Content uploaded by Felisia Angela Chiarello

Author content

All content in this area was uploaded by Felisia Angela Chiarello on Jul 19, 2023

Content may be subject to copyright.

A NON-LOCAL TRAFFIC FLOW MODEL FOR 1-TO-1 JUNCTIONS WITH

BUFFER

F. A. CHIARELLO, J. FRIEDRICH, AND S. G ¨

OTTLICH

Abstract. In this paper, we introduce a non-local PDE-ODE traﬃc model devoted to the descrip-

tion 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.

Keywords: Scalar conservation laws, anisotropic non-local ﬂux, coupled PDE-ODE model, traﬃc

ﬂow.

2020 Mathematics Subject Classiﬁcation: 35L65, 65M08, 76A30

1. Introduction

The aim of this paper is to study a non-local traﬃc ﬂow model for 1-to-1 junctions. The term

‘non-local’ refers to the dependence of the ﬂux function on a convolution term between a kernel

function and the velocity function depending on the conserved quantity. This type of conservation

laws are able to describe several phenomena arising in many ﬁelds of application and for this rea-

son received growing attention in the recent years. The ﬁrst macroscopic traﬃc ﬂow model based

on ﬂuid-dynamics equations has been introduced in the transport literature with the Lighthill,

Whitham and Richards (LWR) model [33, 34], that consists in one scalar equation expressing

conservation of cars. Since then, several approaches have been developed during the last years,

addressing the need for more sophisticated models to better understand the nonlinear traﬃc dy-

namics. ‘Non-local’ versions of the LWR model have been recently proposed in [5, 11, 24]. In

this type of models, the ﬂux is assumed to depend on a weighted mean of the downstream traﬃc

density or velocity. These kind of non-local traﬃc models are intended to describe the behaviour

of drivers that adapt their velocity with respect to what happens in front of them. For this reason,

the ﬂux function depends on a ‘downstream’ convolution term between the density of vehicles or

the velocity, and a kernel function supported on the negative axis, such that drivers only look

forward, not backward. There are general existence and uniqueness results for non-local equations

in [3, 28, 11, 24] for scalar equations in one-space dimension, in [18, 30] for multi-dimensional

scalar equations, and in [2] for the multi-dimensional system case. There are mainly two diﬀerent

approaches to prove the existence of solutions for these non-local models. One is providing suitable

compactness estimates on a sequence of approximate solutions constructed through ﬁnite volume

schemes, as in [3, 2]. Another approach relies on the method of characteristics and ﬁxed-point

theorems, as proposed in [28, 30]. In [3], Kruˇzkov-type entropy conditions are used to prove the

L1−stability with respect to the initial data through the doubling of variable technique, while in

Date: July 19, 2023.

1

2 F. A. CHIARELLO, J. FRIEDRICH, AND S. G ¨

OTTLICH

[28, 30], the uniqueness of weak solutions is obtained directly from the ﬁxed point theorem, without

prescribing any kind of entropy condition.

One key question which naturally arises in the non-local setting from a mathematical, but also

modelling point of view, is the model behavior for the two limits of the non-local interaction range.

Considering an inﬁnite range, the non-local scalar traﬃc ﬂow model tends to a linear transport

equation [11]. Contrary, if the range tends to zero, non-local scalar traﬃc ﬂow models converge

to the local LWR model under speciﬁc assumptions, e.g. [29, 22, 14, 22, 16, 7, 6] or [12] for the

singular nonlocal-to-local limit for weakly-coupled systems.

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].

Here, the models converge still to a linear transport equation which includes a capacity constraint

on the ﬂux induced by the coupling of the roads. In contrast, the limit to zero is only investigated

numerically so far and the correct limit model is still an open problem.

In this paper we aim to model a 1-to-1 junction with a buﬀer, generalizing the ‘local’ model

in [27] to the non-local setting. Further, we want to exploit the limit models for the non-local

range tending to inﬁnity and zero. The paper is organized as follows. In Section 2, we introduce a

new non-local model for 1-to-1 junctions with buﬀer. Then, in Section 3 we perform a numerical

discretisation of our model using an upwind-type numerical scheme. In Section 4, the limit model

as the length of the support ηof the kernel function goes to zero is exploited and in Section 5

the opposite limit model as the length of the support ηof the kernel function goes to inﬁnity is

investigated. In Section 6, we give a conclusion and mention open problems.

2. Modeling equations

As aforementioned, we consider a non-local traﬃc ﬂow model for 1-to-1 junctions that includes

a buﬀer. Therefore, let us recall ﬁrst the 1-to-1 junction model from [10]. This model does not

consider a buﬀer so far and the dynamics for the traﬃc density on each road are given by

(2.1) ∂tρ1(t, x) + ∂xρ1(t, x)V1(t, x) + min{ρ1V2(t, x), ρ2

maxV2(t, x)}= 0,

∂tρ2(t, x) + ∂x(ρ2(t, x)V2(t, x)) = 0.

Here, V1and V2are the parts of the non-local velocity on each road segment deﬁned by

V1(t, x) := Zmin{x+η, 0}

min{x, 0}

v1(ρ1(t, x))ωη(y−x)dy,

V2(t, x) := Zmax{x+η, 0}

max{x, 0}

v2(ρ2(t, x))ωη(y−x)dy.

In particular, the junction point is placed at x= 0, ρ1and ρ2are the traﬃc densities, η > 0 is

the non-local range and support of the kernel function ωη, which will be detailed below, and v1

and v2are suitable velocity functions. The most important feature is the coupling between the two

road segments which is simply induced by the minimum operator and the maximum capacity of

the second road. The latter one expresses the supply of the second road. At the junction itself, the

A NON-LOCAL TRAFFIC FLOW MODEL FOR 1-TO-1 JUNCTIONS WITH BUFFER 3

second road can receive a ﬂux up to ρ2

maxV2(t, 0). Due to non-locality, this capacity restriction must

be considered already in the interval [−η, 0). From a microscopic point of view, this means that as

soon as a driver is aware of an intersection, she/he takes the associated (possible) restrictions into

account.

We now extend this model by placing a buﬀer between the two roads. These buﬀers could, for

example, model a simpliﬁed version (by neglecting the exact geometry) of highway on-ramps or

roundabouts. A buﬀer has a maximum capacity µ, i.e. the maximum rate at which cars can enter

or leave the buﬀer. Furthermore, the buﬀer is full at rmax. This creates a speciﬁc supply that

the buﬀer can provide for the ﬂow of the ﬁrst road. Similar to the situation without a buﬀer, the

supply must be taken into account as soon as a driver sees the buﬀer. As in the situation without

buﬀer, the second road at the junction can take a maximum ﬂow of ρ2

maxV2(t, 0). This results in

the following modeling equations for the buﬀer r(t) and the traﬃc densities:

(2.2)

∂tρ1(t, x) + ∂x(ρ1(t, x)V1(t, x) + min{ρ1V2(t, x), sB(t, x)})=0,

∂tρ2(t, x) + ∂x(ρ2(t, x)V2(t, x)) = 0,

r0(t) = min{sB(t, 0−), ρ1(t, 0−)V2(t, 0)} − min{dB(t), ρ2

maxV2(t, 0)},

where

sB(t, x) =

µRmax{x+η,0}

0ωη(y−x)dy, r(t)∈[0, rmax ),

min{ρ2

maxV2(t, x), µ Rmax{x+η,0}

0ωη(y−x)dy}, r(t) = rmax ,

dB(t) =

µ, r(t)∈(0, rmax],

min{ρ1(t, 0−)V2(t, 0), µ}, r(t)=0.

