A nonlocal Lagrangian traffic flow model and the zero-filter limit

Abstract

In this study, we start from a Follow-the-Leaders model for traffic flow that is based on a weighted harmonic mean (in Lagrangian coordinates) of the downstream car density. This results in a nonlocal Lagrangian partial differential equation (PDE) model for traffic flow. We demonstrate the well-posedness of the Lagrangian model in the $L^1$ sense. Additionally, we rigorously show that our model coincides with the Lagrangian formulation of the local LWR model in the “zero-filter” (nonlocal-to-local) limit. We present numerical simulations of the new model. One significant advantage of the proposed model is that it allows for simple proofs of (i) estimates that do not depend on the “filter size” and (ii) the dissipation of an arbitrary convex entropy.

1 Introduction

The LWR model, developed by Lighthill, Whitham, and Richards [23] more than six decades ago, was the first macroscopic traffic model. The basic form of the LWR model is a hyperbolic conservation law [13], which is a PDE that states that the total number of vehicles on a given stretch of road must remain constant over time. This is expressed mathematically as a continuity equation, which relates the flow of vehicles uV into and out of a given region to the change in the density u of vehicles within that region. The LWR model also includes equations that describe how the speed V of vehicles changes over time t and space x. These equations are based on the assumption that the speed V of a vehicle located at a point x at time t is determined by the density u of vehicles at (t, x), $V=V(u(t,x))$, and that the speed of a vehicle will tend to decrease as the density of surrounding vehicles increases, $V'(\cdot )\le 0$. We refer to uV(u) as the flux function and the conservation law

$$\begin{aligned} \partial _t u+\partial _x \bigl (u V(u)\bigr )=0 \end{aligned}$$

(1.1)

as the original LWR model. There have been many generalisations of the LWR model over the years. For a comprehensive discussion of traffic flow data and the various models used to mathematically represent it, we recommend consulting the book [28].

The original LWR model is based on local PDEs, which means that the speed function V is determined by the values of the car density at a single point x in space. There have been numerous efforts to develop alternative speed functions. In particular, many authors examined nonlocal generalisations of the original LWR model, taking into account the look-ahead distance of drivers in order to better model their behaviour. Some models assume that drivers react to the mean downstream traffic density, while others assume that they react to the mean downstream velocity. The corresponding nonlocal LWR models take the form

$$\begin{aligned} \partial _t u+\partial _x \left( u V\left( \overline{u}\right) \right) =0, \quad \partial _t u+\partial _x \left( u \overline{V(u)}\right) =0, \end{aligned}$$

(1.2)

where, for a given integrable function $v=v(x)$, $\overline{v}(x):=\int _x^\infty \Phi _{\alpha }(y-x)v(y)\,dy$. The anisotropic kernel $\Phi _{\alpha }$ characterizes the nonlocal effect through the “filter size” $\alpha >0$. It is a nonnegative, nonincreasing, and $C^1$ function defined on the nonnegative real numbers, and it has unit mass: $\int \limits _0^\infty \Phi _{\alpha }(x)\, dx =1$. Setting $J_\alpha (x):=\Phi _\alpha (-x)\chi _{(-\infty ,0]}(x)$, the function $\overline{v}(x)$ can be expressed as the convolution $v\star J_\alpha (x)$, noting that $\left\{ J_\alpha (x)\right\} _{\alpha >0}$ is an approximate identify (convolution kernel) that generally is discontinuous at $x=0$. In the formal limit $\alpha \rightarrow 0$ (the “zero-filter” limit), the nonlocal fluxes $u V\left( \overline{u}\right)$ and $u \overline{V(u)}$ converge to the local flux uV(u) of original LWR model (1.1).

The mathematical study of conservation laws with nonlocal flux has gained significant attention in recent years. A comprehensive list of references on this topic is beyond the scope of this text. Instead, we refer the reader to the recent paper [10] (on weak solutions) and the references cited therein. Here we only mention a few references [3, 7, 15, 16] related to nonlocal conservation laws (1.2) that arise as generalisations of the original LWR model. In particular, in [3, 7, 16] the authors establish the well-posedness (of entropy solutions) and convergence of numerical schemes for the first equation in (1.2), as well as a more general version of it. For modifications of these results to account for the second equation in (1.2), see [15].

