# A non-local traffic flow model for 1-to-1 junctions with buffer

F. A. Chiarello
DISIM, University of L’Aquila, Via Vetoio Ed. Coppito 1, 67100 L’Aquila,
Italy<EMAIL_ADDRESS>,  J. Friedrich
IGPM, RWTH Aachen University, Templergraben 55, 52064 Aachen, Germany.
<EMAIL_ADDRESS>and  S. Göttlich
Department of Mathematics, University of Mannheim, B6, 28-29, 68159 Mannheim
<EMAIL_ADDRESS>
###### 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.

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 reason 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 dynamics. ‘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], Kružkov-type entropy conditions are used to
prove the $\mathbf{L^{1}}-$stability with respect to the initial data through
the doubling of variable technique, while in [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 $\eta$ of the
kernel function goes to zero is exploited and in Section 5 the opposite limit
model as the length of the support $\eta$ of 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)
$\displaystyle\partial_{t}\rho_{1}(t,x)+\partial_{x}\left(\rho_{1}(t,x)V_{1}(t,x)+\min\\{\rho_{1}V_{2}(t,x),\,\rho^{2}_{\max}V_{2}(t,x)\\}\right)=0,$
$\displaystyle\partial_{t}\rho_{2}(t,x)+\partial_{x}(\rho_{2}(t,x)V_{2}(t,x))=0.$

Here, $V_{1}$ and $V_{2}$ are the parts of the non-local velocity on each road
segment defined by

$V_{1}(t,x):=\int_{\min\\{x,\,0\\}}^{\min\\{x+\eta,\,0\\}}v_{1}(\rho_{1}(t,x))\omega_{\eta}(y-x)\,dy,$
$V_{2}(t,x):=\int_{\max\\{x,\,0\\}}^{\max\\{x+\eta,\,0\\}}v_{2}(\rho_{2}(t,x))\omega_{\eta}(y-x)\,dy.$

In particular, the junction point is placed at $x=0$, $\rho_{1}$ and
$\rho_{2}$ are the traffic densities, $\eta>0$ is the non-local range and
support of the kernel function $\omega_{\eta}$, which will be detailed below,
and $v_{1}$ and $v_{2}$ are 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 second road can receive a flux up to $\rho^{2}_{\max}V_{2}(t,0)$. Due to
non-locality, this capacity restriction must be considered already in the
interval $[-\eta,0)$. 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 $\mu$, i.e. the maximum rate at which cars can enter or leave the
buffer. Furthermore, the buffer is full at $r_{\max}$. 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 $\rho^{2}_{\max}V_{2}(t,0)$.
This results in the following modeling equations for the buffer $r(t)$ and the
traffic densities:

(2.2)
$\displaystyle\partial_{t}\rho_{1}(t,x)+\partial_{x}\left(\rho_{1}(t,x)V_{1}(t,x)+\min\\{\rho_{1}V_{2}(t,x),\,s_{B}(t,x)\\}\right)=0,$
$\displaystyle\partial_{t}\rho_{2}(t,x)+\partial_{x}(\rho_{2}(t,x)V_{2}(t,x))=0,$
$\displaystyle
r^{\prime}(t)=\min\\{s_{B}(t,0^{-}),\,\rho_{1}(t,0^{-})V_{2}(t,0)\\}-\min\\{d_{B}(t),\,\rho^{2}_{\max}V_{2}(t,0)\\},$

where

$\displaystyle
s_{B}(t,x)=\begin{cases}\mu\int_{0}^{\max\\{x+\eta,0\\}}\omega_{\eta}(y-x)\,dy,\quad&r(t)\in[0,r_{\max}),\\\
\min\\{\rho^{2}_{\max}V_{2}(t,x),\,\mu\int_{0}^{\max\\{x+\eta,0\\}}\omega_{\eta}(y-x)\,dy\\},&r(t)=r_{\max},\end{cases}$
$\displaystyle d_{B}(t)=\begin{cases}\mu,\quad&r(t)\in(0,r_{\max}],\\\
\min\\{\rho_{1}(t,0^{-})V_{2}(t,0),\,\mu\\},&r(t)=0.\end{cases}$

Note that the model (LABEL:eq:model) 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 (LABEL:eq:model) we couple them with the initial data

(2.3)
$\displaystyle\rho_{1}(0,x)=\rho_{1,0}(x)\in\mathbf{BV}((-\infty,0)),\quad\rho_{2}(0,x)=\rho_{2,0}(x)\in\mathbf{BV}((0,+\infty)),\quad
r(0)=r_{0}\geq 0,$

where $\mathbf{BV}([a,b]):=\\{f\in\mathbf{L^{1}}([a,b])|\>TV(f)<+\infty\\}.$
For the well-posedness of the model, we assume the following common
assumptions, see e.g. [5, 11, 10, 23, 24]

  * (H1)

$\quad\omega_{\eta}\in\mathbf{C^{1}}([0,\eta];\mathbb{R}^{+}),\,\eta>0,\,\omega^{\prime}_{\eta}\leq
0,$ $\int_{0}^{\eta}\omega_{\eta}(x)dx=1$;

  * (H2)

$\quad v_{e}:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$, is a smooth non-
increasing function such that $v_{e}(0)=V_{e}^{\max}>0$ and $v_{e}(\rho)=0$
for $\rho\geq\rho_{\max}^{e}$ for $e\in\\{1,2\\}$.