Note that the model (2.2) is inspired by its local counterpart introduced in [27]. For a better

readability we postpone the detailed discussion of the local variant to Section 4.

To close the equations (2.2) we couple them with the initial data

(2.3) ρ1(0, x) = ρ1,0(x)∈BV((−∞,0)), ρ2(0, x) = ρ2,0(x)∈BV((0,+∞)), r(0) = r0≥0,

where BV([a, b]) := {f∈L1([a, b])|T V (f)<+∞}.For the well-posedness of the model, we assume

the following common assumptions, see e.g. [5, 11, 10, 23, 24]

(H1) ωη∈C1([0, η]; R+), η > 0, ω0

η≤0,Rη

0ωη(x)dx = 1;

(H2) ve:R+→R+, is a smooth non-increasing function such that ve(0) = Vmax

e>0 and

ve(ρ) = 0 for ρ≥ρe

max for e∈ {1,2}.

Inspired by [21, Deﬁnition 2.1] and [20, Deﬁnition 4.1], solutions are intended in the following sense:

Deﬁnition 2.1. A couple of functions ρe∈C(R+,L1loc((ae, be))) with e∈ {1,2},and a1=

−∞, b1= 0 and a2= 0, b2= +∞,is called a weak solution of (2.2)-(2.3) if

(1) for e= 1, ρeis a weak solution on (−∞,0) to

∂tρe(t, x) + ∂x(ρe(t, x)V1(t, x) + min{ρeV2(t, x), sB(t, x)}) = 0;

(2) for e= 2, ρeis a weak solution on (0,+∞)to

∂tρe(t, x) + ∂x(ρe(t, x)V2(t, x)) = 0;

4 F. A. CHIARELLO, J. FRIEDRICH, AND S. G ¨

OTTLICH

(3) for all t > 0,it holds

ρ2(t, 0+)V2(t, 0) = min{dB(t), ρ2

maxV2(t, 0)};

(4) for e∈ {1,2}and for a.e. t > 0the function x7→ ρe(t, x)has a bounded total variation.

(5) r(t)is a Carath`eodory solution for, i.e. for a.e. t∈R+

r(t) = r0+Zt

0min{sB(s, 0−), ρ1(s, 0−)V2(s, 0)} − min{dB(s), ρ2

maxV2(s, 0)}ds.

3. Numerical discretisation

For further investigations, we will approximate the solutions by using a numerical scheme. This

scheme is based on the numerical schemes introduced in [10, 24, 23, 25].

For j∈Zand n∈Nand e∈E:= {1,2}, let xe,j−1/2=j∆xbe the cell interfaces, xe,j =

(j+ 1/2)∆xthe cells centers, corresponding to a space step ∆xsuch that η=Nη∆xfor some

Nη∈N, and let tn=n∆tbe the time mesh. In particular, x=xe,−1/2= 0 is a cell interface. Note

that we have j≥0 if e= 2 and j < 0 if e= 1.We aim at constructing a ﬁnite volume approximate

solution ρ∆x

esuch that ρ∆

e(t, x) = ρn

e,j for (t, x)∈[tn, tn+1)×[xe,j−1/2, xe,j+1/2). To this end, we

approximate the initial datum ρ0with the cell averages

ρ0

e,j =1

∆xZxe,j+1/2

xe,j−1/2

ρe,0(x)dx,

j < 0, e = 1,

j≥0, e = 2.

We consider the following numerical ﬂuxes similar to [10, 23]

(3.1) Fn

e,j :=

ρn

e,jVn

e,j + min nρn

e,jVn

2,j, sn

B,j o, e = 1,

ρn

e,jVn

e,j, e = 2,

with

Vn

1,j =

min{−j−2, Nη−1}

X

k=0

γkv1(ρn

1, j+k+1), V n

2,j =

Nη−1

X

k=max{−j−1,0}

γkv2(ρn

2, j+k+1),

sn

B,j =

µ

Nη−1

X

k=−j−1

γk, rn∈[0, rmax),

min

ρ2

maxVn

2,j, µ

Nη−1

X

k=−j−1

γk

, rn=rmax,

dn

B=

µ, rn∈(0, rmax],

min{ρn

1,−1Vn

2,−1, µ}, rn= 0.

where γk=R(k+1)∆x

k∆xωη(x)dx, k = 0, ..., Nη−1 and rnis the approximation of the buﬀer computed

by an explicit Euler scheme. In this way, we deﬁne the following ﬁnite volume scheme:

(3.2) ρn+1

e,j =ρn

e,j −∆t

∆xFn

e,j −Fn

e,j−1,

rn+1 =rn+ ∆tmin{sn

B,0, ρn

1,0Vn

2,0} − min{dn

B, ρ2

maxVn

2,0},

n∈N.

We now show that the numerical scheme satisﬁes a reasonable maximum principle on each road.

Proposition 3.1. Under hypothesis (2.3) and the CFL condition

(3.3) ∆t

∆x≤1

γ0kv0k kρk+ 2 kvk,

A NON-LOCAL TRAFFIC FLOW MODEL FOR 1-TO-1 JUNCTIONS WITH BUFFER 5

with kvk= max{kv1k,kv2k},kv0k= max{kv0

1k,kv0

2k} and kρk= max{ρ1

max, ρ2

max},the sequence

generated by the numerical scheme (3.1)-(3.2) satisﬁes the following maximum principle:

0≤ρn

e,j ≤ρe

max.

Proof. Let us start by showing the positivity, we directly obtain

ρn+1

e,j =ρn

e,j −∆t

∆xFn

e,j −Fn

e,j−1≥ρn

e,j −∆t

∆xkvkρn

e,j ≥0,

with kvk= max{kv1k,kv2k},using min nρn

e,jVn

2,j, sn

B,j o≤ρn

e,jVn

2,j.For the upper bound, it is easy

to see that

Vn

1,j−1−Vn

1,j ≤

γ0v1(ρn

1,j), j ≤ −1,

0, j ≥0,

Vn

2,j−1−Vn

2,j ≤

0, j ≤ −1,

γ0v2(ρn

2,j), j ≥0.

Using v1(ρ1

max) = v2(ρ2

max) = 0 and the mean value theorem, we get

Vn

1,j−1−Vn

1,j ≤

γ0kv0

1k(−ρn

1,j +ρ1

max), j ≤ −1,

0, j ≥0,

Vn

2,j−1−Vn

2,j ≤

0, j ≤ −1,

γ0kv0

2k(−ρn

2,j +ρ2

max), j ≥0.

