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## Abstract

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.
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  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 , Kruˇzkov-type entropy conditions are used to prove the
L1stability 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 . 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  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 , the authors consider measure valued solutions for non-local transport equations
and  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 .
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  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 . 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(yx)dy,
V2(t, x) := Zmax{x+η, 0}
max{x, 0}
v2(ρ2(t, x))ωη(yx)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
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(yx)dy, r(t)[0, rmax ),
min{ρ2
maxV2(t, x), µ Rmax{x+η,0}
0ωη(yx)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 . 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) = r00,
where BV([a, b]) := {fL1([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) ωηC1([0, η]; R+), η > 0, ω0
η0,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 ρρe
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 ρeC(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 Caratheodory solution for, i.e. for a.e. tR+
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 jZand nNand eE:= {1,2}, let xe,j1/2=jxbe the cell interfaces, xe,j =
(j+ 1/2)∆xthe cells centers, corresponding to a space step xsuch that η=Nηxfor some
NηN, and let tn=ntbe the time mesh. In particular, x=xe,1/2= 0 is a cell interface. Note
that we have j0 if e= 2 and j < 0 if e= 1.We aim at constructing a ﬁnite volume approximate
solution ρx
esuch that ρ
e(t, x) = ρn
e,j for (t, x)[tn, tn+1)×[xe,j1/2, 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,j1/2
ρe,0(x)dx,
j < 0, e = 1,
j0, 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{−j2, Nη1}
X
k=0
γkv1(ρn
1, j+k+1), V n
2,j =
Nη1
X
k=max{−j1,0}
γkv2(ρn
2, j+k+1),
sn
B,j =
µ
Nη1
X
k=j1
γk, rn[0, rmax),
min
ρ2
maxVn
2,j, µ
Nη1
X
k=j1
γk
, rn=rmax,
dn
B=
µ, rn(0, rmax],
min{ρn
1,1Vn
2,1, µ}, rn= 0.
where γk=R(k+1)∆x
kxωη(x)dx, k = 0, ..., Nη1 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,j1,
rn+1 =rn+ tmin{sn
B,0, ρn
1,0Vn
2,0} min{dn
B, ρ2
maxVn
2,0},
nN.
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
x1
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:
0ρn
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,j1ρn
e,j t
xkvkρn
e,j 0,
with kvk= max{kv1k,kv2k},using min nρn
e,jVn
2,j, sn
B,j oρn
e,jVn
2,j.For the upper bound, it is easy
to see that
Vn
1,j1Vn
1,j
γ0v1(ρn
1,j), j 1,
0, j 0,
Vn
2,j1Vn
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,j1Vn
1,j
γ0kv0
1k(ρn
1,j +ρ1
max), j 1,
0, j 0,
Vn
2,j1Vn
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,j1ρ1
max Vn
1,jρn
1,j γ0
v0
1
kρ1k+Vn
1,j(ρn
1,j +ρ1
max).
Similarly, writing
sn
B,j1sn
B,j =
µγj1rn[0, rmax),
min{ρ2
maxVn
2,j1, µ PNη1
k=jγk} min{ρ2
maxVn
2,j, µ PNη1
k=j1γk}rn=rmax,
0,
because
min
ρ2
maxVn
2,j1, µ
Nη1
X
k=j
γk
min
ρ2
maxVn
2,j, µ
Nη1
X
k=j1
γk
min
ρ2
maxVn
2,j, µ
Nη1
X
k=j
γk
min
ρ2
maxVn
2,j, µ
Nη1
X
k=j1
γk
=
µPNη1
k=jγkµPNη1
k=j1γk
ρ2
maxVn
2,j µPNη1
k=jγk
+
ρ2
maxVn
2,j µPNη1
k=j1γk
2
µPNη1
k=jγkµPNη1
k=j1γk+
µPNη1
k=jγkµPNη1
k=j1γk
2= 0.