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 $\rho_{e}\in
C(\mathbb{R}^{+},\mathbf{L^{1}}_{loc}((a_{e},b_{e})))$ with $e\in\\{1,2\\},$
and $a_{1}=-\infty,b_{1}=0$ and $a_{2}=0,b_{2}=+\infty,$ is called a weak
solution of (LABEL:eq:model)-(2.3) if

  1. (1)

for $e=1,\,\rho_{e}$ is a weak solution on $(-\infty,0)$ to

$\partial_{t}\rho_{e}(t,x)+\partial_{x}\left(\rho_{e}(t,x)V_{1}(t,x)+\min\\{\rho_{e}V_{2}(t,x),\,s_{B}(t,x)\\}\right)=0;$

  2. (2)

for $e=2,\,\rho_{e}$ is a weak solution on $(0,+\infty)$ to

$\partial_{t}\rho_{e}(t,x)+\partial_{x}(\rho_{e}(t,x)V_{2}(t,x))=0;$

  3. (3)

for all $t>0,$ it holds

$\rho_{2}(t,0^{+})V_{2}(t,0)=\min\\{d_{B}(t),\,\rho^{2}_{\max}V_{2}(t,0)\\};$

  4. (4)

for $e\in\\{1,2\\}$ and for a.e. $t>0$ the function $x\mapsto\rho_{e}(t,x)$
has a bounded total variation.

  5. (5)

$r(t)$ is a Carathèodory solution for, i.e. for a.e. $t\in\mathbb{R}^{+}$

$r(t)=r_{0}+\int_{0}^{t}\left(\min\\{s_{B}(s,0^{-}),\,\rho_{1}(s,0^{-})V_{2}(s,0)\\}-\min\\{d_{B}(s),\,\rho^{2}_{\max}V_{2}(s,0)\\}\right)\,ds.$

##  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 $j\in\mathbb{Z}$ and $n\in\mathbb{N}$ and $e\in E:=\\{1,2\\}$, let
$x_{e,j-1/2}=j\Delta x$ be the cell interfaces, $x_{e,j}=(j+1/2)\Delta x$ the
cells centers, corresponding to a space step $\Delta x$ such that
$\eta=N_{\eta}\Delta x$ for some $N_{\eta}\in\mathbb{N}$, and let
$t_{n}=n\Delta t$ be the time mesh. In particular, $x=x_{e,-1/2}=0$ is a cell
interface. Note that we have $j\geq 0$ if $e=2$ and $j<0$ if $e=1.$ We aim at
constructing a finite volume approximate solution $\rho^{\Delta x}_{e}$ such
that $\rho^{\Delta}_{e}(t,x)=\rho^{n}_{e,j}$ for
$(t,x)\in[t^{n},t^{n+1})\times[x_{e,j-1/2},x_{e,j+1/2})$. To this end, we
approximate the initial datum $\rho_{0}$ with the cell averages

$\rho_{e,j}^{0}=\frac{1}{\Delta
x}\int_{x_{e,j-1/2}}^{x_{e,j+1/2}}\rho_{e,0}(x)\,dx,\qquad\begin{cases}j<0,\quad
e=1,\\\ j\geq 0,\quad e=2.\end{cases}$

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

(3.1)
$F^{n}_{e,j}:=\begin{cases}\rho^{n}_{e,j}V^{n}_{e,j}+\min\left\\{\rho^{n}_{e,j}V^{n}_{2,j},s^{n}_{B,j}\right\\},\quad&e=1,\\\
\rho^{n}_{e,j}V^{n}_{e,j},\quad&e=2,\end{cases}$

with

$\displaystyle V^{n}_{1,j}$
$\displaystyle=\sum_{k=0}^{\min\\{-j-2,\,N_{\eta}-1\\}}\gamma_{k}v_{1}(\rho^{n}_{1,\,j+k+1}),\quad
V^{n}_{2,j}=\sum_{k=\max\\{-j-1,\,0\\}}^{N_{\eta-1}}\gamma_{k}v_{2}(\rho^{n}_{2,\,j+k+1}),$
$\displaystyle s^{n}_{B,j}$
$\displaystyle=\begin{cases}\mu\displaystyle{\sum_{k=-j-1}^{N_{\eta-1}}}\gamma_{k},\quad&r^{n}\in[0,r_{\max}),\\\
\min\left\\{\rho^{2}_{\max}V^{n}_{2,j},\,\mu\displaystyle{\sum_{k=-j-1}^{N_{\eta-1}}}\gamma_{k}\right\\},&r^{n}=r_{\max},\end{cases}\quad
d^{n}_{B}=\begin{cases}\mu,\quad&r^{n}\in(0,r_{\max}],\\\
\min\\{\rho^{n}_{1,-1}V^{n}_{2,-1},\,\mu\\},&r^{n}=0.\end{cases}$

where $\gamma_{k}=\int_{k\Delta x}^{(k+1)\Delta x}\omega_{\eta}(x)\,dx,\
k=0,...,N_{\eta}-1$ and $r^{n}$ is the approximation of the buffer computed by
an explicit Euler scheme. In this way, we define the following finite volume
scheme:

(3.2) $\begin{aligned} \rho^{n+1}_{e,j}&=\rho^{n}_{e,j}-\frac{\Delta t}{\Delta
x}\left(F^{n}_{e,j}-F^{n}_{e,j-1}\right),\\\ r^{n+1}&=r^{n}+\Delta
t\left(\min\\{s^{n}_{B,0},\rho^{n}_{1,0}V^{n}_{2,0}\\}-\min\\{d^{n}_{B},\rho^{2}_{\max}V^{n}_{2,0}\\}\right),\end{aligned}\quad
n\in\mathbb{N}.$

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) $\frac{\Delta t}{\Delta
x}\leq\frac{1}{\gamma_{0}\left\|v^{\prime}\right\|\left\|\rho\right\|+2\left\|v\right\|},$

with
$\left\|v\right\|=\max\\{\left\|v_{1}\right\|,\left\|v_{2}\right\|\\},\,\left\|v^{\prime}\right\|=\max\\{\left\|v_{1}^{\prime}\right\|,\left\|v_{2}^{\prime}\right\|\\}$
and $\left\|\rho\right\|=\max\\{\rho^{1}_{\max},\,\rho^{2}_{\max}\\},$ the
sequence generated by the numerical scheme (3.1)-(3.2) satisfies the following
maximum principle:

