Nonlocal reaction traffic flow model with on-off ramps

1.   Introduction

1.1.   Scope

Models of conservation laws with nonlocal flux have been used to describe traffic flow dynamics in which drivers adapt their velocity with respect to what happens to the cars in front of them [3,5,9,11,18]. In this type of models, the flux function depends on a downstream convolution term between the density or the velocity of vehicles and a kernel function with support on the negative axis. However, the above models cannot be used to study the traffic flow on the highway with ramps since they did not include their presence. Indeed, ramps are an important element of traffic systems and develop some complex traffic phenomena, see [12,15,16,19,20,21,22].

In this work, we propose a new nonlocal traffic model which includes the effects of the inflow and output flow over the on- and off- ramps respectively. We start by considering a modified local reaction traffic model proposed in [16],

where the non-negative functions $S_{\mathrm{on}}$ and $S_{\mathrm{off}}$ are the source and sink term, respectively, defined by

where $q_{\mathrm{on}}\in \mathbb{R}^{+}$, and $q_{\mathrm{off}}\in \mathbb{R}^{+}$ the rate (number of vehicles per unit time per unit length) of the on- and off-ramp respectively, as in [20,21],

with $q^{\mathrm{ramp}}_{\mathrm{on}}(t)$ the expected inflow flow of the on-ramp and $q^{\mathrm{ramp}}_{\mathrm{off}}(t)$ the expected output flow of the off-ramp, $L_\mathrm{on}$ and $L_\mathrm{off}$ are the lengths of the on- and off-ramps respectively, whose spatial positions are described by the indicator functions $\mathbf{1}_{\mathrm{on}}(x)$ and $\mathbf{1}_{\mathrm{off}}(x),$ defined as

For simplicity we consider $L_\mathrm{on} = L_\mathrm{off} = L$ in the whole paper.

In order to obtain a non-local version of the model (1.1), we first rewrite the flux function $f(\rho) = \rho v(\rho)$ in its non-local version, where drivers react adapting their velocity with respect to what happens in front of them, see [1,3,5,11],

On the on-ramp the idea is that at position $x$ the flow merging in the traffic way is inversely proportional to the average density around position $x+\delta$, see Fig. 1, i.e, we write

where

with $\eta\in[0, 1]$ and $\delta\in[-\eta, \eta]$. {Similarly to [5], here the parameter $\eta$ represents the radius of the support of the kernel function $\omega_{\eta, \delta}$, while $\delta$ is the point at which the maximum is attained. This choice of the kernel models the fact that drivers on the on-ramp can see what happens on the backward and forward on the main road.} However, in the numerical test section we will see that the choice of the non-local term (1.4) does not guarantee that the proposed model satisfies a Maximum Principle, see Example 3. In order to overcome this difficulty, we consider a first variant of (1.4) taking

Note that this term contains a product which differentiates it from the original model, this choice is also assumed in the multilane model studied in [8]. An alternative to avoid the double product in the previous equation (1.5) is the following

{In both models with (1.5) and (1.6), if the main road is crowded only few vehicles can enter to the main road.}

The purpose of this work is the study of the well-posedness of a nonlocal reaction traffic flow model with source term given by (1.5) and (1.6).

1.2.   Related work

In [2,3,4,5,6,9,11] the authors studied a nonlocal conservation law to model vehicular traffic flow in the case $S_{\mathrm{on}} = S_{\mathrm{off}} = 0$, i.e., without on- and off-ramps. The need to design more realistic models has led to the development of multi-lane vehicular traffic models among which we can highlight the following. In [14] it is introduced a new local model for multilane dense vehicular traffic by means of a system of a weakly coupled scalar conservation laws. In [10] the authors consider the model proposed in [14] but with a more general source term and they allow for the presence of space discontinuities both in the speed law and in the number of lanes; in these two local models the source term accounts for the lane change rate and the key assumption is that the drivers prefer to drive faster, and that the tendency of a vehicle to change lanes is proportional to the difference in velocity between neighboring lanes. In [8] a multilane model with local and non-local flux combined with a source term that also incorporates a nonlocality is studied; here, the non-local source term describes the lane changing rate depending on a (nonlinear) evaluation of the velocity. In particular, the lane changing rate is proportional to the difference in the velocity between two adjacent lanes, but the velocities are evaluated in a neighbourhood of the current position, moreover, this rate is proportional also to the density in the receiving lane, meaning that if that lane is crowded only a few vehicles can actually change lane.

