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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.
ESAIM: M2AN 56 (2022) 213–235 ESAIM: Mathematical Modelling and Numerical Analysis
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:
The authors. Published by EDP Sciences, SMAI 2022
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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 𝑒𝐸the density of cars is given by 𝜌𝑒(𝑡, 𝑥) and for a given initial state 𝜌𝑒(0, 𝑥)𝑒𝐸the
dynamics are governed by conservation laws of the form
𝜕𝑡𝜌𝑒(𝑡, 𝑥) + 𝜕𝑥𝑓𝑒(𝑡, 𝑥, {𝜌𝑘(𝑡, ·)}𝑘𝐸)=0 𝑒𝐸, 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, ·)}𝑘𝐸) = 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 𝑣𝑉have 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
𝑣1(𝜌(𝑡, 𝑦))𝜔𝜂(𝑦𝑥)𝑑𝑦 (2.3)
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,𝑒𝐸, and the kernel
function 𝜔𝜂:
𝑣𝑒𝐶2([0, 𝜌max
𝑒]; R+): 𝑣
𝑒0, 𝑣𝑒(𝜌max
𝑒) = 0,
𝜔𝜂𝐶1([0, 𝜂]; R+) : 𝜔
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 𝑒𝐸as 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 < 𝑏𝑒𝑎𝑒𝑒𝐸.
This assumption ensures that the nonlocal range is restricted by the length 𝑏𝑒𝑎𝑒of 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 ={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}𝑘𝐸) = 𝜌𝑒(𝑡, 𝑥)𝑉𝑒(𝑡, 𝑥) + 𝑔𝑒{𝜌𝑖(𝑡, ·)}𝑖∈ℐ ,{𝑉𝑜(𝑡, 𝑥)}𝑜∈𝒪, 𝑒 ,
𝜌𝑒(𝑡, 𝑥)𝑉𝑒(𝑡, 𝑥), 𝑒 𝒪,(2.5)
𝑉𝑒(𝑡, 𝑥) :=
min{𝑥,0}𝑣𝑒(𝜌𝑒(𝑡, 𝑦))𝜔𝜂(𝑦𝑥)𝑑𝑦, 𝑒 ,
max{𝑥,0}𝑣𝑒(𝜌𝑒(𝑡, 𝑦))𝜔𝜂(𝑦𝑥)𝑑𝑦, 𝑒 𝒪.(2.6)
In addition, we couple equations (2.1), (2.5) with the initial conditions
𝜌𝑒,0𝐿1BV((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
Following ([26], Def. 4.2.4) we define a weak solution for (2.1) and (2.5) at a single junction as:
Definition 2.3 (Weak solution).A collection of functions 𝜌𝑒𝐶(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
𝑣𝑒(𝜌𝑒(𝑡, 𝑦))𝜔𝜂(𝑦𝑥)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
𝑖=1 R+0
𝜌𝑖(𝑡, 𝑥)𝜕𝑡𝜑𝑖(𝑡, 𝑥) + 𝑓𝑖(𝑡, 𝑥, {𝜌𝑘(𝑡, ·)}𝑘𝐸)𝜕𝑥𝜑𝑖(𝑡, 𝑥)𝑑𝑥𝑑𝑡
𝑜=𝑀+1 R++
𝜌𝑜(𝑡, 𝑥)𝜕𝑡𝜑𝑜(𝑡, 𝑥) + 𝑓𝑜(𝑡, 𝑥, {𝜌𝑘(𝑡, ·)}𝑘𝐸)𝜕𝑥𝜑𝑜(𝑡, 𝑥)𝑑𝑥𝑑𝑡= 0
for every collection of test function 𝜑𝑖𝐶1
0((0,)×(−∞,0]; R), 𝑖 = 1, . . . , 𝑀 and 𝜑𝑜𝐶1
[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
𝑓𝑖(𝑡, 0,{𝜌𝑘(𝑡, ·)}𝑘𝐸) =
𝑓𝑜(𝑡, 0+,{𝜌𝑘(𝑡, ·)}𝑘𝐸).(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 𝜌𝑒𝐶(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
𝑜𝑖 {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) +
𝛼𝑖,𝑜𝑉𝑜(𝑡, 𝑥))= 0.