$0\leq\rho^{n}_{e,j}\leq\rho^{e}_{\max}.$

###### Proof.

Let us start by showing the positivity, we directly obtain

$\rho^{n+1}_{e,j}=\rho^{n}_{e,j}-\frac{\Delta t}{\Delta
x}\left(F^{n}_{e,j}-F^{n}_{e,j-1}\right)\geq\rho^{n}_{e,j}-\frac{\Delta
t}{\Delta x}\left\|v\right\|\rho^{n}_{e,j}\geq 0,$

with $\left\|v\right\|=\max\\{\left\|v_{1}\right\|,\left\|v_{2}\right\|\\},$
using
$\min\left\\{\rho^{n}_{e,j}V^{n}_{2,j},s^{n}_{B,j}\right\\}\leq\rho^{n}_{e,j}V^{n}_{2,j}.$
For the upper bound, it is easy to see that

$\displaystyle
V^{n}_{1,j-1}-V^{n}_{1,j}\leq\begin{cases}\gamma_{0}v_{1}(\rho^{n}_{1,j}),\quad&j\leq-1,\\\
0,\quad&j\geq 0,\end{cases}\quad
V^{n}_{2,j-1}-V^{n}_{2,j}\leq\begin{cases}0,\quad&j\leq-1,\\\
\gamma_{0}v_{2}(\rho^{n}_{2,j}),\quad&j\geq 0.\end{cases}$

Using $v_{1}(\rho^{1}_{\max})=v_{2}(\rho^{2}_{\max})=0$ and the mean value
theorem, we get

$\displaystyle
V^{n}_{1,j-1}-V^{n}_{1,j}\leq\begin{cases}\gamma_{0}\left\|v_{1}^{\prime}\right\|(-\rho^{n}_{1,j}+\rho^{1}_{\max}),\quad&j\leq-1,\\\
0,\quad&j\geq 0,\end{cases}$ $\displaystyle
V^{n}_{2,j-1}-V^{n}_{2,j}\leq\begin{cases}0,\quad&j\leq-1,\\\
\gamma_{0}\left\|v_{2}^{\prime}\right\|(-\rho^{n}_{2,j}+\rho^{2}_{\max}),\quad&j\geq
0.\end{cases}$

Considering the case $j\leq-1,$ multiplying the first inequality by
$\rho^{1}_{\max}$ and subtracting $V^{n}_{1,j}\rho^{n}_{1,j},$ we get

(3.4)
$V^{n}_{1,j-1}\rho^{1}_{\max}-V^{n}_{1,j}\rho^{n}_{1,j}\leq\left(\gamma_{0}\left\|v_{1}^{\prime}\right\|\left\|\rho_{1}\right\|+V^{n}_{1,j}\right)(-\rho^{n}_{1,j}+\rho^{1}_{\max}).$

Similarly, writing

$\displaystyle
s^{n}_{B,j-1}-s^{n}_{B,j}=\begin{cases}-\mu\gamma_{-j-1}\quad&r^{n}\in[0,r_{\max}),\\\
\min\\{\rho^{2}_{\max}V^{n}_{2,j-1},\mu\sum_{k=-j}^{N_{\eta}-1}\gamma_{k}\\}-\min\\{\rho^{2}_{\max}V^{n}_{2,j},\mu\sum_{k=-j-1}^{N_{\eta}-1}\gamma_{k}\\}\quad&r^{n}=r_{\max},\end{cases}\leq
0,$

because

$\displaystyle\min\left\\{\rho^{2}_{\max}V^{n}_{2,j-1},\mu\sum_{k=-j}^{N_{\eta}-1}\gamma_{k}\right\\}-\min\left\\{\rho^{2}_{\max}V^{n}_{2,j},\mu\sum_{k=-j-1}^{N_{\eta}-1}\gamma_{k}\right\\}$
$\displaystyle\leq\min\left\\{\rho^{2}_{\max}V^{n}_{2,j},\mu\sum_{k=-j}^{N_{\eta}-1}\gamma_{k}\right\\}-\min\left\\{\rho^{2}_{\max}V^{n}_{2,j},\mu\sum_{k=-j-1}^{N_{\eta}-1}\gamma_{k}\right\\}$
$\displaystyle=\frac{\mu\sum_{k=-j}^{N_{\eta}-1}\gamma_{k}-\mu\sum_{k=-j-1}^{N_{\eta}-1}\gamma_{k}-\left|\rho^{2}_{\max}V^{n}_{2,j}-\mu\sum_{k=-j}^{N_{\eta}-1}\gamma_{k}\right|+\left|\rho^{2}_{\max}V^{n}_{2,j}-\mu\sum_{k=-j-1}^{N_{\eta}-1}\gamma_{k}\right|}{2}$
$\displaystyle\leq\frac{\mu\sum_{k=-j}^{N_{\eta}-1}\gamma_{k}-\mu\sum_{k=-j-1}^{N_{\eta}-1}\gamma_{k}+\left|\mu\sum_{k=-j}^{N_{\eta}-1}\gamma_{k}-\mu\sum_{k=-j-1}^{N_{\eta}-1}\gamma_{k}\right|}{2}=0.$

Then, we get

(3.5) $\displaystyle\min\\{\rho^{1}_{\max}V^{n}_{2,j-1},s^{n}_{B,j-1}\\}$
$\displaystyle-\min\\{\rho^{n}_{1,j}V^{n}_{2,j},s^{n}_{B,j}\\}$
$\displaystyle\leq\min\\{\rho^{1}_{\max}V^{n}_{2,j},s^{n}_{B,j}\\}-\min\\{\rho^{n}_{1,j}V^{n}_{2,j},s^{n}_{B,j}\\}$
$\displaystyle\leq\left(\rho^{1}_{\max}-\rho^{n}_{1,j}\right)V^{n}_{2,j}.$