Regarding to vehicular traffic flow models taking into account the presence of ramps, we can mention [16], where the authors study the (local) first order nonlinear conservation law (1.1). A (local) second order model is proposed in [21] to study the effects of on- and off-ramps on a main road during two rush periods. Likewise, other works about the study of effects of ramps in vehicular traffic flow models are referenced in [21]. In particular, in [7] the authors consider a Lighthill-Witham-Richards (LWR) traffic flow model on a junction composed by one mainline, an on-ramp and an off-ramp, which are connected by a node. Moreover, in [13] a non-local gas-kinetic traffic model including ramps is proposed, the model allows to simulate syncronized congested traffic and reproduces realistic phenomena of vehicular traffic by variations of the on-ramp flow. A new modeling methodology for merging and diverging flows is studied in [17], the methodology includes coupling effects between main and ramps flows and a new formulation for the modeling of traffic friction is also introduced.

1.3.   Outline of the paper

This work is organized as follows. In Section 2 we present the proposed mathematical model with all the considered assumptions on it. Afterwards, we introduce an upwind-type scheme with two different source terms and derive important properties such as maximum principle, $\mathbf{L^{1}}-$ bound and $\mathbf{BV}$ estimates. Furthermore, we derive the $\mathbf{L^{1}}-$Lipschitz continuous dependence of solutions to (2.1) on the initial data and the terms $q_{\mathrm{on}}$ and $q_{\mathrm{off}}$ in Section 3. In Section 4, we present numerical examples illustrating the behavior of the solutions of our model.

2.   Mathematical model

The main goal of this work is to study the well-posedness of the non-local reaction traffic model

where $S_{\mathrm{on}}(\cdot, \cdot, \rho, \rho*\omega_{\eta, \delta})$ defined in (1.5) or (1.6), $S_{\mathrm{off}}$ defined by (1.3) and initial condition

From now on we call Model 0 the equations (2.1)-(1.4)-(2.2), Model 1 the equations (2.1)-(1.5)-(2.2), and Model 2 (2.1)-(1.6)-(2.2). Let us assume the following assumptions:

We recall the definition of weak entropy solution for (2.1).

Definition 2.1. Let $\rho_{0}\in (\mathbf{L^{1}}\cap \mathbf{BV})( \mathbb{R};[0, {\rho_{\max}}]).$ We say that $\rho\in\mathbf{C}([0, T]; \mathbf{L^{1}}( \mathbb{R};[0, {\rho_{\max}}])),$ with $\rho(t, \cdot)\in \mathbf{BV}( \mathbb{R};[0, {\rho_{\max}}])$ for $t\in[0, T]$, is a weak solution to (2.1) with initial datum $\rho_{0}$ if for any $\varphi\in\mathbf{C_c^{{1}}}([0, T[\times \mathbb{R}; \mathbb{R})$

where $V(t, x) = v((\rho*\omega)(t, x))$ and $S_{\mathrm{on}}$ is as in (1.5) or (1.6).

Definition 2.2. Let $\rho_{0}\in (\mathbf{L^{1}}\cap \mathbf{BV})( \mathbb{R};[0, {\rho_{\max}}]).$ We say that $\rho\in\mathbf{C}([0, T]; \mathbf{L^{1}}( \mathbb{R};[0, {\rho_{\max}}])),$ with $\rho(t, \cdot)\in \mathbf{BV}( \mathbb{R};[0, {\rho_{\max}}])$ for $t\in[0, T]$, is a entropy weak solution to (2.1) with initial datum $\rho_{0}$ if for any $\varphi\in\mathbf{C_c^{{1}}}([0, T[\times \mathbb{R}; \mathbb{R})$ and for all $k\in \mathbb{R}$

Our main result is given by the following theorem, which states the well-posedness of problem (2.1) to (2.2) with source term given by (1.5) or (1.6). In order to simplify the computations we consider $\rho_{\max} = 1$ from now on.