(ii) For all 𝑜 {𝑀+ 1, . . . , 𝑀 +𝑁}we have
𝑓𝑜(𝑡, 0+,{𝜌𝑘(𝑡, ·)}𝑘𝐸) =
𝛼𝑖,𝑜𝑓𝑖(𝑡, 0,{𝜌𝑘(𝑡, ·)}𝑘𝐸).
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 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.,
0𝑓𝑒(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:
0𝑔𝑒{𝜌𝑖(𝑡, ·)}𝑖∈ℐ ,{𝑉𝑜(𝑡, 𝑥)}𝑜∈𝒪𝜌𝑒(𝑡, 𝑥)
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,...)𝐶𝑉𝑘(𝑡, 𝑥)and 𝑘 𝒪.(2.12)
(4) The function 𝑔𝑒is Lipschitz continuous in 𝜌𝑒with Lipschitz constant 𝐿max𝑒𝐸𝑣𝑒.
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.
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.,
0𝑔1(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)
3𝑉3(𝑡, 𝑥)
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 𝑒being 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)
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
3𝜌𝑒(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 𝑗Z, 𝑛 Nand 𝑒𝐸, let 𝑥𝑒,𝑗1/2=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 𝑁𝜂N, 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 𝑗0 iff 𝑒 𝒪 and 𝑗 < 0 iff 𝑒 .
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,𝑗1/2, 𝑥𝑒,𝑗+1/2) and 𝑒𝐸. The initial datum 𝜌𝑒,0in (2.7) is approximated by the cell averages
𝑒,𝑗 =1
𝛥𝑥 𝑥𝑒,𝑗+1/2
𝜌𝑒,0(𝑥)𝑑𝑥, 𝑗 < 0, 𝑒 ,
𝑗0, 𝑒 𝒪.
Following [11,24], we consider the numerical flux function
𝑒,𝑗 := 𝜌𝑛
𝑒,𝑗 𝑉𝑛
𝑒,𝑗 +𝑔𝑒𝜌𝑛
𝑒,𝑗 ,{𝜌𝑛
𝑖,1}𝑖∈ℐ ,{𝑉𝑛
𝑜,𝑗 }𝑜∈𝒪, 𝑒
𝑒,𝑗 𝑉𝑛
𝑒,𝑗 , 𝑒 𝒪,(3.1a)
𝑒,𝑗 =min{−𝑗2,𝑁𝜂1}
𝑘=0 𝛾𝑘𝑣𝑒(𝜌𝑛
𝑒,𝑗+𝑘+1 ), 𝑒 ,
𝑒,𝑗+𝑘+1 ), 𝑒 𝒪,(3.1b)
𝜔𝜂(𝑥)𝑑𝑥, 𝑘 = 0, . . . , 𝑁𝜂1,(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.
In this way, we can define the following finite volume numerical scheme
𝑒,𝑗 =𝜌𝑛
𝑒,𝑗 𝜆𝐹𝑛
𝑒,𝑗 𝐹𝑛
𝑒,𝑗1with 𝜆:= 𝛥𝑡
𝛥𝑥, 𝑛 N,𝑗 < 0, 𝑒 ,
𝑗0, 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
𝑒,𝑗 𝑣max
𝑒𝑗Z, 𝑛 N, 𝑒 𝐸.
We set
𝑣:= max
𝑒𝐸𝑣𝑒,𝑣:= max
𝑒,𝜌:= max
and consider the following CFL condition:
𝛾0𝑣‖‖𝜌+ 2𝑣·(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
𝑒,𝑗 ), 𝑗 1,
0, 𝑗 0,𝑒 ,
0, 𝑗 1,
𝑒,𝑗 ), 𝑗 0,𝑒 𝒪.
In addition, we have
𝑒,𝑗 𝜌𝑛
𝑒,𝑗 )(𝜌max
𝑒,𝑗 ), 𝑗 1,
𝑒,𝑗 (𝜌max
𝑒,𝑗 ), 𝑗 0,𝑒 ,
𝑒,𝑗 (𝜌max
𝑒,𝑗 ), 𝑗 1,
𝑒,𝑗 )(𝜌max
𝑒,𝑗 ), 𝑗 0,𝑒 𝒪.