Adding the two inequalities (3.4)-(3.5), due to the CFL condition (3.3), we
have $\rho^{n+1}_{1,j}\leq\rho^{1}_{\max}.$
For $j\geq 0,$ we can write the following bound

$\rho^{2}_{\max}V^{n}_{2,j-1}-\rho^{n}_{2,j}V^{n}_{2,j}\leq(\gamma_{0}\left\|v^{\prime}\right\|\left\|\rho\right\|+\left\|v\right\|)(\rho^{2}_{\max}-\rho^{n}_{j}),$

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

$\min\\{d_{B}^{n},\rho^{2}_{\max}V^{n}_{2,-1}\\}\leq\rho^{2}_{\max}V^{n}_{2,-1}.$

Then, we get

$\rho^{n+1}_{2,j}\leq\rho^{n}_{2,j}+\frac{\Delta t}{\Delta
x}\left(\rho^{2}_{\max}V^{n}_{2,j-1}-\rho^{n}_{2,j}V^{n}_{2,j}\right)\leq\rho^{2}_{\max}.$

This concludes the proof. ∎

######  Remark 3.1.

We point out that the maximum principle in Proposition 3.1 is uniform with
respect to $\eta$ and this allows us to say that if they exist, the limit
solutions as $\eta\to 0$ and $\eta\to+\infty$ satisfy the maximum principle,
too.

In the following we are in particularly interested in the model hierarchies
induced by the non-local range $\eta$. Therefore, we will investigate the
models for $\eta\to 0$ and $\eta\to\infty$ 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 $\eta>0$ related to the support of the kernel is fixed and the re-
scaled kernel function is considered

$\omega_{\eta}(x)=\frac{1}{\eta}\omega\left(\frac{x}{\eta}\right).$

Owing to the assumption $\int_{\mathbb{R}}\omega_{\eta}(x)dx=1,$ when $\eta\to
0^{+}$ the family $\omega_{\eta}$ converges weakly∗ in the sense of measures
to the Dirac delta and we formally obtain the corresponding local conservation
law:

$\textbf{Non-local:
}\begin{cases}\partial_{t}\rho_{\eta}+\partial_{x}(\rho_{\eta}(v(\rho_{\eta})\ast\omega_{\eta}))=0,\\\
\rho_{\eta}(0,x)=\bar{\rho}(x),\end{cases}\to\textbf{ Local:
}\begin{cases}\partial_{t}\rho+\partial_{x}(\rho v(\rho))=0,\\\
\rho(0,x)=\bar{\rho}(x),\end{cases}$

with $v:\mathbb{R}\to\mathbb{R}$ being a Lipschitz continuous vector-valued
function. The above derivation is just formal and has to be rigorously
justified. The question is whether the solution $\rho_{\eta}$ of 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. $v(\rho_{\eta}\ast\omega_{\eta}),$
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 datum, 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.

###  4.1. A local buffer model

A local buffer model which strongly inspired the non-local model
(LABEL:eq:model) 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)
$\displaystyle\partial_{t}\rho_{1}(t,x)+\partial_{x}\left(\rho_{1}(t,x)v_{1}(\rho_{1}(t,x))\right)=0,$
$\displaystyle\partial_{t}\rho_{2}(t,x)+\partial_{x}(\rho_{2}(t,x)v_{2}(\rho_{2}(t,x)))=0.$

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

$\displaystyle q_{1}(t)=\min\\{s_{B}(t),D_{1}(\rho_{1}(t,0^{-}))\\},\qquad
q_{2}(t)=\min\\{d_{B}(t),S_{2}(\rho_{2}(t,0^{+})\\}$ $\displaystyle
s_{B}(t)=\begin{cases}\mu,&\text{if }r(t)\in[0,r_{\max}),\\\
\min\\{S_{2}(\rho_{2}(t,0^{+}),\mu\\},&\text{if }r(t)=r_{\max},\end{cases}$
$\displaystyle d_{B}(t)=\begin{cases}\mu,&\text{if }r(t)\in(0,r_{\max}],\\\
\min\\{D_{1}(\rho_{1}(t,0^{-})),\mu\\},&\text{if }r(t)=0.\end{cases}$

Here, $S_{i}$ and $D_{i}$ with $i\in\\{1,2\\}$ are the usual supply and demand
functions [32] defined for a flux $f_{i}$ with a single maximum at $\sigma$ by

$\displaystyle D_{i}(\rho)=\begin{cases}f_{i}(\rho),&\rho\leq\sigma,\\\
f_{i}(\sigma),&\rho>\sigma,\end{cases}\qquad
S_{i}(\rho)=\begin{cases}f_{i}(\sigma),&\rho\leq\sigma,\\\
f_{i}(\rho),&\rho>\sigma.\end{cases}$

Note that, here $f_{i}(\rho):=\rho v_{i}(\rho)$ holds. Finally, the
development of the buffer is described by

$r^{\prime}(t)=q_{1}(t)-q_{2}(t).$

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 (LABEL:eq:model), 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 $x$ and the non-local velocity are not
directly coupled, the flow is not allowed to exceed
$\rho_{\max}^{2}V_{2}(t,0^{+})$ in contrast to $S(\rho_{2}(0^{+},t))$ at the
junction point. Moreover, extending this fact to the whole interval
$[-\eta,0)$ with similar ideas as in [10, 23], we end up with the model
(LABEL:eq:model).
Even though the local model (LABEL:eq:model0herty) strongly inspired the non-
local one, it is not a priori clear which local solution will be obtained as
$\eta\to 0$. For example, the 1-to-1 model (LABEL:eq:1to1NObuffer) 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 (LABEL:eq:model), it becomes apparent that the
solutions might be different to the solution of the local model
(LABEL:eq:model0herty). This is induced by the different definition of the
supply and demand functions mentioned above. Hence, when supply or demand in
the model (LABEL:eq:model0herty) are active differences can occur, since, in
general,