Theorem 2.1. Let $\rho_{0}\in\left(\mathbf{L^{1}}\cap \mathbf{BV}\right)\left( \mathbb{R};[0, 1]\right)$. Assume $v\in\mathbf{C^{{2}}}\left([0, 1]; \mathbb{R}^{+}\right).$ Then, for all $T>0$, the problem $(2.1)$ has a unique solution $\rho\in\mathbf{C^{{0}}}\left([0, T];\mathbf{L^{1}}( \mathbb{R};[0, 1])\right)$ in the sense of Definition 2.2. Moreover, the following estimates hold: for any $t\in[0, T]$

where

3.   Existence of entropy solution

3.1.   Numerical discretization

We take a space step $\Delta x$ such that $\eta = N\Delta x$, for some $N\in \mathbb{N}$, and a time step $\Delta t$ subject to a CFL condition which will be specified later. For any $j\in \mathbb{Z}$, let $x_{j-1/2} = j\Delta x$ be a cells interfaces, $x_{j} = \bigg(j+\frac{1}{2}\bigg)\Delta x$ the cells centers. We consider ramps with length $L$ and take $L = \ell\Delta x, \ \text{for some} \ \ell\in \mathbb{Z}^{+}$ such that $\underline{x}_{\mathrm{on}} = x_{\underline{k}_{\mathrm{on}}+1/2}$, $\overline{x}_{\mathrm{on}} = x_{\underline{k}_{\mathrm{on}}+1/2+\ell}$, $\underline{x}_{\mathrm{off}} = x_{\underline{k}_{\mathrm{off}}+1/2}$ and $\overline{x}_{\mathrm{off}} = x_{\underline{k}_{\mathrm{off}}+1/2+\ell}$, for some $\underline{k}_{\mathrm{on}}, \underline{k}_{\mathrm{off}}\in \mathbb{Z}$. With this notation, we define the subdomains $\Omega_{\mathrm{on}} = [\underline{x}_{\mathrm{on}}, \overline{x}_{\mathrm{on}}]$, $\Omega_{\mathrm{off}} = [\underline{x}_{\mathrm{off}}, \overline{x}_{\mathrm{off}}]$, and we put $\Omega_{\mathrm{on}}^{k} = [\underline{k}_{\mathrm{on}}+1, \underline{k}_{\mathrm{on}}+\ell]$ and $\Omega_{\mathrm{off}}^{k} = [\underline{k}_{\mathrm{off}}+1, \underline{k}_{\mathrm{off}}+\ell]$. We fix $T>0$, and set $N_{T}\in \mathbb{N}$ such that $N_{T}\Delta t\leq T<\left(N_{T}+1\right)\Delta t$ and define the time mesh as $t^{n} = n\Delta t$ for $n = 0, \ldots, N_{T}$. Set $\lambda = \Delta t/\Delta x$. The initial data is approximated for $j\in \mathbb{Z},$ as follows:

We define a piecewise constant approximate solution $\rho_{\Delta}(t, x)$ to (2.1) as

The $S_{\mathrm{on}}$ terms (1.5) and (1.6) are discretized via

The $S_{\mathrm{off}}$ term is discretizated via

where we denote

The approximate solution $\rho_{\Delta}$ is obtained via an upwind-type scheme together with operator splitting to account for the reaction term, see Algorithm 3.1.

Algorithm 3.1 (Upwind scheme).

● Input: approximate solution vector $\{\rho_j^n\}_{j\in \mathbb{Z}}$ for $t = t^n$

● do $j\in \mathbb{Z}$

● enddo

● do $j\in \mathbb{Z}$

● $S_{\mathrm{on}, j}^{n+1/2} \leftarrow S_{\mathrm{on}}\left(t^{n+1/2}, x_{j}, \rho_{j}^{n+1/2}, R_{\mathrm{on}, j}^{n+1/2}\right),$ using $(3.2)$ or $(3.3)$,

● $S_{\mathrm{off}, j}^{n+1/2} \leftarrow S_{\mathrm{off}}\left(t^{n+1/2}, x_{j}, \rho_{j}^{n+1/2}\right),$ using $(3.4)$,

● enddo

● Output: approximate solution vector $\{\rho_j^{n+1}\}_{j\in \mathbb{Z}}$ for $t = t^{n+1} = t^n+\Delta t$.

The terms $R_{j+1/2}^{n}$, $R_{\mathrm{on}, j}^{n+1/2}$ for $j\in \mathbb{Z}$ and $n = 0, \ldots, N_{T}-1$ denotes the discrete convolution operators in the velocity and source term and they are defined, respectively, by

Here we denote $\gamma_{p} = \int_{x_{p-1/2}}^{x_{p+1/2}}\omega_{\eta}(y-x) \mathrm{d} y$, for $p\in[0, \lfloor \eta/\Delta x \rfloor-1]$ and $\hat{\gamma}_{h} = \int_{x_{h-1/2}}^{x_{h+1/2}}\omega_{\eta, \delta}(y-x) \mathrm{d} y$, for $h\in[\lfloor (\delta-\eta)/\Delta x \rfloor, \lfloor (\delta+\eta)/\Delta x \rfloor-1]$.