Proof. Let us consider the case 𝑒 :
𝑒,𝑗 =
𝑒,𝑗 ),
𝑗 𝑁𝜂1,
𝑘=1 (𝛾𝑘𝛾𝑘1)𝑣𝑒(𝜌𝑛
𝑒,𝑗+𝑘) + 𝛾0𝑣𝑒(𝜌𝑛
𝑒,𝑗 ),𝑁𝜂𝑗 2,
𝑒,1), 𝑗 =1,
0, 𝑗 0,
𝑒,𝑗 ), 𝑗 1,
0, 𝑗 0.
Using 𝑣𝑒(𝜌max
𝑒) = 0 and the mean value theorem yields inequality (3.3) for 𝑒 . 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:
1,𝑗 𝜌max
1for 𝑗 1and 0𝜌𝑛
𝑒,𝑗 𝜌max
𝑒for 𝑗0, 𝑒 {2,3} 𝑛N.
Proof. We start with the incoming road and the lower bound which can be obtained by applying (2.14):
1,𝑗 =𝜌𝑛
1,𝑗 +𝜆𝐹𝑛
1,𝑗 𝜆𝜌𝑛
1,𝑗 𝑉𝑛
1,𝑗 𝜆𝑔1(𝜌𝑛
1,𝑗 , 𝑉 𝑛
2,𝑗 , 𝑉 𝑛
3,𝑗 )
1,𝑗 (1 𝜆𝑉𝑛
1,𝑗 +𝛼1,2𝑉𝑛
2,𝑗 +𝛼1,3𝑉𝑛
3,𝑗 )
To obtain the upper bound we use the third and fourth property on 𝑔𝑒of Assumption 2.5 and the estimates
from Lemma 3.1:
1,𝑗 +𝜆𝑉𝑛
1,𝑗 𝜌𝑛
1,𝑗 +𝜆𝑔1(𝜌max
1, 𝑉 𝑛
2,𝑗1, 𝑉 𝑛
1,𝑗 , 𝑉 𝑛
2,𝑗 , 𝑉 𝑛
3,𝑗 )
1,𝑗 +𝜆𝛾0𝑣‖‖𝜌+𝑉𝑛
1,𝑗 (𝜌max
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)
1,𝑗 +𝜆𝛾0𝑣‖‖𝜌+𝑉𝑛
1,𝑗 +𝐿(𝜌max
1,𝑗 ) (3.6)
1,𝑗 +𝜆(𝛾0𝑣‖‖𝜌+ 2𝑣) (𝜌max
1,𝑗 )
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}:
𝑒,0(1 𝜆𝑣)0.
Similarly with Lemma 3.1,
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:
1,𝑗 𝜌𝑛
1,𝑗 +𝜆𝛾0𝑣‖‖𝜌+𝑉𝑛
1,𝑗 (𝜌max
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.
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
In order to prove a BV bound in space we consider the following total variation:
𝑖,𝑗+1 𝜌𝑛
𝑖,𝑗 +
𝑖∈ℐ 𝜌𝑛
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
2.62.9 satisfy
𝑔𝑒({𝜌𝑖(𝑡, 𝑥)}𝑖∈ℐ ,{𝑉𝑜(𝑡, 𝑥)}𝑜∈𝒪)𝜌𝑒(𝑡, 𝑥)
𝛼𝑖,𝑜𝑉𝑜(𝑡, 𝑥)
if 𝜌𝑖(𝑡, 𝑥)𝜌max
𝑜,𝑜 𝒪, 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.
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 = (0,)×(−∞,0) and 𝒪= (0,)×(0,). Define 𝑒= if 𝑒 and 𝑒= 𝒪if
𝑒 𝒪. In addition, let 𝜌𝑒𝐿(Ω𝑒)be a collection of functions which are either of compact support on 𝑒or
identical to zero and let 𝐾𝑒be a compact subset of 𝑒. Then, we obtain for the velocities (2.6)the following:
𝜂→∞ 𝐾𝑒
𝑒(𝑡, 𝑥)|𝑑𝑡𝑑𝑥 = 0, 𝑒 ,lim
𝜂→∞ 𝐾𝑒
𝑒(𝑡, 𝑥)𝑣𝑒(0)|𝑑𝑡𝑑𝑥 = 0, 𝑒 𝒪,
𝜂→∞ 𝐾𝑖
𝑒(𝑡, 𝑥)𝑣𝑒(0)|𝑑𝑡𝑑𝑥 = 0, 𝑒 𝒪, 𝑖 .