$\displaystyle
D_{1}(\rho_{1}(t,0^{-}))\neq\rho_{1}(t,0^{-})v_{2}(\rho_{2}(t,0^{+})=\lim_{\eta\to
0}\rho_{1}(t,0^{-})V_{2}(t,0^{+}),$ $\displaystyle
S_{2}(\rho_{2}(t,0^{+}))\leq\rho^{2}_{\max}v_{2}(\rho_{2}(t,0^{+}))=\lim_{\eta\to
0}\rho_{\max}^{2}V_{2}(t,0^{+}).$

The fact that the model (LABEL:eq:model), in general, does not converge to
solutions of (LABEL:eq:model0herty) is also shown by the following example:

######  Example 4.1.

Let us consider constant initial data $\rho_{1}$ and $\rho_{2}$ on each road
with the same flux function on both roads. Further, let
$S(\rho_{2})<D(\rho_{1})$ and $S(\rho_{2})<\mu$ hold. In this case, we can
exclude that the solution of (LABEL:eq:model) converges to the solution of the
local model (LABEL:eq:model0herty). Therefore, we show that the buffer is
different for the model (LABEL:eq:model0herty) and (LABEL:eq:model). We start
with the local case. Here, we have

$q_{1}(t)=\min\\{\mu,D(\rho_{1})\\},\quad q_{2}(t)=\min\\{S(\rho_{2}),\mu\\}.$

Moreover, due to $S(\rho_{2})<D(\rho_{1})<\mu,$

$r^{\prime}(t)=q_{1}(t)-q_{2}(t)>0,$

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

$r^{\prime}(t)=\min\\{\mu,\rho_{1}(t,0^{-})V_{2}(t,0)\\}-\min\\{\mu,\rho_{1}(t,0^{-})V_{2}(t,0),\rho^{2}_{\max}V_{2}(t,0)\\}.$

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

$\rho_{1}(t,x)\leq\rho^{1}_{\max}=\rho^{2}_{\max},$

and hence,

$r^{\prime}(t)=\min\\{\mu,\rho_{1}(t,0^{-})V_{2}(t,0^{-})\\}-\min\\{\mu,\rho_{1}(t,0^{-})V_{2}(t,0)\\}=0.$

Therefore, the buffer does not increase for every $\eta>0$, this also holds in
the limit $\eta\to 0.$ 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 (LABEL:eq:model0herty) 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 (LABEL:eq:model0herty) seems not to be the correct limit
model of (LABEL:eq:model), let us formally derive a potential candidate for
the limit model of (LABEL:eq:model). The limit of $\omega_{\eta}$ to the Dirac
delta needs to be considered in the non-local velocities and the supply
function. Hence, the model (LABEL:eq:model) formally converges to

(4.2)
$\displaystyle\partial_{t}\rho_{1}(t,x)+\partial_{x}\left(\rho_{1}(t,x)v_{1}(\rho_{1}(t,x))\right)=0,$
$\displaystyle\partial_{t}\rho_{2}(t,x)+\partial_{x}(\rho_{2}(t,x)v_{2}(\rho_{2}(t,x)))=0,$
$\displaystyle
r^{\prime}(t)=\min\\{s_{B}(t,0^{-}),\,\rho_{1}(t,0^{-})v_{2}(\rho_{2}(t,0^{+})\\}-\min\\{d_{B}(t),\,\rho^{2}_{\max}v_{2}(\rho_{2}(t,0^{+}))\\},$

where

$\displaystyle s_{B}(t,x)=\begin{cases}\mu,\quad&r(t)\in[0,r_{\max}),\\\
\min\\{\rho^{2}_{\max}v_{2}(\rho_{2}(t,x)),\,\mu\\},&r(t)=r_{\max},\end{cases}$
$\displaystyle d_{B}(t)=\begin{cases}\mu,\quad&r(t)\in(0,r_{\max}],\\\
\min\\{\rho_{1}(t,0^{-})v_{2}(\rho_{2}(t,0+)),\,\mu\\},&r(t)=0.\end{cases}$

Comparing the dynamics to (LABEL:eq:model0herty) 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

$\displaystyle
D_{1}(\rho_{1}(t,0^{-}))\neq\rho_{1}(t,0^{-})v_{2}(\rho_{2}(t,0^{+})\quad\text{and}\quad
S_{2}(\rho_{2}(t,0^{+}))\leq\rho^{2}_{\max}v_{2}(\rho_{2}(t,0^{+}))$

holds.
Note that we will note prove the well-posedness of (LABEL:eq:model0) or the
limit from (LABEL:eq:model) to (LABEL:eq:model0) rigorously. Nevertheless, we
investigate the limit and model (LABEL:eq:model0) numerically in the
following. In particular, a numerical convergence to model (LABEL:eq:model0)
can be obtained.

###  4.3. Numerical examples

During this section the space step size for all examples is chosen as $\Delta
x=10^{-3}$ and the time step is chosen according to the CFL condition (3.3).
We deal with the following example: $v_{1}(\rho)=1-\rho$,
$v_{2}(\rho)=1-5\rho/3$, constant initial data of $\rho_{1,0}\equiv 0.75$ and
$\rho_{2,0}\equiv 0.5$ with $r_{\max}=\infty$ and $r_{0}=0$. If not stated
otherwise, we use $\mu=0.15$. 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., $r_{\max}<\infty$. 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 (LABEL:eq:model) for different kernels in the situation
described above for $\eta=0.5$. Here, we consider a constant kernel
$\omega_{\eta}(x)=1/\eta$, a linear decreasing kernel, i.e.
$\omega_{\eta}(x)=2(\eta-x)/\eta^{2}$ and a quadratic decreasing kernel, i.e.
$\omega_{\eta}(x)=3(\eta^{2}-x^{2})/(2\eta^{3})$. 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 $\mu$ of 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 $x\approx-0.5$ which results in a steeper increase. However, the
constant kernel behaves differently. There are two effects visible: First,
there are additional increases depending on $\eta=0.5$. This effect is also
present without a buffer model, i.e. in the model (LABEL:eq:1to1NObuffer),
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 ($x\approx-0.25$) 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.