Remark 3.1. If $0\leq\rho_{j}^{n+1/2}\leq 1$ for all $j\in \mathbb{Z}$, then for all $n\in\{0, \ldots, N_{T}-1\}$,

$\left\|R_{\mathrm{on}}^{n+1/2}\right\|_{\mathbf{L^{\infty}}(\Omega_{\mathrm{on}}^{k})}\leq1$. Indeed, we have that

Remark 3.2. The discrete convolution operator $R_{\mathrm{on}, j}^{n+1/2}$ satisfies

The proof of this property can be seen in [8] Lemma 3.2.

3.2.   Existence of solution Model 1

In order to prove the existence of solution of model (2.1)-(1.5), in the next lemmas we will show some properties of the approximate solutions constructed by the Algorithm 3.1.

Lemma 3.1 (Maximum principle). Let $\rho_{0}\in\mathbf{L^{\infty}}( \mathbb{R};[0, 1])$. Let hypotheses $({{\text{H}}_{\text{1}}})$ and the following Courant-Friedrichs-Levy (CFL) condition hold

with $\mathcal{Q}_T$ defined in $(2.4)$ then for all $t>0$ and $x\in \mathbb{R}$ the piece-wise constant approximate solution $\rho_{\Delta}$ constructed through Algorithm 3.1 is such that

Proof. The proof is made by induction. Let us assume that $0\leq\rho_{j}^{n}\leq1$ for all $j\in \mathbb{Z}$. Consider the convective step (3.5) of Algorithm 3.1, by CFL condition (3.7) we have $0\leq\rho_{j}^{n+1/2}\leq 1$ for $j\in \mathbb{Z}$ (see Theorem 3.3 of [9]).

Now focus on the remaining step, involving the source term.

Because of CFL condition (3.7), the last right-hand side is a convex combination of $\rho_{j}^{n+1/2}$ and one. Then $\rho_{j}^{n+1}\in\left[\rho_{j}^{n+1/2}, 1\right]$ and since $\rho_{j}^{n+1/2}\in[0, 1]$, we therefore conclude that $0\leq\rho_{j}^{n+1}\leq1$, for $j\in \mathbb{Z}.$

Lemma 3.2 ($\mathbf{L^{1}}-Bound$). Let $\rho_{0}\in\mathbf{L^{1}}( \mathbb{R}, [0, 1])$. Let $({{\rm{H}}_{\rm{1}}})$ and the CFL condition (3.7) hold. Then, the piece-wise constant approximate solution $\rho_{\Delta}$ constructed through Algorithm 3.1 satisfies, for all $T>0$,

with

Proof. For the conservative form of the scheme (3.5), it is satisfied

Now, we going to work $\mathbf{L^{1}}$ norm for relaxation step (3.6). By Remark 3.1 and CFL condition (3.7) we have

multiplying this inequality by $\Delta x$ and summing over all $j\in \mathbb{Z}$ we obtain

Thus, by a standard iterative procedure we can deduce

3.3.   BV estimates

We first prove the Lipschitz continuity of the source terms (3.2) in its second, third and fourth argument and of (3.4) in its second and third argument.

Lemma 3.3. The map $S_{\mathrm{on}}$ defined in $(3.2)$ is Lipschitz continuous in second, third and fourth argument with Lipschitz constant $\left\|q_{\mathrm{on}}\right\|_{\mathbf{L^{\infty}}([0, T])}$, and the map $S_{\mathrm{off}}$ defined in $(3.4)$ is Lipschitz continuous in second and third argument with Lipschitz constant $\left\|q_{\mathrm{off}}\right\|_{\mathbf{L^{\infty}}([0, T])}$.

Proof. Let us start with term (3.2). We denote

then

Now, we prove the Lipschitz continuity of $S_{\mathrm{off}}$ term (3.4). Denoting

$\mathcal{S}_{\mathrm{off}} = S_{\mathrm{off}}(t, x, \rho)-S_{\mathrm{off}}(t, \tilde{x}, q_{\mathrm{off}}, \tilde{\rho}),$ we get

Thus, we have completed the proof.