Proof. We start with 𝑒 . If 𝜌𝑒(𝑡, ·)0 we directly obtain
𝑒(𝑡, 𝑥) = min{𝑥+𝜂,0}
𝜔𝜂(𝑦𝑥)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
𝑒(𝑡, 𝑥) = min{𝑏𝑒(𝑡),𝑥+𝜂}
𝜔𝜂(𝑦𝑥)𝑣𝑒(𝜌𝑒(𝑡, 𝑦))𝑑𝑦 +min{𝑎𝑒(𝑡),𝑥+𝜂}
𝜔𝜂(𝑦𝑥)𝑣𝑒(𝜌𝑒(𝑡, 𝑦))𝑑𝑦 +𝑎𝑒(𝑡)
𝜔𝜂(0)𝑣𝑒(0) (|𝑏𝑒(𝑡)max{𝑥, 𝑎𝑒(𝑡)}| +|𝑎𝑒(𝑡)min{𝑥, 𝑎𝑒(𝑡)}| +|max{𝑥, 𝑏𝑒(𝑡)}|)
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 .
If we consider 𝑒 𝒪,𝑥 > 0 and 0 𝑎𝑒(𝑡)< 𝑏𝑒(𝑡), we get
𝑒(𝑡, 𝑥) = min{𝑏𝑒(𝑡),𝑥+𝜂}
𝜔𝜂(𝑦𝑥)𝑣𝑒(𝜌𝑒(𝑡, 𝑦))𝑑𝑦 +min{𝑎𝑒(𝑡),𝑥+𝜂}
𝜔𝜂(𝑦𝑥)𝑣𝑒(𝜌𝑒(𝑡, 𝑦))𝑑𝑦 +𝑎𝑒(𝑡)
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
𝜔𝜂(𝑦𝑥)𝑑𝑦 = 1 max{𝑥,𝑏(𝑡)}
𝜔𝜂(𝑦𝑥)𝑑𝑦 1𝜔𝜂(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 𝒪. 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 .
In addition, we have 𝑉𝜂
𝑒(𝑡, 𝑥)𝑣𝑒(0) (𝑡, 𝑥)𝑒, 𝑒 𝐸with 𝑣𝑒(0) being an integrable function on each
compact subset of 𝑒. 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 𝜌0BV(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
𝜕𝑡𝜌+𝜕𝑥(min{𝜌𝑣2(0), 𝜌max
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
𝑙𝑜𝑐 norm.
We start from the entropy inequality and add and subtract 𝑣2(0) at the right place, such that for a fixed
𝜅Rand 𝜑𝐶1
0([0,)×R;R+), we obtain
(|𝜌𝜅|𝜑𝑡+|𝜌𝜅|𝑉1𝜑𝑥+ sgn(𝜌𝜅)(˜𝑔(𝜌)˜𝑔(𝜅))(𝑉2±𝑣2(0))𝜑𝑥
sgn(𝜌𝜅)𝜅𝜕𝑥𝑉1𝜑sgn(𝜌𝜅𝑔(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 :
|𝜌𝜅|𝑉1𝜑𝑥𝑑𝑥𝑑𝑡 (𝜌+|𝜅|)𝜑𝑥𝑇
|𝑉1|𝑑𝑥𝑑𝑡 0
sgn(𝜌𝜅)(˜𝑔(𝜌)˜𝑔(𝜅))(𝑉2𝑣2(0))𝜑𝑥𝑑𝑥𝑑𝑡 2𝜌‖‖𝜑𝑥𝑇
Again, thanks to the compactness of the test function and condition (2.4) of the kernel, we have
sgn(𝜌𝜅)𝜅𝜕𝑥𝑉1𝜑𝑑𝑥𝑑𝑡 16|𝜅|‖𝜑‖‖𝑣𝑇 𝑅𝜔𝜂(0) 0,
sgn(𝜌𝜅𝑔(𝜅)𝜕𝑥𝑉2𝜑)𝑑𝑥𝑑𝑡 16𝜌‖‖𝜑‖‖𝑣𝑇 𝑅𝜔𝜂(0) 0.
