nonlocal distribution model can be found in [26] and can

be summarized as: for the 1-to-2 junction

𝑓1(𝑡, 0−, 𝜌1, 𝜌2, 𝜌3) = min{𝐷1(𝜌1(𝑡, 0−)), 𝑆2(𝜌2(𝑡, 0+))/𝛼1,2, 𝑆3(𝜌3(𝑡, 0+))/𝛼1,3}

𝑓𝑒(𝑡, 0+, 𝜌1, 𝜌𝑒) = 𝛼1,𝑒𝑓1(𝑡, 0−, 𝜌1, 𝜌2, 𝜌3), 𝑒 ∈ {2,3}

and for the 2-to-1 junction:

𝑓𝑒(𝑡, 0−, 𝜌𝑒, 𝜌𝑒−) = min{𝐷𝑒(𝜌𝑒(𝑡, 0−)), 𝑞𝑒,3/𝑞𝑒−,3𝐷𝑒−(𝜌𝑒−(𝑡, 0−)), 𝑞𝑒,3𝑆3(𝜌3(𝑡, 0+))}

𝑓3(𝑡, 0+, 𝜌1, 𝜌2, 𝜌3) = 𝑓1(𝑡, 0−, 𝜌1, 𝜌2) + 𝑓2(𝑡, 0−, 𝜌2, 𝜌1).

In order to solve the local models numerically we use the Godunov scheme with the coupling conditions as

presented in [29].

Figure 4shows the approximate solutions of the local supply and demand approach and the nonlocal maximum

ﬂux model at 𝑇= 20 and for 𝜂= 0.05 on road 4. It can be seen that both models lead to diﬀerent solutions at

road 4 as the end of the traﬃc jam is located further downstream in the nonlocal models. Considering smaller

values of 𝜂(and also 𝛥𝑥) this behavior does not change. Hence, this example provides numerical evidence that

in the network case for the nonlocal maximum ﬂux model no convergence for 𝜂→0 to the local network model

can be expected. We remark that in [11] it was observed that the simple 1-to-1 junction tends towards the

3We note that the choice of the velocity function in (5.1) ensures the existence of a unique maximum point.

232 J. FRIEDRICH ET AL.

Table 3. Traﬃc measures for the nonlocal maximum ﬂux model and the local model with the

coupling conditions from [29] at 𝑇= 20.

Model Outﬂow 𝑇 𝑇 𝑇 Congestion

𝜂= 0.5 4.6774 44.577 16.144

𝜂= 0.25 4.3651 46.971 19.114

𝜂= 0.1 4.1546 49.033 21.611

𝜂= 0.05 4.0719 49.924 22.752

[29] 3.7862 52.692 26.09

Figure 5. Approximate solutions of the local model with the coupling conditions from [26]

and the nonlocal distribution model (nl. distr. model) with diﬀerent values of 𝜂at 𝑇= 1 (left

column) and 𝑇= 20 (right column) for the roads 1 (top row) and 4 (bottom row).

vanishing viscosity solution. This seems also not the case for the nonlocal maximum ﬂux model. The vanishing

viscosity solution of the considered network behaves very similar to the solution obtained by the supply and

demand approach and hence convergence can be ruled out.

Remark 5.1. Note that for the vanishing viscosity approach no distribution and priority parameters have to

be prescribed and the maximum densities on all roads have to be equal. However, an approximate solution can

be obtained by the numerical scheme presented in [46].

Let us now compare the traﬃc measures for the nonlocal maximum ﬂux model with 𝜂∈ {0.5,0.25,0.1,0.05}and

the corresponding local model. Obviously, the nonlocal models perform better than the local model regarding

the traﬃc measures while the advantages increase even further with larger nonlocal range 𝜂.

NETWORK MODELS FOR NONLOCAL TRAFFIC FLOW 233

Table 4. Traﬃc measures for the nonlocal distribution model and the local model with the

coupling conditions from [26] at 𝑇= 20.

Model Outﬂow 𝑇 𝑇 𝑇 Congestion

𝜂= 0.5 2.1531 62.9 48.744

𝜂= 0.25 2.1485 63.345 48.219

𝜂= 0.1 2.1455 63.742 47.96

𝜂= 0.05 2.1446 63.89 47.9

[26] 2.1434 64.102 47.782

Now, we compare the nonlocal distribution model to the corresponding local model with coupling conditions

from [26]. For small time and large 𝜂the nonlocal solution provides diﬀerent dynamics as the local model.

Even though these eﬀects are less for larger times and/or smaller values of 𝜂as displayed in Figure 5. Here,

we selected exemplary two roads, namely road 1 and road 4. For small time periods we see that the solution

with large 𝜂is diﬀerent to the local solution but has a kind of smoothing eﬀect across the transition area.

For 𝜂small at 𝑇= 1 and 𝑇= 20, the nonlocal approximate solution suggests a numerical convergence to the

local approximate solution. It is interesting to see that for larger times and 𝜂the nonlocal eﬀects become less

signiﬁcant. In particular, road 1 is the only road displaying visible diﬀerences between the local and nonlocal

solution while the other roads behave similarly to road 4. Nevertheless, even on road 1 both models result in a

traﬃc jam but they diﬀer how the transition from free ﬂow to traﬃc jam is created.

In addition, the traﬃc measures displayed in Table 4computed at the ﬁnal time 𝑇= 20 support the

observation that in this example the nonlocal distribution model model behaves similarly as its local counterpart.

Nevertheless, we intend to stress that these observations are only due to the choice of the speciﬁc example. We

notice that in other scenarios the diﬀerences between the nonlocal distribution model and its local counterpart

are more signiﬁcant for large times and 𝜂, even though for smaller values of 𝜂the solution seems to numerically

converge to the local one. In our numerical study we do not ﬁnd any example to rule out the convergence as we

do for the nonlocal maximum ﬂux model.

6. Conclusion

We have introduced a network model for nonlocal traﬃc. The modeling is essentially based on coupling

conditions for 1-to-1, 2-to-1 and 1-to-2 junctions. Using a ﬁnite volume numerical scheme we can show a

maximum principle and the existence of weak solutions for the network model. Further investigations include

the consideration of the limiting behaviour for 𝜂→ ∞. The numerical simulations demonstrate the ideas of the

proposed junction models. We also investigate the limit 𝜂→0 numerically and notice that in case of the nonlocal

maximum ﬂux model convergence to its local counterpart can be ruled out while in the nonlocal distribution

model convergence can be observed.

Future work will include the investigation of a general 𝑀-to-𝑁junction. We also intend to derive coupling

conditions for other nonlocal modeling equations, such as the second order traﬃc ﬂow model proposed in [12].

Acknowledgements. The ﬁnancial support of the DFG project GO1920/10 is acknowledged.

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