The Lipschitz continuity of the source term proved in Lemma 3.3 is one of the key ingredients in order to prove the following total variation bound on the numerical approximation.

Proposition 3.1 ($\mathbf{BV}$ estimate in space). Let $\rho_{0}\in\left(\mathbf{L^{1}}\cap \mathbf{BV}\right)\left( \mathbb{R};[0, 1]\right).$ Assume that the hypotheses $({{\rm{H}}_{\rm{1}}})$ and CFL condition $(3.7)$ hold. Then, for $n = 0, \ldots, N_{T}-1$ the following estimate holds

with $\mathcal{Q}_{T}$ like in $(2.4)$ and $\mathcal{H}$ like in $(2.5)$.

Proof. Let us compute

By the Lipschitz continuity of the source term proved in Lemma 3.3 and the property of the discrete convolution operator given in Remark 3.2, we get

Now, for convective part (3.5) we follow [9] and get

with $\mathcal{L} = \left(\|v\|_{\mathbf{L^{\infty}}([0, 1])}+\|v'\|_{\mathbf{L^{\infty}}([0, 1])}\right)$. Plugging the inequality above in (3.10) we obtain

which applied recursively yields

with $\mathcal{H} = 2\left\|q_{\mathrm{on}}\right\|_{\mathbf{L^{\infty}}([0, T])}+\left\|q_{\mathrm{off}}\right\|_{\mathbf{L^{\infty}}([0, T])}+\omega_{\eta}(0)\mathcal{L}$.

Proposition 3.2 ($\mathbf{BV}$ estimate in space and time). Let hypotheses $({{\rm{H}}_{\rm{1}}})$ hold, $\rho_{0}\in\left(\mathbf{L^{1}}\cap \mathbf{BV}\right)\left( \mathbb{R};[0, 1]\right)$. If the CFL condition $(3.7)$ holds, then, for every $T>0$ the following discrete space and time total variation estimate is satisfied:

with

Proof.

By $\mathbf{BV}$ estimate in space (3.11), we have

On the other hand, observe that

We then estimate separately each term on the right hand side of the inequality (3.14).

By the definition of the relaxation step (3.6), for the first term on right hand side of (3.14) we have

then multiplying by $\Delta x$ and summing over all $j\in \mathbb{Z}$,

Now we analyze the second term of the right hand side (3.14). Since the numerical flux defined in (3.5) is Lipschitz continuous in both arguments with Lipschitz constant $\mathcal{L}$, defined by (2.6), we obtain

multiplying by $\Delta x$, summing over all $j\in \mathbb{Z}$ and by the Remark 3.2 we get

Collecting together (3.16) and (3.17), and by using Lemma 3.2 and Proposition 3.1 we have,

Then, collecting together (3.13) and (3.18) we get

3.4.   Discrete entropy inequality

In order to define an entropy inequality we define, for $\kappa\in[0, 1]$, and the numerical fluxes

with $a\vee b = \max\{a, b\}$, and $a\wedge b = \min\{a, b\}$.

Lemma 3.4. Let $\rho_{0}\in\left(\mathbf{L^{1}}\cap \mathbf{BV}\right)\left( \mathbb{R};[0, 1]\right)$. Assume that hypotheses $({{\rm{H}}_{\rm{1}}})$ and CFL condition $(3.7)$ hold. Then, the approximate solution $\rho_{\Delta}$ constructed by Algorithm 3.1 satisfies the following discrete entropy inequality: for $j\in \mathbb{Z}$, for $n = 0, \ldots, N_{T}-1$ and for any $\kappa\in[0, 1]$,

Proof. We set

Clearly $\rho_{j}^{n+1/2} = \mathscr{G}_{j}(\rho_{j-1}^{n}, \rho_{j}^{n})$.

The map $\mathscr{G}_{j}$ is a monotone non-decreasing function with respect to each variable under the CFL condition (3.7) since we have

Moreover, we have the following identity

Then, by monotonicity, the definition of scheme (3.5) and by using

$\left|a+b\right|\geq|a|+{\rm{sgn}}(a)b$, we get

The following Theorem states the $\mathbf{L^{1}}$-Lipschitz continuous dependence of solution to (2.1) on both the initial datum and the $q_{\mathrm{on}}$ and $q_{\mathrm{off}}$ functions.