$-1.4$$-1.2$$-1$$-0.8$$-0.6$$-0.4$$-0.2$$0$$0.75$$0.8$$0.85$$0.9$$x$$\rho_{1}$constantlinearquadratic(LABEL:eq:1to1NObuffer)$0$$0.2$$0.4$$0.5$$0.55$$0.6$$x$$\rho_{2}$$0$$0.2$$0.4$$0.6$$0.8$$1$$0$$0.5$$1$$\cdot
10^{-2}$$t$$r$ Figure 1. Approximate solution of the non-local buffer model
with non-local range of $\eta=0.5$ with different kernels and the approximate
solution of (LABEL:eq:1to1NObuffer) 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.

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 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 $\eta\to 0$. For the local
models (LABEL:eq:model0herty) and (LABEL:eq:model0) 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
$\mu$. Hence, for a decreasing sequence of $\eta$, 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. $\mu=0.05$, gives the same dynamics
for all models.

$-2$$-1.8$$-1.6$$-1.4$$-1.2$$-1$$-0.8$$-0.6$$-0.4$$-0.2$$0$$0.75$$0.8$$0.85$$0.9$$x$$\rho_{1}$(LABEL:eq:model0herty)(LABEL:eq:model0)(LABEL:eq:model)
$\eta=0.5$(LABEL:eq:model) $\eta=0.05$(LABEL:eq:model)
$\eta=0.005$$0$$0.2$$0.4$$0.5$$0.55$$0.6$$x$$\rho_{2}$$0$$0.2$$0.4$$0.6$$0.8$$1$$0$$0.5$$1$$1.5$$2$$\cdot
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 (LABEL:eq:model0herty) and (LABEL:eq:model0) at $T=1$ and a capacity of
$\mu=0.1$ at $T=1$: approximate solutions of road 1, left, of road 2, top
right and the evolution of the buffer, bottom right.

In contrast, increasing $\mu$ demonstrates the possible convergence to
(LABEL:eq:model0) instead of (LABEL:eq:model0herty). In Figure 3, the same
example as before, but with $\mu=0.15$ is considered. In the local model
(LABEL:eq:model0herty) the outflow of the first road is higher than in the
local model (LABEL:eq:model0). This results in a lower density, but fuller
buffer. The decreasing sequence of $\eta$ shows that (LABEL:eq:model0) might
be the limit of the non-local buffer model (LABEL:eq:model).

$-2$$-1.8$$-1.6$$-1.4$$-1.2$$-1$$-0.8$$-0.6$$-0.4$$-0.2$$0$$0.75$$0.8$$0.85$$0.9$$x$$\rho_{1}$(LABEL:eq:model0herty)(LABEL:eq:model0)(LABEL:eq:model)
$\eta=0.5$(LABEL:eq:model) $\eta=0.05$(LABEL:eq:model)
$\eta=0.005$$0$$0.2$$0.4$$0.5$$0.55$$0.6$$x$$\rho_{2}$$0$$0.2$$0.4$$0.6$$0.8$$1$$0$$2$$4$$6$$\cdot
10^{-2}$$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 (LABEL:eq:model0herty) and (LABEL:eq:model0) at $T=1$ and a capacity of
$\mu=0.15$ at $T=1$: approximate solutions of road 1, left, of road 2, top
right and the evolution of the buffer, bottom right.

######  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 (LABEL:eq:model0herty) which also maximizes the flux through the
junction for every case, in particular, if the buffer is not active.

##  5\. An infinite non-local reach

Besides the limit $\eta\to 0$ also the case $\eta\to\infty$ 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 $V_{\max}$. The limit for non-local
traffic networks without buffers has 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)
$\displaystyle\partial_{t}\rho_{1}+\partial_{x}\min\\{\rho_{1}v,\mu_{1}\\}=0$
$\displaystyle\partial_{t}\rho_{2}+\partial_{x}\min\\{\rho_{1}v,\mu_{2}\\}=0$
$\displaystyle r^{\prime}(t)=\min\\{\rho_{1}(t,0^{-})v,\mu_{1}\\}-d_{B}(t)$
$\displaystyle
d_{B}(t)=\begin{cases}\min\\{\rho_{1}(t,0^{-})v,\mu_{1},\mu_{2}\\},\quad&\text{if
}r(t)=0,\\\ \mu_{2},\quad&\text{if }r(t)>0.\end{cases}$

Note that we consider an unlimited buffer for simplicity. Further, the buffer
have different capacities, but the same processing velocity. In the case
$\mu_{1}=\mu_{2}$ and $r_{0}=0$ we can view the problem as a single processor
as the buffer will never fill up. For the initial data

(5.2) $\rho_{0}(x)=\begin{cases}\rho_{1},\quad&x<0,\\\
\rho_{2},&x>0,\end{cases}$

The solution is

  * •

a linear transport with velocity $v$, if $\rho_{i}<\mu_{i}/v$ $i\in\\{1,2\\}$,
or

  * •

a rarefaction wave with intermediate state $\mu_{2}/v$, if
$\rho_{1}>\mu_{1}/v$ and $\rho_{2}<\mu_{2}/v$.

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 $\mu_{1}=\mu_{2}$ and $r_{0}=0$, is
the limit of the 1-to-1 model (LABEL:eq:1to1NObuffer) without a buffer.