Considering the case j≤ −1,multiplying the ﬁrst inequality by ρ1

max and subtracting Vn

1,jρn

1,j,we

get

(3.4) Vn

1,j−1ρ1

max −Vn

1,jρn

1,j ≤γ0

v0

1

kρ1k+Vn

1,j(−ρn

1,j +ρ1

max).

Similarly, writing

sn

B,j−1−sn

B,j =

−µγ−j−1rn∈[0, rmax),

min{ρ2

maxVn

2,j−1, µ PNη−1

k=−jγk} − min{ρ2

maxVn

2,j, µ PNη−1

k=−j−1γk}rn=rmax,

≤0,

because

min

ρ2

maxVn

2,j−1, µ

Nη−1

X

k=−j

γk

−min

ρ2

maxVn

2,j, µ

Nη−1

X

k=−j−1

γk

≤min

ρ2

maxVn

2,j, µ

Nη−1

X

k=−j

γk

−min

ρ2

maxVn

2,j, µ

Nη−1

X

k=−j−1

γk

=

µPNη−1

k=−jγk−µPNη−1

k=−j−1γk−

ρ2

maxVn

2,j −µPNη−1

k=−jγk

+

ρ2

maxVn

2,j −µPNη−1

k=−j−1γk

2

≤

µPNη−1

k=−jγk−µPNη−1

k=−j−1γk+

µPNη−1

k=−jγk−µPNη−1

k=−j−1γk

2= 0.

Then, we get

(3.5)

min{ρ1

maxVn

2,j−1, sn

B,j−1} − min{ρn

1,jVn

2,j, sn

B,j }

≤min{ρ1

maxVn

2,j, sn

B,j } − min{ρn

1,jVn

2,j, sn

B,j }

≤ρ1

max −ρn

1,jVn

2,j.

6 F. A. CHIARELLO, J. FRIEDRICH, AND S. G ¨

OTTLICH

Adding the two inequalities (3.4)-(3.5), due to the CFL condition (3.3), we have ρn+1

1,j ≤ρ1

max.

For j≥0,we can write the following bound

ρ2

maxVn

2,j−1−ρn

2,jVn

2,j ≤(γ0

v0

kρk+kvk)(ρ2

max −ρn

j),

that follows analogously to what has been done above. In particular, the bound holds for j= 0,

since for the inﬂow on road 2 we have

min{dn

B, ρ2

maxVn

2,−1} ≤ ρ2

maxVn

2,−1.

Then, we get

ρn+1

2,j ≤ρn

2,j +∆t

∆xρ2

maxVn

2,j−1−ρn

2,jVn

2,j≤ρ2

max.

This concludes the proof.

Remark 3.1. We point out that the maximum principle in Proposition 3.1 is uniform with respect

to ηand this allows us to say that if they exist, the limit solutions as η→0and η→+∞satisfy

the maximum principle, too.

In the following we are in particularly interested in the model hierarchies induced by the non-local

range η. Therefore, we will investigate the models for η→0 and η→ ∞ in the next sections.

4. Limit to zero

The singular local limit is a very interesting topic regarding non-local conservation laws. The

problem is deﬁned as follows. In the scalar case, a parameter η > 0 related to the support of the

kernel is ﬁxed and the re-scaled kernel function is considered

ωη(x) = 1

ηωx

η.

Owing to the assumption RRωη(x)dx = 1,when η→0+the family ωηconverges weakly∗in the

sense of measures to the Dirac delta and we formally obtain the corresponding local conservation

law:

Non-local:

∂tρη+∂x(ρη(v(ρη)∗ωη)) = 0,

ρη(0, x) = ¯ρ(x),

→Local:

∂tρ+∂x(ρv(ρ)) = 0,

ρ(0, x) = ¯ρ(x),

with v:R→Rbeing a Lipschitz continuous vector-valued function. The above derivation is just

formal and has to be rigorously justiﬁed. The question is whether the solution ρηof the non-local

Cauchy problem converges to the entropy admissible solution of the local Cauchy problem in some

suitable topology. The rigorous derivation of this limit with the convolution inside v, i.e. v(ρη∗ωη),

was initially posed in [3] motivated by numerical evidence, later corroborated in [5]. The answer

is positive if we consider an even kernel function and a smooth compactly supported initial da-

tum, as proved in [36]. The solution of one-dimensional scalar non-local conservation laws with an

anisotropic kernel converges to the entropy solution of the corresponding local conservation laws

also in the case of monotone initial data, see [29, 22], for an exponential-type kernel [14, 22, 7] or

considering a positive initial datum and a convex kernel [16]. Nevertheless, in general, the solution

of the non-local Cauchy problem does not converge to the solution of the local one and in [17] the

authors show three counterexamples in this sense, while in [15], the role of numerical viscosity in

the numerical investigation of such non-local-to-local limit is highlighted.

A NON-LOCAL TRAFFIC FLOW MODEL FOR 1-TO-1 JUNCTIONS WITH BUFFER 7

4.1. A local buﬀer model. A local buﬀer model which strongly inspired the non-local model

(2.2) is originally introduced in [27]. For completeness we recall the model of [27]. Note that this

model is derived only for the same ﬂux function on each road, but an extension to diﬀerent ﬂux

functions is straightforward. The modeling equations are given by

(4.1) ∂tρ1(t, x) + ∂x(ρ1(t, x)v1(ρ1(t, x))) = 0,

∂tρ2(t, x) + ∂x(ρ2(t, x)v2(ρ2(t, x))) = 0.

Since we consider a local model we only need to describe the ﬂuxes at the intersection, which

couples the two equations. Following [27] this coupling is given by

q1(t) = min{sB(t), D1(ρ1(t, 0−))}, q2(t) = min{dB(t), S2(ρ2(t, 0+)}

sB(t) =

µ, if r(t)∈[0, rmax),

min{S2(ρ2(t, 0+), µ},if r(t) = rmax,

dB(t) =

µ, if r(t)∈(0, rmax],

min{D1(ρ1(t, 0−)), µ},if r(t)=0.

Here, Siand Diwith i∈ {1,2}are the usual supply and demand functions [32] deﬁned for a ﬂux

fiwith a single maximum at σby

Di(ρ) =

fi(ρ), ρ ≤σ,

fi(σ), ρ > σ,

Si(ρ) =

fi(σ), ρ ≤σ,

fi(ρ), ρ > σ.

Note that, here fi(ρ) := ρvi(ρ) holds. Finally, the development of the buﬀer is described by

r0(t) = q1(t)−q2(t).

For a more detailed discussion of the model and an extension to more general junctions we refer

the reader to [27].

Comparing the local model with (2.2), we see that the non-local version adapts the structure

of the local model. Besides the non-local velocity, one of the main diﬀerences between the two

models is the deﬁnition of the supply and demand functions. Since the non-local model allows for

a higher ﬂow, as the local density at xand the non-local velocity are not directly coupled, the ﬂow

is not allowed to exceed ρ2

maxV2(t, 0+) in contrast to S(ρ2(0+, t)) at the junction point. Moreover,

extending this fact to the whole interval [−η, 0) with similar ideas as in [10, 23], we end up with

the model (2.2).