Theorem 3.1 (Uniqueness). Let $\rho$ and $\tilde{\rho}$ be two solutions to problem $(2.1)$ in the sense of Definition 2.2, with initial data $\rho_0, \ \tilde{\rho}_{0}\in\mathbf{L^{1}}\cap \mathbf{BV}\left( \mathbb{R};[0, 1]\right)$, with on-ramp rate $q_{\mathrm{on}}$, $\tilde{q}_{\mathrm{on}}$ and off-ramp rate $q_{\mathrm{off}}$, $\tilde{q}_{\mathrm{off}}$, respectively. Assume $v\in\mathbf{C^{{2}}}\left([0, 1], \mathbb{R}^{+}\right)$. Then, for a.e. $t\in[0, T]$,

Proof. The proof follows closely Theorem 5.6 of [8].

By using Kružkov's doubling of variables technique we get

where

Let us now estimate all the terms appearing in the right hand side of the above inequality. We start bounding $\tilde{\mathcal{S}}_{\mathrm{on}}$ and $\tilde{\mathcal{S}}_{\mathrm{off}}$ terms:

where

First we are going to bound $\tilde{\mathcal{S}}_{\mathrm{on}}^{1}$ term,

thus

Observe that

since $\int_{ \mathbb{R}}\omega_{\eta}(x) \mathrm{d} x = 1$. Then,

Now we are going to bound $\tilde{\mathcal{S}}_{\mathrm{on}}^{2}$.

Integrating in time and space we have

Bounding $\tilde{\mathcal{S}}_{\mathrm{on}}^{3}$,

thus

Therefore, we get the following estimate

Regarding $\tilde{\mathcal{S}}_{\mathrm{off}}$ term, we proceed in a similar way like above and we get

where

Then,

and

Thus, we get

{Next, focus on $\mathcal{V}$, by using the following estimate

we obtain}

Next, we pass to $\mathcal{V}_{x}$. Following [8] we compute

thus

where

Collecting together (3.19), (3.20), (3.21) and (3.22) we get

where

An application of Gronwall Lemma to (3.23) completes the proof.

3.5.   Proof of Theorem 2.1

The convergence of the approximate solutions constructed by Algorithm 3.1 towards the unique weak entropy solution can be proven by applying Helly's compactness theorem. The latter can be applied due to Lemma 3.1 and Proposition 3.2 and states that there exists a sub-sequence of approximate solution $\rho_{\Delta}$ that converges in $\mathbf{L^{1}}$ to a function $\rho\in\mathbf{L^{\infty}}\left([0, T]\times \mathbb{R};[0, 1]\right)$. Following a Lax-Wendroff type argument, we can show that the limit function $\rho$ is a weak entropy solution of (2.1) in the sense of Definition 2.2. Together with the uniqueness result in Theorem 3.1. this concludes the proof of Theorem 2.1.

3.6.   Existence for Model 2

In this section we consider the problem (2.1) with the $S_{\mathrm{on}}$ (1.6). In Algorithm 3.1 we substitute $S_{\mathrm{on}}$ term in the reaction step (3.6) by (3.3), thus now the term (3.6) is given by

Lemma 3.5 (Maximum Principle). Let $\rho_{0}\in\mathbf{L^{\infty}}( \mathbb{R};[0, 1])$. Let hypotheses $({{\rm{H}}_{\rm{1}}})$ and CFL condition $(3.7)$ hold, then for all $t>0$ and $x\in \mathbb{R}$ the piece-wise constant approximate solution $\rho_{\Delta}$ constructed through Algorithm 3.1 is such that

Proof. The proof is made by induction. We assume that $0\leq\rho_{j}^{n}\leq1$ for all $j\in \mathbb{Z}$. Consider the step (3.5) of Algorithm 3.1, by CFL condition (3.7) we have $0\leq\rho_{j}^{n+1/2}\leq 1$ for $j\in \mathbb{Z}$.

Now focus on the remaining step, involving the source term.

now we can proceed as in Lemma 3.1.

Lemma 3.6. Let $\rho_{0}\in\mathbf{L^{1}}( \mathbb{R}, [0, 1])$. Let $({{\rm{H}}_{\rm{1}}})$ and the CFL condition $(3.7)$ hold. Then, the piece-wise constant approximate solution $\rho_{\Delta}$ constructed through Algorithm 3.1 satisfies,

where $\mathcal{C}_{1}$ like in $(3.8)$.