Now, we consider the more general case, $\mu_{1}\neq\mu_{2}$. For simplicity,
the buffer is initially empty and we consider only the case
$\rho_{i}<\mu_{i}/v$ $i\in\\{1,2\\}$, 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 $\mu_{2}$. In fact, we have

  * •

for $\rho_{1}\leq\mu_{2}/v$ a linear transport and

  * •

for $\rho_{1}>\mu_{2}/v$ a rarefaction wave with intermediate state
$\mu_{2}/v$

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

###  5.2. Limit models

Now, we derive the limit models of (LABEL:eq:model). Therefore, we assume that
the initial data $\rho_{1,0}$ and $\rho_{2,0}$ are of compact support. Note
that due to the finite speed of propagation also the solutions
$\rho_{1}(t,\cdot)$ and $\rho_{2}(t,\cdot)$ are compactly supported. We can
use the following result proven in [23, Lemma 4.1]: in the $L^{1}_{loc}$ limit
the non-local velocities converge to

(5.3) $\displaystyle V_{1}(t,x)\to 0\qquad\text{and}\qquad V_{2}(t,x)\to
v_{2}(0)\qquad\text{for }\eta\to\infty.$

Using similar arguments we can deduce

(5.4) $\displaystyle
s_{B}(t,x)\to\begin{cases}\mu\quad&r(t)\in[0,r_{\max}),\\\
\min\\{\rho_{\max}^{2}v_{2}(0),\mu\\}\quad&r(t)=r_{\max},\end{cases}\qquad\text{and}$
(5.5) $\displaystyle d_{B}(t)\to\begin{cases}\mu\quad&r(t)\in(0,r_{\max}],\\\
\min\\{\rho_{1}(t,0^{-})v_{2}(0),\mu\\}\quad&r(t)=0,\end{cases}\qquad\text{for
}\eta\to\infty.$

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 $\mu$ and of the second road by
$\rho_{\max}^{2}v_{2}(0)$.

#### Case 1: $\mu\leq\rho_{\max}^{2}v_{2}(0)$

In this case the limit equations are given by

$\displaystyle\partial_{t}\rho_{1}(t,x)+\partial_{x}\min\\{\rho_{1}(t,x)v_{2}(0),\mu\\}=0$
$\displaystyle\partial_{t}\rho_{2}(t,x)+\partial_{x}\rho_{2}(t,x)v_{2}(0)=0,$

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. $\mu_{2}\geq\rho_{\max}^{2}v_{2}(0)$, the
dynamics are described by

$\displaystyle\partial_{t}\rho_{2}(t,x)+\partial_{x}\min\\{\rho_{2}(t,x)v_{2}(0),\mu_{2}\\}=0.$

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. $\mu\leq\mu_{2}$. 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 $t^{*}\geq 0$ holds, the buffer stays empty
for $t\geq t^{*}$. 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.

(5.6) $\displaystyle\rho_{1,0}(x)=\begin{cases}\rho_{1},\qquad&\text{if
}x\in[a,b],\\\ 0,&\text{ else}\end{cases}\qquad\rho_{2,0}(x)=0,\qquad r(0)=0,\
r_{\max}=\infty.$

for $a<b<0$. The buffer model has a capacity of $\mu$ on 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 $\rho_{1}$:

  1. (1)

$\mu/v_{2}(0)>\rho_{1}$: Here, we have a linear transport.

  2. (2)

$\rho_{1}>\mu/v_{2}(0)$: The solution of the buffer model is a rarefaction
wave with constant state $\mu/v_{2}(0)$ at $b$ and a shock at $a$, which moves
onto the second road. Both the shock at $a$ and the rarefaction wave move at
the speed $\mu/\rho_{1}<v_{2}(0)$ on the first road. On the second road the
speed increases to $v_{2}(0)$.

####  Case 2 $\mu>\rho_{\max}^{2}v_{2}(0)$

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

$\displaystyle\begin{cases}\partial_{t}\rho_{1}+\partial_{x}\min\\{\rho_{1}v_{2}(0),\mu\\}=0,\quad&\text{
if }r(t)\in[0,r_{\max})\\\
\partial_{t}\rho_{1}+\partial_{x}\min\\{\rho_{1}v_{2}(0),\rho_{\max}^{2}v_{2}(0)\\}=0,\quad&\text{
if }r(t)=r_{\max}.\end{cases}$

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

$\displaystyle\partial_{t}\rho_{2}+\partial_{x}\rho_{2}v_{2}(0)=0.$

More importantly, the inflow onto road 2 is given by

$\displaystyle\begin{cases}\rho_{\max}^{2}v_{2}(0),\quad&\text{ if
}r(t)\in(0,r_{\max}],\\\
\min\\{\rho_{1}(t,0-)v_{2}(0),\rho_{\max}^{2}v_{2}(0)\\},\quad&\text{ if
}r(t)=0.\end{cases}$

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

$\displaystyle
r^{\prime}(t)=\begin{cases}\min\\{\mu,\rho_{1}(t,0-)v_{2}(0)\\}-\min\\{\rho_{1}(t,0-)v_{2}(0),\rho_{\max}^{2}v_{2}(0)\\},\quad&\text{if
}r(t)=0,\\\
\min\\{\mu,\rho_{1}(t,0-)v_{2}(0)\\}-\rho_{\max}^{2}v_{2}(0),\quad&\text{if
}r(t)\in(0,r_{\max}),\\\
\min\\{\rho_{\max}^{2}v_{2}(0),\rho_{1}(t,0-)v_{2}(0)\\}-\rho_{\max}^{2}v_{2}(0),\quad&\text{if
}r(t)=r_{\max}.\end{cases}$

As can be seen, the dynamics are more involved and strongly dependent on the
buffer size. Nevertheless, for $r(t)\in[0,r_{\max})$ the limit equations can
be viewed as the supply chain model (5.1) by assigning the artificial capacity
$\mu_{2}=\rho_{\max}^{2}v_{2}(0)$ to the second road. In particular, the
capacity of the buffer is given by $\rho_{\max}^{2}v_{2}(0)$ and is equal to
the capacity of the next processor as in (5.1). Note that the dynamics change
when $r(t)=r_{\max}$. 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. (1)

$\mu/v_{2}(0)>\rho_{\max}^{2}>\rho_{1}$: Here, the density is low enough not
to affect the buffer, so the solution is simply the linear transport.