Even though the local model (4.1) strongly inspired the non-local one, it is not a priori clear which

local solution will be obtained as η→0. For example, the 1-to-1 model (2.1) without a buﬀer

seems to converge to the vanishing viscosity solution of a discontinuous conservation law in certain

cases, see [10].

For the non-local model (2.2), it becomes apparent that the solutions might be diﬀerent to the

solution of the local model (4.1). This is induced by the diﬀerent deﬁnition of the supply and

demand functions mentioned above. Hence, when supply or demand in the model (4.1) are active

diﬀerences can occur, since, in general,

D1(ρ1(t, 0−)) 6=ρ1(t, 0−)v2(ρ2(t, 0+) = lim

η→0ρ1(t, 0−)V2(t, 0+),

8 F. A. CHIARELLO, J. FRIEDRICH, AND S. G ¨

OTTLICH

S2(ρ2(t, 0+)) ≤ρ2

maxv2(ρ2(t, 0+)) = lim

η→0ρ2

maxV2(t, 0+).

The fact that the model (2.2), in general, does not converge to solutions of (4.1) is also shown by

the following example:

Example 4.1. Let us consider constant initial data ρ1and ρ2on each road with the same ﬂux

function on both roads. Further, let S(ρ2)< D(ρ1)and S(ρ2)< µ hold. In this case, we can

exclude that the solution of (2.2) converges to the solution of the local model (4.1). Therefore, we

show that the buﬀer is diﬀerent for the model (4.1) and (2.2). We start with the local case. Here,

we have

q1(t) = min{µ, D(ρ1)}, q2(t) = min{S(ρ2), µ}.

Moreover, due to S(ρ2)< D(ρ1)< µ,

r0(t) = q1(t)−q2(t)>0,

this means that the buﬀer is increasing. On the contrary, in the non-local case we have

r0(t) = min{µ, ρ1(t, 0−)V2(t, 0)} − min{µ, ρ1(t, 0−)V2(t, 0), ρ2

maxV2(t, 0)}.

Due to the choice of the same ﬂux function and thanks to the maximum principle we get

ρ1(t, x)≤ρ1

max =ρ2

max,

and hence,

r0(t) = min{µ, ρ1(t, 0−)V2(t, 0−)} − min{µ, ρ1(t, 0−)V2(t, 0)}= 0.

Therefore, the buﬀer does not increase for every η > 0, this also holds in the limit η→0.For this

reason, the solution of the non-local problem does not, in general, converge to the solution of the

local one.

Even though convergence to the model (4.1) can be excluded for certain initial data, there are

still cases where we can observe (numerically) a convergence.

4.2. Limit model. Since the local model (4.1) seems not to be the correct limit model of (2.2),

let us formally derive a potential candidate for the limit model of (2.2). The limit of ωηto the

Dirac delta needs to be considered in the non-local velocities and the supply function. Hence, the

model (2.2) formally converges to

(4.2)

∂tρ1(t, x) + ∂x(ρ1(t, x)v1(ρ1(t, x))) = 0,

∂tρ2(t, x) + ∂x(ρ2(t, x)v2(ρ2(t, x))) = 0,

r0(t) = min{sB(t, 0−), ρ1(t, 0−)v2(ρ2(t, 0+)} − min{dB(t), ρ2

maxv2(ρ2(t, 0+))},

where

sB(t, x) =

µ, r(t)∈[0, rmax),

min{ρ2

maxv2(ρ2(t, x)), µ}, r(t) = rmax,

dB(t) =

µ, r(t)∈(0, rmax],

min{ρ1(t, 0−)v2(ρ2(t, 0+)), µ}, r(t) = 0.

A NON-LOCAL TRAFFIC FLOW MODEL FOR 1-TO-1 JUNCTIONS WITH BUFFER 9

Comparing the dynamics to (4.1) we see that the outﬂow of the ﬁrst road and the inﬂow to the

second road and therefore the development of the buﬀer are diﬀerent. The reason is similar as

before that

D1(ρ1(t, 0−)) 6=ρ1(t, 0−)v2(ρ2(t, 0+) and S2(ρ2(t, 0+)) ≤ρ2

maxv2(ρ2(t, 0+))

holds.

Note that we will note prove the well-posedness of (4.2) or the limit from (2.2) to (4.2) rigorously.

Nevertheless, we investigate the limit and model (4.2) numerically in the following. In particular,

a numerical convergence to model (4.2) can be obtained.

4.3. Numerical examples. During this section the space step size for all examples is chosen as

∆x= 10−3and the time step is chosen according to the CFL condition (3.3). We deal with the

following example: v1(ρ)=1−ρ,v2(ρ)=1−5ρ/3, constant initial data of ρ1,0≡0.75 and

ρ2,0≡0.5 with rmax =∞and r0= 0. If not stated otherwise, we use µ= 0.15. Both roads are in

a congested state but due to the lower maximum density of the second road, the latter one is even

more congested. This results in a backwards moving traﬃc jam on the ﬁrst road and, depending

on the capacity, in an increasing buﬀer.

Remark 4.1. We note that the obtained results are very similar in case of a limited buﬀer, i.e.,

rmax <∞. As soon as the buﬀer gets full, an additional wave is induced moving backwards on the

ﬁrst road. To simplify the discussion and concentrate on the eﬀects induced by the capacity, we

stick to an inﬁnite buﬀer.

Before considering the limit process, we start with a comparison of the dynamics given by (2.2)

for diﬀerent kernels in the situation described above for η= 0.5. Here, we consider a constant

kernel ωη(x)=1/η, a linear decreasing kernel, i.e. ωη(x) = 2(η−x)/η2and a quadratic decreasing

kernel, i.e. ωη(x) = 3(η2−x2)/(2η3). The approximate solutions are shown in Figure 1. Since most

of the interactions take place on the ﬁrst street, we have increased the visual representation of this

road. Due to the bottleneck situation between the two roads, the buﬀer and the congestion on the

ﬁrst road are increasing. In particular, for all models the ﬂow on the ﬁrst road is higher than the

capacity µof the buﬀer. As can be seen, the linear and quadratic kernels behave very similar and

provide a rather continuous increase of the density until drivers are aware of the junction around

x≈ −0.5 which results in a steeper increase. However, the constant kernel behaves diﬀerently.

There are two eﬀects visible: First, there are additional increases depending on η= 0.5. This

eﬀect is also present without a buﬀer model, i.e. in the model (2.1), with constant kernel. The

approximate solution without a buﬀer can also be seen in Figure 1. The obtained behavior is a

speciﬁc feature of the kernel, since drivers become aware of the bottleneck at x=−0.5 and the

traﬃc directly ahead, which results in a traﬃc jam. This traﬃc jam then resolves but there is a

point in front of the junction (x≈ −0.25) where the current ﬂow rate is inﬂuenced by the capacity

of the buﬀer which increases the density. This is the second eﬀect which can be obtained. Due to

the non-locality this eﬀects propagate backwards. The eﬀect of the capacity on the ﬂow can also

be observed for the other two kernels, even though it is smoothed out slightly.