Then, we get
(3.5)
min{ρ1
maxVn
2,j1, sn
B,j1} 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 j0,we can write the following bound
ρ2
maxVn
2,j1ρn
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,j1ρn
2,jVn
2,jρ2
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 η0and 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 η0 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 η0+the family ωηconverges weaklyin 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:RRbeing 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(ρηωη),
was initially posed in  motivated by numerical evidence, later corroborated in . The answer
is positive if we consider an even kernel function and a smooth compactly supported initial da-
tum, as proved in . 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 . Nevertheless, in general, the solution
of the non-local Cauchy problem does not converge to the solution of the local one and in  the
authors show three counterexamples in this sense, while in , 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 . For completeness we recall the model of . 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  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  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
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 η0. 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 .
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
η0ρ1(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
η0ρ2
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 η0.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= 103and the time step is chosen according to the CFL condition (3.3). We deal with the
following example: v1(ρ)=1ρ,v2(ρ)=15ρ/3, constant initial data of ρ1,00.75 and
ρ2,00.5 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, a linear decreasing kernel, i.e. ωη(x) = 2(ηx)2and a quadratic decreasing
kernel, i.e. ωη(x) = 3(η2x2)/(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
1.41.210.80.60.40.2 0
0.76
0.78
0.8
0.82
0.84
0.86
0.88
0.9
x
ρ1
constant
linear
(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
·102
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 η0. 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.  and the references therein. In , 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
21.81.61.41.210.80.60.40.2 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
2·102
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.
21.81.61.41.210.80.60.40.2 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
·102
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 η0 also the case η is of great interest. This limit represents the
situation in which drivers have perfect knowledge about the downstream traﬃc. In , 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 . Here, the limit equations share similarities to a model which is
usually applied in the context of supply chains . 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 , 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 ρ1µ2/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
maxv2(0).
Case 1: µρ2
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. µµ2.
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 . Note that once r(t) = 0 for some
t0 holds, the buﬀer stays empty for tt. 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)(ρ1ρ2
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) = µv2(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
54321 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(t1/3).
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 η0, 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
4321 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 T1, 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 η0, we can
exclude the convergence to the local model of  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 Probabilita 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 ottlich under
grant GO 1920/12-1.
References
 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.
 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.
 P. Amorim, R. Colombo, and A. Teixeira. On the numerical integration of scalar nonlocal conservation laws.
ESAIM M2AN, 49(1):19–37, 2015.
 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.
 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.
 A. Bressan and W. Shen. On traﬃc ﬂow with nonlocal ﬂux: a relaxation representation. Archive for Rational
Mechanics and Analysis volume, 237, 2020.
 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.
 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.
 F. Chiarello and G. Coclite. Nonlocal scalar conservation laws with discontinuous ﬂux. Netw. Heterog. Media,
18(1):380–398, 2023.
 F. A. Chiarello, J. Friedrich, P. Goatin, S. 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
 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.
 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.
 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.
 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.
 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.
 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.
 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.
 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.
 C. D’Apice, S. 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.
 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.
 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.
 J. Friedrich, S. ottlich, A. Keimer, and L. Pﬂug. Conservation laws with nonlocal velocity–the singular limit
problem. arXiv preprint arXiv:2210.12141, 2022.
 J. Friedrich, S. ottlich, and M. Osztfalk. Network models for nonlocal traﬃc ﬂow. ESAIM: Mathematical
Modelling and Numerical Analysis, 56(1):213–235, 2022.
 J. Friedrich, O. Kolb, and S. 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.
 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.
 K. Hameister. A dual tailored branch-and-bound algorithm for quadratic mixed-integer problems applied to
production models with buﬀers. Verlag Dr. Hut, unchen, 2018. Mannheim, Univ., Diss., 2019.
 M. Herty, J.-P. Lebacque, and S. Moutari. A novel model for intersections of vehicular traﬃc ﬂow. Netw. Heterog.
Media, 4(4):813, 2009.
 A. Keimer and L. Pﬂug. Existence, uniqueness and regularity results on nonlocal balance laws. J. Diﬀerential
Equations, 263(7):4023–4069, 2017.
 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.
 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.
 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.
 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.
 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.
 P. Richards. Shock waves on the highway. Operations research, 4(1):42–51, 1956.
 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
 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.