  2. (2)

$\mu/v_{2}(0)>\rho_{1}>\rho_{\max}^{2}$: The solution is first given by a
linear transport equation. When the density reaches the buffer at
$t^{*}=-b/v_{2}(0)$, the buffer starts to fill at the rate:

$r^{\prime}(t)=v_{2}(0)(\rho_{1}-\rho_{\max}^{2}).$

The inflow on the second road is then $\rho_{\max}^{2}v_{2}(0)$. This results
in a density of $\rho_{\max}^{2}$.

  3. (3)

$\rho_{1}>\mu/v_{2}(0)>\rho_{\max}^{2}$: Instead of a linear transport, a
rarefaction wave with intermediate state $\mu/v_{2}(0)$ and a shock is
generated. Again, after some time (i.e. when the rarefaction wave reaches
$x=0$), the buffer begins to fill with

$r^{\prime}(t)=\mu-v_{2}(0)\rho_{\max}^{2}.$

On road 2 the density is $\rho_{\max}^{2}$ as before.

###  5.3. Numerical examples

In this subsection we consider some numerical results for the different cases
discussed before. For the numerical tests we use $\Delta x=0.01$.

We start with an explicit example for the initial condition (5.6) and the
parameters chosen such that $\rho_{1}>\mu/v_{2}(0)>\rho_{\max}^{2}$ holds.
More precisely, we choose the initial conditions

$\displaystyle\rho_{1,0}(x)=\begin{cases}1,\qquad&\text{if }x\in[-5,-1/3],\\\
0,&\text{ else}\end{cases}\qquad\rho_{2,0}(x)=0,\qquad r(0)=0,\
r_{\max}=\infty,$

$\mu=0.75$, $v_{1}(\rho)=1-\rho$, $v_{2}(\rho)=1-2\rho$ and hence
$\rho_{\max}^{2}=0.5$. The explicit solution for the limit model for
$\eta\to\infty$ is for $t<1/3$ given by:

$\displaystyle\rho_{1}(t,x)=\begin{cases}0.75,\qquad&\text{if
}x\in[-1/3,-1/3+t],\\\ 1,\qquad&\text{if }x\in[-5+0.75t,-1/3],\\\ 0,&\text{
else}\end{cases}\qquad\rho_{2,0}(x)=0,\qquad r(t)=0,$

and for $t\in[1/3,56/9)$ by

$\displaystyle\rho_{1}(t,x)$ $\displaystyle=\begin{cases}0.75,\qquad&\text{if
}x\in[-1/3,0],\\\ 1,\qquad&\text{if }x\in[-5+0.75t,-1/3],\\\ 0,&\text{
else}\end{cases}\qquad\rho_{2,0}(x)=\begin{cases}0.5,\qquad&\text{if
}x\in[0,t-1/3],\\\ 0,&\text{ else}\end{cases},$ $\displaystyle r(t)$
$\displaystyle=0.25(t-1/3).$

We compare the exact limit solution with a numerical approximation of the non-
local buffer model computed with $\eta=2$ for a constant and linear decreasing
kernel. Figure 4 shows also the possible limit model for $\eta\to 0$, i.e.,
(LABEL:eq:model0). It can be seen that for the different models induced by
$\eta$ more mass seems to be transported on the first road, into the buffer
and trough the junction with increasing $\eta$. In the following, we again
restrict ourselves to a linear decreasing kernel.

$-5$$-4$$-3$$-2$$-1$$0$$0$$0.2$$0.4$$0.6$$0.8$$1$$x$$\rho_{1}$(LABEL:eq:model0)constantlinear$\eta\rightarrow\infty$$0$$1$$2$$3$$0$$0.2$$0.4$$x$$\rho_{2}$$0$$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 $\eta=2$ 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.

The limit for $\eta\to\infty$ 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 $\eta\to\infty$ requires rather large values of $\eta$.
Further, with increasing $\eta$ more mass is transported on the first and
second road.

$-4$$-3$$-2$$-1$$0$$0$$0.2$$0.4$$0.6$$0.8$$1$$x$$\rho_{1}$$\eta=10$$\eta=25$$\eta=75$$\eta=300$$\eta\rightarrow\infty$$0$$1$$2$$3$$0$$0.2$$0.4$$x$$\rho_{2}$$0$$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 $\eta$ at $T=3$: approximate solutions of road 1, left, of road 2,
top right and the evolution of the buffer, bottom right.

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 $r_{\max}=0.15$ for $\eta=200$ at $T=2$.
In the latter case, the buffer increases until about $T=1$. Before $T\approx
1$, 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.

-4-3-2-10$0$$0.2$$0.4$$0.6$$0.8$$1$$x$$\rho_{1}$Initial data(LABEL:eq:model) $r_{\max}=0.15$(LABEL:eq:model) $r_{\max}=\infty$$0$$0.5$$1$$1.5$$2$$0$$0.2$$0.4$$0.6$$x$$\rho_{2}$$0$$0.5$$1$$1.5$$2$$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.

##  6\. Conclusion

We presented a non-local buffer model for a 1-to-1 junction with a suitable
numerical discretisation. We have proved that the numerical scheme preserves
the maximum principle uniform in $\eta$. Further, we investigate the model
hierarchies induced by the non-local term. For $\eta\to 0$, 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 $\eta\to\infty$. 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 extending the junction model to more complex junctions and
network structures.

## Acknowledgement

F.A.C. is member of Gruppo Nazionale per l’Analisi Matematica, la Probabilità
e le loro Applicazioni (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 Göttlich under grant GO
1920/12-1.

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