Even though the eﬀect of the increasing density for the constant kernel is explainable, we want to

consider less eﬀects on the dynamics to better understand the behavior. Further, as the dynamics

10 F. A. CHIARELLO, J. FRIEDRICH, AND S. G ¨

OTTLICH

−1.4−1.2−1−0.8−0.6−0.4−0.2 0

0.76

0.78

0.8

0.82

0.84

0.86

0.88

0.9

x

ρ1

constant

linear

quadratic

(2.1)

0 0.2 0.4

0.5

0.55

0.6

x

ρ2

0 0.2 0.4 0.6 0.8 1

0

0.5

1

·10−2

t

r

Figure 1. Approximate solution of the non-local buﬀer model with non-local

range of η= 0.5 with diﬀerent kernels and the approximate solution of (2.1) with a

constant kernel at T= 1: approximate solutions of road 1, left, of road 2, top right

and the evolution of the buﬀer, bottom right.

for the linear and quadratic kernel are rather similar, we restrict ourselves to the linear kernel for

the remaining part of this section.

Next, we turn our attention to the numerical limit η→0. For the local models (4.1) and (4.2)

we use a Godunov type scheme, where the ﬂuxes at the junction point are deﬁned by the coupling.

In the above mentioned example the capacity is active. This holds for the local case, too, such

that the two local models produce the same solution. Further, this results in the same outﬂow of

the ﬁrst road in all models given by µ. Hence, for a decreasing sequence of η, the convergence can

be numerically obtained, as can be seen in Figure 2. Small diﬀerences can still be seen on road

2, as the inﬂow on this road is still diﬀerent for the models. We note that this diﬀerence may

disappear in the limit. Nevertheless, choosing an even smaller capacity, e.g. µ= 0.05, gives the

same dynamics for all models. In contrast, increasing µdemonstrates the possible convergence to

(4.2) instead of (4.1). In Figure 3, the same example as before, but with µ= 0.15 is considered.

In the local model (4.1) the outﬂow of the ﬁrst road is higher than in the local model (4.2). This

results in a lower density, but fuller buﬀer. The decreasing sequence of ηshows that (4.2) might

be the limit of the non-local buﬀer model (2.2).

Remark 4.2. We remark that even without a buﬀer the non-local model does not necessarily

converge to the solution which maximizes the ﬂux through the junction. In particular, for a local

1-to-1 junction without a buﬀer there a several possible Riemann solvers available at the interface,

see e.g. [1] and the references therein. In [10], it has been already discovered that the non-local

1-to-1 model without a buﬀer seems to converge to the vanishing viscosity solution and not to the

solution given by supply and demand which maximizes the ﬂux through the junction. Hence, we

cannot expect a convergence to the model (4.1) which also maximizes the ﬂux through the junction

for every case, in particular, if the buﬀer is not active.

A NON-LOCAL TRAFFIC FLOW MODEL FOR 1-TO-1 JUNCTIONS WITH BUFFER 11

−2−1.8−1.6−1.4−1.2−1−0.8−0.6−0.4−0.2 0

0.74

0.76

0.78

0.8

0.82

0.84

0.86

0.88

0.9

x

ρ1

(4.1)

(4.2)

(2.2) η= 0.5

(2.2) η= 0.05

(2.2) η= 0.005

0 0.2 0.4

0.5

0.55

0.6

x

ρ2

0 0.2 0.4 0.6 0.8 1

0

0.5

1

1.5

2·10−2

t

r

Figure 2. Approximate solutions of the non-local buﬀer model with diﬀerent

non-local ranges and the approximate solutions of the local models (4.1) and (4.2)

at T= 1 and a capacity of µ= 0.1 at T= 1: approximate solutions of road 1, left,

of road 2, top right and the evolution of the buﬀer, bottom right.

−2−1.8−1.6−1.4−1.2−1−0.8−0.6−0.4−0.2 0

0.74

0.76

0.78

0.8

0.82

0.84

0.86

0.88

0.9

x

ρ1

(4.1)

(4.2)

(2.2) η= 0.5

(2.2) η= 0.05

(2.2) η= 0.005

0 0.2 0.4

0.5

0.55

0.6

x

ρ2

0 0.2 0.4 0.6 0.8 1

0

2

4

6

·10−2

t

r

Figure 3. Approximate solutions of the non-local buﬀer model with diﬀerent

non-local ranges and the approximate solutions of the local models (4.1) and (4.2)

at T= 1 and a capacity of µ= 0.15 at T= 1: approximate solutions of road 1,

left, of road 2, top right and the evolution of the buﬀer, bottom right.

5. An infinite non-local reach

Besides the limit η→0 also the case η→ ∞ is of great interest. This limit represents the

situation in which drivers have perfect knowledge about the downstream traﬃc. In [11], it has been

proven that on a single road a non-local traﬃc ﬂow model tends to the solution of a linear transport

equation with transport velocity Vmax. The limit for non-local traﬃc networks without buﬀers has

12 F. A. CHIARELLO, J. FRIEDRICH, AND S. G ¨

OTTLICH

been already investigated in [23]. Here, the limit equations share similarities to a model which is

usually applied in the context of supply chains [4]. The key characteristic is that cars travel with

a constant velocity until a certain capacity is reached. This is very similar to what happens when

goods pass through a processor, cf. the supply chain models introduced in [4, 19]. Therefore, we

start by discussing the solution concept of this model.

5.1. A supply chain model. Following [4, 19] two processors with one buﬀer in between can be

described by:

(5.1)

∂tρ1+∂xmin{ρ1v, µ1}= 0

∂tρ2+∂xmin{ρ1v, µ2}= 0

r0(t) = min{ρ1(t, 0−)v, µ1} − dB(t)

dB(t) =

min{ρ1(t, 0−)v, µ1, µ2},if r(t) = 0,

µ2,if r(t)>0.

Note that we consider an unlimited buﬀer for simplicity. Further, the buﬀer have diﬀerent capaci-

ties, but the same processing velocity. In the case µ1=µ2and r0= 0 we can view the problem as

a single processor as the buﬀer will never ﬁll up. For the initial data

(5.2) ρ0(x) =

ρ1, x < 0,

ρ2, x > 0,

The solution is

•a linear transport with velocity v, if ρi< µi/v i ∈ {1,2}, or

•a rarefaction wave with intermediate state µ2/v, if ρ1> µ1/v and ρ2< µ2/v.

In the latter case the capacity of the buﬀer is not suﬃcient to handle the complete ﬂux, such that

the ﬂux decreases to the capacity. In [23], it is shown that this situation, i.e. (5.1) with µ1=µ2

and r0= 0, is the limit of the 1-to-1 model (2.1) without a buﬀer.