Proof. By (3.26) and CFL condition (3.7) we have

this cases reduce to (3.9) and we can proceed as in Lemma 3.2.

3.7.   BV estimates

Lemma 3.7. The map $S_{\mathrm{on}}$ given in $(3.25)$ is Lipschitz continuous in second, third and fourth argument with Lipschitz constant $\left\|q_{\mathrm{on}}\right\|_{\mathbf{L^{\infty}}([0, T])}$.

Proof.

where

by the definition of $S_{\mathrm{on}}$ term and by using the estimation

we have

Pass now to $\mathcal{S}_{2}$:

Next, we analyze the $\mathcal{S}_{3}$ term:

Proposition 3.3 ($\mathbf{BV}$ estimate in space). Let $\rho_{0}\in\left(\mathbf{L^{1}}\cap \mathbf{BV}\right)\left( \mathbb{R};[0, 1]\right).$ Assume that the hypotheses $({{\rm{H}}_{\rm{1}}})$ and CFL condition $(3.7)$ hold. Then, for $n = 0, \ldots, N_{T}-1$ the following estimate holds

with $\mathcal{H}$ like in $(2.5)$.

Proof. Due to the results obtained in Lemma 3.7, the proof is analogous to that one of Proposition 3.1.

Proposition 3.4 ($\mathbf{BV}$ estimate in space and time). Let hypotheses $({{\rm{H}}_{\rm{1}}})$ hold, $\rho_{0}\in\left(\mathbf{L^{1}}\cap \mathbf{BV}\right)\left( \mathbb{R};[0, 1]\right)$. If the CFL condition $(3.7)$ holds, then, for every $T>0$ the following discrete space and time total variation estimate is satisfied:

with$\mathcal{C}_{xt}(T)$ defined in $(3.12)$.

Proof. For this proof we need to compute the following estimate,

Here we need to consider two cases, which are described below:

$\textbf{Case 1:}$ $\max\left\{\rho_{j}^{n+1/2}, R_{\mathrm{on}, j}^{n+1/2}\right\} = \rho_{j}^{n+1/2}.$ In this case we get the following estimate

$\textbf{Case 2:}$ $\max\left\{\rho_{j}^{n+1/2}, R_{\mathrm{on}, j}^{n+1/2}\right\} = R_{\mathrm{on}, j}^{n+1/2}.$ Observe that since $R_{\mathrm{on, j}}^{n+1/2}\leq1$, this implies that $0\leq\left|1-R_{\mathrm{on, j}}^{n+1/2}\right|\leq1\leq1+\left|\rho_{j}^{n+1/2}\right|$, from what we get the following estimate

Note that both cases reduces to (3.15) and therefore the rest of the proof is analogous to Proposition 3.2.

4.   Numerical experiments

In this section we present some numerical examples to describe the effects that the ramps have on a road. We solve Model 1 and Model 2 by means Algorithm 3.1 with the term $S_{\mathrm{on}}$ computed as (3.2) and (3.3), respectively. In all numerical examples below, we consider one on-ramp and one off-ramp, both ramps with length $L = 0.1$, the on-ramp is located from $x = 1.0$ until $x = 1.1$, the off-ramp is located from $x = 3$ until $x = 3.1$ and we consider the following kernel functions

for convective and reactive terms respectively, with $\eta\in[0, 1]$ and $\delta\in[-\eta, \eta]$.

4.1.   Example 1:

Dynamic of Model 1 vs. Model 2.

In this example we show numerically the behavior of the density of vehicles in a main road with the presence of one on-ramp and one off-ramp. We solve (2.1) numerically in the interval $[-1, 9]$ in simulated times $T = 0.5, \ T = 2, \ T = 5, \ T = 7$. We consider $\Delta x = 1/1000$, $\eta = 0.05, \ \delta = -0.01$, a constant initial condition $\rho_{0}(x) = 0.3$, and the rate of the on- and off-ramp are given by $q_{\mathrm{on}}(t) = 1.2, \ q_{\mathrm{off}}(t) = 0.8$, respectively.

In Fig. 2 we can see that when vehicles enter the ramp, the density of vehicles on the main road increases and a shock wave with negative speed is formed, after that, a rarefaction wave appears and when some vehicles leave the main road through off-ramp a shock wave with positive speed is formed. In particular we can observe a difference between the maximum density that is reached in each model, which may be due to the presence of the term $1-\rho$ in the Model 1.