Here, we use that
|𝜕𝑥𝑉1| min{𝑥+𝜂,0}
𝑣1(𝜌(𝑡, 𝑦))𝜔
+𝑣1(𝜌(𝑡, min{𝑥, 0}))𝜔𝜂(0)
+𝑣1(𝜌(𝑡, min{𝑥+𝜂, 0}))𝜔𝜂(0)
+ 2𝑣𝜔𝜂(0)
and analogously we get |𝜕𝑥𝑉2| 4𝑣𝜔𝜂(0). We are left with
(|𝜌𝜅|𝜑𝑡+ sgn(𝜌𝜅)(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),
with 𝑉𝜂(𝑡, 𝑥) = 𝑥+𝜂
𝑥𝑣(𝜌(𝑡, 𝑦))𝜔𝜂(𝑦𝑥)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),
(|𝜌𝜅|𝜑𝑡+|𝜌𝜅|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
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 𝜌𝐿𝜌max
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 2.62.9 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:
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
}= 0,
Example 2.8,𝑒 {1,2}:
𝜕𝑡𝜌𝑒+𝜕𝑥(min{𝜌𝑒(𝑡, 𝑥),max{𝑞𝑒,3𝜌max
3, 𝜌max
3𝜌𝑒(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
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
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) = ¯𝜌𝑒𝑥(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(𝜂𝑥)/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.
total travel time
𝑒=1 𝑇
𝜌𝑒(𝑡, 𝑥)𝑑𝑥𝑑𝑡,
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
𝑓7(𝑡, 1, 𝜌7(𝑡, 1))𝑑𝑡,
congestion measure
𝐶𝑀 =
𝑒=1 𝑇
max 0,1
𝜌𝑒(𝑡, 𝑥)𝑓𝑒(𝑡, 𝑥, 𝜌𝑒(𝑡, 𝑥))
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
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)
𝑓𝑒(𝑡, 𝑥, {𝜌𝑘}𝑘𝐸) = 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.
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 𝑒𝐸,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 𝜌𝜎𝑒,
𝜎𝑒𝑣𝑒(𝜎𝑒),if 𝜌 > 𝜎𝑒,𝑆𝑒(𝜌) = 𝜎𝑒𝑣𝑒(𝜎𝑒),if 𝜌𝜎𝑒,
𝜌𝑣𝑒(𝜌),if 𝜌>𝜎𝑒,(5.2)
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 𝜂0 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.
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.
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 𝜂0 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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... Some works about non-local traffic models deal with networks, for example [8,10,9,23,13,35,31]. In [8], the authors consider measure valued solutions for non-local transport equations and [31] deals with non-local conservation laws on bounded domains while [10,9,35,13] include 1-to-1 junctions. ...
... In particular in [10,9], the existence and well-posedness of solutions at a 1-to-1 junction is shown, where the roads are allowed to differ in the speed limits and maximum road capacities. The limit of those models for a non-local range tending to infinity is investigated in [23]. ...
... For the well-posedness of the model, we assume the following common assumptions, see e.g. [5,11,10,23,24] (H1) (1) for e = 1, ρ e is a weak solution on (−∞, 0) to ∂ t ρ e (t, x) + ∂ x (ρ e (t, x)V 1 (t, x) + min{ρ e V 2 (t, x), s B (t, x)}) = 0; ...
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In this paper, we introduce a non-local PDE-ODE traffic model devoted to the description of a 1-to-1 junction with buffer. We present a numerical method to approximate solutions and show a maximum principle which is uniform in the non-local interaction range. Further, we exploit the limit models as the support of the kernel tends to zero and to infinity. We compare them with other already existing models for traffic and production flow and present numerical examples.
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In this work we present a rather general approach to approximate the solutions of nonlocal conservation laws. In a first step, we approximate the nonlocal term with an appropriate quadrature rule applied to the spatial discretization. Then, we apply a numerical flux function on the reduced problem. We present explicit conditions which such a numerical flux function needs to fulfill. These conditions guarantee the convergence to the weak entropy solution of the considered model class. Numerical examples validate our theoretical results and demonstrate that the approach can be applied to other nonlocal problems.