Now, we consider the more general case, µ16=µ2. For simplicity, the buﬀer is initially empty

and we consider only the case ρi< µi/v i ∈ {1,2}, since this case is the most important one for

our upcoming limit cases. Explicit formulas for the solution in all cases can be found in [19, Eq.

(4.21)]. The solution now depends on the outgoing ﬂux of the ﬁrst processor. If it is small enough,

all goods can be transported, otherwise only up to the capacity µ2. In fact, we have

•for ρ1≤µ2/v a linear transport and

•for ρ1> µ2/v a rarefaction wave with intermediate state µ2/v

as a solution. In the latter case, the buﬀer starts to ﬁll.

5.2. Limit models. Now, we derive the limit models of (2.2). Therefore, we assume that the

initial data ρ1,0and ρ2,0are of compact support. Note that due to the ﬁnite speed of propagation

also the solutions ρ1(t, ·) and ρ2(t, ·) are compactly supported. We can use the following result

proven in [23, Lemma 4.1]: in the L1

loc limit the non-local velocities converge to

V1(t, x)→0 and V2(t, x)→v2(0) for η→ ∞.(5.3)

A NON-LOCAL TRAFFIC FLOW MODEL FOR 1-TO-1 JUNCTIONS WITH BUFFER 13

Using similar arguments we can deduce

sB(t, x)→

µ r(t)∈[0, rmax),

min{ρ2

maxv2(0), µ}r(t) = rmax,

and(5.4)

dB(t)→

µ r(t)∈(0, rmax],

min{ρ1(t, 0−)v2(0), µ}r(t) = 0,

for η→ ∞.(5.5)

To discuss the limit models in detail, we distinguish two cases depending on the maximum ﬂux on

each road. In the limit the maximum ﬂux of the ﬁrst road is given by the capacity µand of the

second road by ρ2

maxv2(0).

Case 1: µ≤ρ2

maxv2(0). In this case the limit equations are given by

∂tρ1(t, x) + ∂xmin{ρ1(t, x)v2(0), µ}= 0

∂tρ2(t, x) + ∂xρ2(t, x)v2(0) = 0,

where the inﬂow on road 2 is given by (5.5) and hence depends on the buﬀer size. If we assign

an artiﬁcial capacity to the second road which is greater or equal to the maximum ﬂow, i.e. µ2≥

ρ2

maxv2(0), the dynamics are described by

∂tρ2(t, x) + ∂xmin{ρ2(t, x)v2(0), µ2}= 0.

Obviously, this does not change the dynamics on the second road, but now the similarities with

the model (5.1) become apparent. However, the inﬂow on road 2 and the capacity of the buﬀer

given in (5.5) do not match the ’artiﬁcial’ capacity of the second road and are smaller, i.e. µ≤µ2.

Thus, the model can be seen as an extension of the production model (5.1), which allows a smaller

capacity of the buﬀer. Such models have been studied in [26]. Note that once r(t∗) = 0 for some

t∗≥0 holds, the buﬀer stays empty for t≥t∗. In this case, the complete limit dynamics coincide

with (5.1). In contrast, if r(t)>0, the buﬀer decreases.

Example 5.1. To get a better understanding of the dynamics of the limit model, we focus on a

speciﬁc initial condition with an empty initial buﬀer and inﬁnite capacity of the buﬀer, i.e.

ρ1,0(x) =

ρ1,if x∈[a, b],

0,else

ρ2,0(x)=0, r(0) = 0, rmax =∞.(5.6)

for a < b < 0. The buﬀer model has a capacity of µon the ﬁrst road. This capacity is less than

the capacity on the second road. Hence, the second road can take all the ﬂux coming from the ﬁrst

road and the buﬀer does not increase. Further, due to the empty initial buﬀer, the buﬀer remains

empty. The solution depends on the value of ρ1:

(1) µ/v2(0) > ρ1: Here, we have a linear transport.

(2) ρ1> µ/v2(0): The solution of the buﬀer model is a rarefaction wave with constant state

µ/v2(0) at band a shock at a, which moves onto the second road. Both the shock at aand

the rarefaction wave move at the speed µ/ρ1< v2(0) on the ﬁrst road. On the second road

the speed increases to v2(0).

14 F. A. CHIARELLO, J. FRIEDRICH, AND S. G ¨

OTTLICH

Case 2 µ>ρ2

maxv2(0). In this case not only the inﬂow onto road 2 depends on the buﬀer size but

the ﬂux on the ﬁrst road, too. The model equations for the ﬁrst road are given by

∂tρ1+∂xmin{ρ1v2(0), µ}= 0,if r(t)∈[0, rmax)

∂tρ1+∂xmin{ρ1v2(0), ρ2

maxv2(0)}= 0,if r(t) = rmax.

and the second road is the same to the previous case:

∂tρ2+∂xρ2v2(0) = 0.

More importantly, the inﬂow onto road 2 is given by

ρ2

maxv2(0),if r(t)∈(0, rmax],

min{ρ1(t, 0−)v2(0), ρ2

maxv2(0)},if r(t) = 0.

This results in the following development of the buﬀer in the limit:

r0(t) =

min{µ, ρ1(t, 0−)v2(0)} − min{ρ1(t, 0−)v2(0), ρ2

maxv2(0)},if r(t)=0,

min{µ, ρ1(t, 0−)v2(0)} − ρ2

maxv2(0),if r(t)∈(0, rmax),

min{ρ2

maxv2(0), ρ1(t, 0−)v2(0)} − ρ2

maxv2(0),if r(t) = rmax .

As can be seen, the dynamics are more involved and strongly dependent on the buﬀer size. Nev-

ertheless, for r(t)∈[0, rmax) the limit equations can be viewed as the supply chain model (5.1) by

assigning the artiﬁcial capacity µ2=ρ2

maxv2(0) to the second road. In particular, the capacity of

the buﬀer is given by ρ2

maxv2(0) and is equal to the capacity of the next processor as in (5.1). Note

that the dynamics change when r(t) = rmax. Models with ﬁnite buﬀers are not studied in [4, 19].

Example 5.2. Let us return to the setting (5.6) for the second case. Now, the capacity of the ﬁrst

road is higher than that of the second one. This allows for an increasing buﬀer and interesting

dynamics:

(1) µ/v2(0) > ρ2

max > ρ1: Here, the density is low enough not to aﬀect the buﬀer, so the solution

is simply the linear transport.

(2) µ/v2(0) > ρ1> ρ2

max: The solution is ﬁrst given by a linear transport equation. When the

density reaches the buﬀer at t∗=−b/v2(0), the buﬀer starts to ﬁll at the rate:

r0(t) = v2(0)(ρ1−ρ2

max).

The inﬂow on the second road is then ρ2

maxv2(0). This results in a density of ρ2

max.

(3) ρ1> µ/v2(0) > ρ2

max: Instead of a linear transport, a rarefaction wave with intermediate

state µ/v2(0) and a shock is generated. Again, after some time (i.e. when the rarefaction

wave reaches x= 0), the buﬀer begins to ﬁll with

r0(t) = µ−v2(0)ρ2

max.