4.2.   Example 2:

limit $\eta\to0$ in Model 2.

In this example we take a look at the limit case $\eta\to0$ and investigate the convergence of the Model 2 to the solution of the local problem (1.1)-(1.3). In particular, we consider the initial condition $\rho_{0}(x) = 0.3$ for $x\in[0, 1]$, $q_{\mathrm{on}}(t) = 1.2, \ q_{\mathrm{off}}(t) = 0.8$ at $T = 5$ with fixed $\Delta x = 1/1000$ and $\eta\in\{0.1, 0.05, 0.01, 0.004\}$, and $\delta = 0$. To evaluate the convergence, we compute the $\mathbf{L^{1}}$ distance between the approximate solution obtained for the proposed upwind-type scheme by means Algorithm 3.1 with a given $\eta$ and the result of a classical Godunov scheme for the corresponding local problem. In Table 1, we can observe that the $\mathbf{L^{1}}$ distance goes to zero when $\eta\to0.$ The results are illustrated in Fig. 3.

4.3.   Example 3:

Maximum principle.

In this example we verify that the Algorithm 3.1 with the terms $S_{\mathrm{on}}$ (3.2) and (3.3) satisfy the maximum principle, i.e., we verify numerically that Lemmas 1 and 5 respectively, are fulfilled. On the other hand, we also verify that the Algorithm 3.1 with a discretization of the term $S_{\mathrm{on}}$ (1.4), which we called Model 0, does not satisfy a maximum principle. For this purpose we consider the initial condition given by

$q_{\mathrm{on}}(t) = 1, \ q_{\mathrm{off}}(t) = 0.2$ at $T = 0.3$, with $\Delta x = 1/100$, $\eta = 0.05,$ and $\delta = -0.01$. We can see in Fig. 4 (a) that the Model 0 does not satisfy a maximum principle unlike Model 1 and Model 2. The Fig 4(b) is a zoom of (a) in which we can appreciate in a better form that Model 0 does not satisfy a maximum principle.

4.4.   Example 4:

Free main road.

In this example we consider a free main road, i.e, we consider a initial condition $\rho_{0} = 0$, boundary conditions $\rho_{0}(t) = 0.4$ for all $t>0$ and absorbing conditions at $x = 5$. We also consider the rate of the on-ramp $q_{\mathrm{on}}(t) = \frac{1}{2}\left(\sin(\pi t)+1\right)$ and the rate of the off-ramp $q_{\mathrm{off}}(t) = 0.2$. We solve (2.1) numerically in the interval $[-1, 5]$ in different times, namely $T = 1, \ T = 2, \ T = 5, \ T = 7$ and consider $\Delta x = 1/1000$, $\eta = 0.1, \ \delta = -0.02$. In Fig. 5 we can see the dynamic of the model 2.1 approximated by means of Model 1 and Model 2.

5.   Conclusion and perspectives

In this paper we introduced a nonlocal balance law to model vehicular traffic flow including on- and off-ramps. We presented three different models called Model 0, Model 1 and Model 2 and we proved existence and uniqueness of solutions for Model 1 and Model 2. We approximated the problem through a upwind-type numerical scheme, providing a Maximum principle, $\mathbf{L^{1}}$ and $\mathbf{BV}$ estimates for approximate solutions. Numerical simulations illustrate the dynamics of the studied models and show that Model 0 does not satisfy a maximum principle. A limit model as the kernel support tends to zero is numerically investigated. In a future work, we would like to consider a nonlocal version of the second order model proposed in [21].

Acknowledgments

FAC is member of GNFM (Gruppo Nazionale per la Fisica Matematica) of INdAM (Istituto Nazionale di Alta Matematica), Italy. LMV is supported by ANID-Chile through Fondecyt project 1181511 and by Centro de Modelamiento Matemático (CMM), ACE210010 and FB210005, BASAL funds for center of excellence from ANID-Chile. FAC, HDC and LMV are supported by the INRIA Associated Team "Efficient numerical schemes for non-local transport phenomena" (NOLOCO; 2018–2020) and by project MATH-Amsud 22-MATH-05 "NOTION: NOn-local conservaTION laws for engineering, biological and epidemiological applications: theoretical and numerical". HDC was partially supported by the National Agency for Research and Development, ANID-Chile through Scholarship Program, Doctorado Becas Chile 2021, 21210826.