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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.
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(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