On road 2 the density is ρ2

max as before.

A NON-LOCAL TRAFFIC FLOW MODEL FOR 1-TO-1 JUNCTIONS WITH BUFFER 15

−5−4−3−2−1 0

0

0.2

0.4

0.6

0.8

1

x

ρ1

(4.2)

constant

linear

η→ ∞

0 1 2 3

0

0.2

0.4

x

ρ2

0 1 2 3

0

0.2

0.4

0.6

t

r

Figure 4. Approximate solution of the non-local buﬀer model with a constant

and linear kernel for η= 2 with the limit models to zero and to inﬁnity at T= 3:

approximate solutions of road 1, left, of road 2, top right and the evolution of the

buﬀer, bottom right.

5.3. Numerical examples. In this subsection we consider some numerical results for the diﬀerent

cases discussed before. For the numerical tests we use ∆x= 0.01.

We start with an explicit example for the initial condition (5.6) and the parameters chosen such

that ρ1> µ/v2(0) > ρ2

max holds. More precisely, we choose the initial conditions

ρ1,0(x) =

1,if x∈[−5,−1/3],

0,else

ρ2,0(x)=0, r(0) = 0, rmax =∞,

µ= 0.75, v1(ρ) = 1 −ρ,v2(ρ) = 1 −2ρand hence ρ2

max = 0.5. The explicit solution for the limit

model for η→ ∞ is for t < 1/3 given by:

ρ1(t, x) =

0.75,if x∈[−1/3,−1/3 + t],

1,if x∈[−5+0.75t, −1/3],

0,else

ρ2,0(x)=0, r(t)=0,

and for t∈[1/3,56/9) by

ρ1(t, x) =

0.75,if x∈[−1/3,0],

1,if x∈[−5+0.75t, −1/3],

0,else

ρ2,0(x) =

0.5,if x∈[0, t −1/3],

0,else

,

r(t)=0.25(t−1/3).

We compare the exact limit solution with a numerical approximation of the non-local buﬀer model

computed with η= 2 for a constant and linear decreasing kernel. Figure 4 shows also the possible

limit model for η→0, i.e., (4.2). It can be seen that for the diﬀerent models induced by ηmore mass

seems to be transported on the ﬁrst road, into the buﬀer and trough the junction with increasing

η. In the following, we again restrict ourselves to a linear decreasing kernel. The limit for η→ ∞

16 F. A. CHIARELLO, J. FRIEDRICH, AND S. G ¨

OTTLICH

−4−3−2−1 0

0

0.2

0.4

0.6

0.8

1

x

ρ1

η= 10

η= 25

η= 75

η= 300

η→ ∞

0 1 2 3

0

0.2

0.4

x

ρ2

0 1 2 3

0

0.2

0.4

0.6

0.8

t

r

Figure 5. Approximate solution of the non-local buﬀer model with increasing

values of ηat T= 3: approximate solutions of road 1, left, of road 2, top right and

the evolution of the buﬀer, bottom right.

is shown in Figure 5. It can be seen that in particular the approximation of the shock and the

rarefaction wave of the exact solution for η→ ∞ requires rather large values of η. Further, with

increasing ηmore mass is transported on the ﬁrst and second road. During the discussion of the

limit models in Section 5.2, we also encountered speciﬁc cases of the production model (5.1) that

have not yet been studied in the literature. Therefore, we consider slight modiﬁcations of the initial

setup in (5.6). In Figure 6, we consider the same situation as above but with an unlimited and a

limited buﬀer of rmax = 0.15 for η= 200 at T= 2. In the latter case, the buﬀer increases until

about T= 1. Before T≈1, the solution is the same as discussed above. Afterwards the dynamics

with a limited buﬀer change, which can be seen in the diﬀerent speeds for the shock on the ﬁrst

road. For a full buﬀer, the speed drops from 0.75 to 0.5. On the second road the solutions for both

models coincide.

6. Conclusion

We presented a non-local buﬀer model for a 1-to-1 junction with a suitable numerical discreti-

sation. We have proved that the numerical scheme preserves the maximum principle uniform in η.

Further, we investigate the model hierarchies induced by the non-local term. For η→0, we can

exclude the convergence to the local model of [27] which originally inspired the non-local buﬀer

model. We propose a diﬀerent local model which seems to be (numerically) the correct limit of

the non-local buﬀer model. Moreover, we investigate the limit of η→ ∞. Here, drivers drive

with constant speed up to a certain capacity. This shares similarities to supply chain models with

two processors and one buﬀer. Nevertheless, the obtained limit dynamics are more general than

the supply chain models studied in the literature so far. Again the convergence can be obtained

numerically.

Future work could include rigorously prooﬁng the limits of the non-local buﬀer model and ex-

tending the junction model to more complex junctions and network structures.

A NON-LOCAL TRAFFIC FLOW MODEL FOR 1-TO-1 JUNCTIONS WITH BUFFER 17

-4 -3 -2 -1 0

0

0.2

0.4

0.6

0.8

1

x

ρ1

Initial data

(2.2) rmax = 0.15

(2.2) rmax =∞

0 0.511.5 2

0

0.2

0.4

0.6

x

ρ2

0 0.511.5 2

0

0.2

0.4

t

r

Figure 6. Approximate solution of the non-local buﬀer model at T= 2 with a

ﬁnite and inﬁnite buﬀer: approximate solutions of road 1, left, of road 2, top right

and the evolution of the buﬀer, bottom right.

Acknowledgement

F.A.C. is member of Gruppo Nazionale per l’Analisi Matematica, la Probabilit`a e le loro Appli-

cazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). F.A.C. was partially

supported by the INdAM-GNAMPA project CUP E53C22001930001. Jan Friedrich is supported

by the German Research Foundation (DFG) under grant FR 4850/1-1, and Simone G¨ottlich under

grant GO 1920/12-1.

References

[1] S. M. Adimurthi and G. D. Veerappa Gowda. Optimal entropy solutions for conservation laws with discontinuous

ﬂux-functions. J. Hyperbolic Diﬀer. Equ., 2(4):783–837, 2005.

[2] A. Aggarwal, R. M. Colombo, and P. Goatin. Nonlocal systems of conservation laws in several space dimensions.

SIAM J. Numer. Anal., 53(2):963–983, 2015.

[3] P. Amorim, R. Colombo, and A. Teixeira. On the numerical integration of scalar nonlocal conservation laws.

ESAIM M2AN, 49(1):19–37, 2015.

[4] D. Armbruster, P. Degond, and C. Ringhofer. A model for the dynamics of large queuing networks and supply

chains. SIAM J. Appl. Math., 66(3):896–920, 2006.

[5] S. Blandin and P. Goatin. Well-posedness of a conservation law with non-local ﬂux arising in traﬃc ﬂow modeling.

Numer. Math., 132(2):217–241, 2016.

[6] A. Bressan and W. Shen. On traﬃc ﬂow with nonlocal ﬂux: a relaxation representation. Archive for Rational

Mechanics and Analysis volume, 237, 2020.

[7] A. Bressan and W. Shen. Entropy admissibility of the limit solution for a nonlocal model of traﬃc ﬂow. Commun.

Math. Sci., 19(5):1447–1450, 2021.

[8] F. Camilli, R. De Maio, and A. Tosin. Measure-valued solutions to nonlocal transport equations on networks. J.

Diﬀer. Equ., 264(12):7213–7241, 2018.

[9] F. Chiarello and G. Coclite. Nonlocal scalar conservation laws with discontinuous ﬂux. Netw. Heterog. Media,

18(1):380–398, 2023.

[10] F. A. Chiarello, J. Friedrich, P. Goatin, S. G¨ottlich, and O. Kolb. A non-local traﬃc ﬂow model for 1-to-1

junctions. European J. Appl. Math., 31(6):1029–1049, 2020.

18 F. A. CHIARELLO, J. FRIEDRICH, AND S. G ¨

OTTLICH

[11] F. A. Chiarello and P. Goatin. Global entropy weak solutions for general non-local traﬃc ﬂow models with

anisotropic kernel. ESAIM Math. Model. Numer. Anal., 52(1):163–180, 2018.

[12] F. A. Chiarello and A. Keimer. On the singular limit problem of nonlocal balance laws — applications to nonlocal

lane-changing traﬃc ﬂow models. Submitted, 2023.

[13] J. Chien and W. Shen. Stationary wave proﬁles for nonlocal particle models of traﬃc ﬂow on rough roads.

NoDEA, 26(6):Paper No. 53, 2019.

[14] G. M. Coclite, N. De Nitti, A. Keimer, and L. Pﬂug. Singular limits with vanishing viscosity for nonlocal

conservation laws. Nonlinear Anal., 211:112370, 2021.

[15] M. Colombo, G. Crippa, M. Graﬀe, and L. V. Spinolo. Recent results on the singular local limit for nonlocal

conservation laws. arXiv:1902.06970, 2019.

[16] M. Colombo, G. Crippa, E. Marconi, and L. Spinolo. Nonlocal traﬃc models with general kernels: singular limit,

entropy admissibility, and convergence rate. Arch. Rational. Mech. Anal., 247, 2023.

[17] M. Colombo, G. Crippa, and L. V. Spinolo. On the singular local limit for conservation laws with nonlocal ﬂuxes.

Arch. Rational. Mech. Anal., 233:1131–1167, 2019.

[18] R. M. Colombo, M. Herty, and M. Mercier. Control of the continuity equation with a non local ﬂow. ESAIM

Control Optim. Calc. Var., 17(2):353–379, 2011.

[19] C. D’Apice, S. G¨ottlich, M. Herty, and B. Piccoli. Modeling, simulation, and optimization of supply chains.

Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2010. A continuous approach.

[20] M. Delle Monache and P. Goatin. Scalar conservation laws with moving constraints arising in traﬃc ﬂow mod-

eling: An existence result. J. Diﬀer. Equ., 257(11):4015–4029, 2014.

[21] M. L. Delle Monache, J. Reilly, S. Samaranayake, W. Krichene, P. Goatin, and A. M. Bayen. A PDE-ODE model

for a junction with ramp buﬀer. SIAM J. Appl. Math., 74(1):22–39, 2014.

[22] J. Friedrich, S. G¨ottlich, A. Keimer, and L. Pﬂug. Conservation laws with nonlocal velocity–the singular limit

problem. arXiv preprint arXiv:2210.12141, 2022.

[23] J. Friedrich, S. G¨ottlich, and M. Osztfalk. Network models for nonlocal traﬃc ﬂow. ESAIM: Mathematical

Modelling and Numerical Analysis, 56(1):213–235, 2022.

[24] J. Friedrich, O. Kolb, and S. G¨ottlich. A Godunov type scheme for a class of LWR traﬃc ﬂow models with

non-local ﬂux. Netw. Heterog. Media, 13(4):531–547, 2018.

[25] J. Friedrich, S. Sudha, and S. Rathan. Numerical schemes for a class of nonlocal conservation laws: a general

approach. Netw. Heterog. Media, 18(3):1335–1354, 2023.

[26] K. Hameister. A dual tailored branch-and-bound algorithm for quadratic mixed-integer problems applied to

production models with buﬀers. Verlag Dr. Hut, M¨unchen, 2018. Mannheim, Univ., Diss., 2019.

[27] M. Herty, J.-P. Lebacque, and S. Moutari. A novel model for intersections of vehicular traﬃc ﬂow. Netw. Heterog.

Media, 4(4):813, 2009.

[28] A. Keimer and L. Pﬂug. Existence, uniqueness and regularity results on nonlocal balance laws. J. Diﬀerential

Equations, 263(7):4023–4069, 2017.

[29] A. Keimer and L. Pﬂug. On approximation of local conservation laws by nonlocal conservation laws. J. Appl.

Math. Anal. Appl., 475(2):1927 – 1955, 2019.

[30] A. Keimer, L. Pﬂug, and M. Spinola. Existence, uniqueness and regularity of multi-dimensional nonlocal balance

laws with damping. J. Math. Anal. Appl., 466(1):18 – 55, 2018.

[31] A. Keimer, L. Pﬂug, and M. Spinola. Nonlocal scalar conservation laws on bounded domains and applications

in traﬃc ﬂow. SIAM J. Math. Anal., 50(6):6271–6306, 2018.

[32] J. P. Lebacque. The Godunov scheme and what it means for ﬁrst order traﬃc ﬂow models. Proc. 13th Intrn.

Symp. Transportation and Traﬃc Theory, 1996.

[33] M. Lighthill and G. Whitham. On kinematic waves. i. ﬂood movement in long rivers. Proceedings of the Royal

Society of London. Series A. Mathematical and Physical Sciences, 229(1178):281–316, 1955.

[34] P. Richards. Shock waves on the highway. Operations research, 4(1):42–51, 1956.

[35] W. Shen. Traveling waves for conservation laws with nonlocal ﬂux for traﬃc ﬂow on rough roads. Netw. Heterog.

Media, 14(4):709–732, 2019.

A NON-LOCAL TRAFFIC FLOW MODEL FOR 1-TO-1 JUNCTIONS WITH BUFFER 19

[36] K. Zumbrun. On a nonlocal dispersive equation modeling particle suspensions. Quart. Appl. Math., 57(3):573–

600, 1999.

(Felisia Angela Chiarello)

DISIM, University of L’Aquila, Via Vetoio Ed. Coppito 1, 67100 L’Aquila, Italy.

Email address:felisiaangela.chiarello@univaq.it

(Jan Friedrich)

IGPM, RWTH Aachen University, Templergraben 55, 52064 Aachen, Germany.

Email address:friedrich@igpm.rwth-aachen.de

(Simone G¨ottlich)

Department of Mathematics, University of Mannheim, B6, 28-29, 68159 Mannheim

Email address:goettlich@uni-mannheim.de