In general [11], solutions of nonlocal conservation laws like $\partial _t u_\alpha +\partial _x \bigl (u_\alpha V(u_\alpha \star J_\alpha )\bigr )=0$, where $J_\alpha$ is an arbitrary approximate identity and V is a Lipschitz function, do not converge to the entropy solution of the corresponding local conservation law as the “filter size” $\alpha$ approaches zero. The counterexamples in [11] do not exclude the possibility that convergence may still hold in specific cases. In particular, the case where $V'(\cdot )\le 0$, the initial function is nonnegative, and the convolution kernel $J_\alpha$ is anisotropic, specifically supported on the negative axis $(-\infty , 0]$. This case corresponds to nonlocal traffic flow PDEs, like the first one in (1.2). Recently, under assumptions like these, positive results have been obtained for the zero-filter limit [4, 5, 9, 12, 20].

Traffic flow models can be divided into two categories: macroscopic models, which describe the flow of vehicles on a roadway as a continuous fluid, and microscopic models, which describe the motion and interactions of individual vehicles. LWR-type PDEs are examples of macroscopic traffic flow models, while microscopic models are often described using systems of differential equations, such as the Follow-the-Leaders (FtL) model. In the FtL model, the velocity of each vehicle is determined by the velocity of the vehicle in front of it. There is a (rigorous) connection between FtL models and hyperbolic conservation laws, which has been studied in detail in the literature, see [14, 18] and the references therein. In [6, 8, 24, 26], the authors provide links between nonlocal FtL models and macroscopic LWR-type equations (1.2).

Before we present our own model, it is helpful to briefly describe the nonlocal FtL models of [8, 24, 26]. Let $x_i(t)$, $i\in \mathbb {Z}$, be the position of the ith car, ordering them so that $x_{i+1}(t)\ge x_i(t)+\ell$, where $\ell$ is the (common) length of the cars. Set

$$\begin{aligned} u_i(t):= \frac{\ell }{x_{i+t}(t)-x_i(t)}, \end{aligned}$$

(1.3)

which is the local discrete density (or “car saturation”) perceived by the driver of car $i\in {\mathbb {Z}}$. One of the nonlocal FtL models of [24] asks that the car positions $x_i(t)$ satisfy the following system of differential equations:

$$\begin{aligned} x_i'(t)=V\left( \overline{u_i}(t)\right) , \quad i \in \mathbb {Z}, \,\, t>0, \end{aligned}$$

(1.4)

where

$$\begin{aligned} \overline{u_i}(t) :=\sum _{j=0}^\infty \Phi _{ij\alpha }(t) u_{i+j}(t),\quad \Phi _{ij\alpha }(t):=\int \limits _{x_{i+j}(t)}^{x_{i+1+j}(t)} \Phi _{\alpha }(\zeta -x_i(t))\, d\zeta , \quad i\in \mathbb {Z}. \end{aligned}$$

(1.5)

In other words, the velocity of each vehicle is not only determined by the vehicle directly in front of it, but also by the other vehicles in the surrounding (downstream) area. Replacing (1.4) by

$$\begin{aligned} x_i'(t)=\overline{V(u_i(t))}, \quad i \in \mathbb {Z}, \,\, t>0, \end{aligned}$$

(1.6)

we obtain a slightly different FtL model. While drivers under model (1.4) react to the mean downstream traffic saturation, drivers under model (1.6) react to the mean downstream velocity.

Nonlocal FtL model (1.4), (1.5) uses a weighted arithmetic mean of the (downstream) car-density values to calculate the speed. There are several ways to aggregate a sequence of numbers. While the arithmetic mean is a simple average calculated by adding up the values in a set and dividing by the number of values, the harmonic mean is calculated by taking the reciprocal of the arithmetic mean of the reciprocals of the values in a set. In view of the well-known harmonic mean-arithmetic mean inequality [27, p. 126], the harmonic mean is generally a more conservative estimate of the average value in a set; roughly speaking, the harmonic mean takes into account the “size” of the values in the set, while the arithmetic mean does not.

In this paper we propose a nonlocal FtL model based on a weighted harmonic mean in the Lagrangian coordinates. The governing differential equations are of the form

$$\begin{aligned} x_i'(t)= V\left( \,\left[ \, \overline{\frac{1}{u_i(t)}} \,\right] ^{-1}\,\right) , \quad \overline{\frac{1}{u_i(t)}} :=\sum _{j=0}^\infty \frac{\Phi _{ij\alpha }}{u_{i+j}(t)}, \qquad i \in \mathbb {Z}, \,\, t>0. \end{aligned}$$

(1.7)

Now the weights are determined by

$$\begin{aligned} \Phi _{ij \alpha }:=\int \limits _{z_{i+j}}^{z_{i+1+j}} \Phi _{\alpha }(\zeta -z_i)\, d\zeta , \quad j=0,1,2,\ldots , \end{aligned}$$

(1.8)

where $z_i:=i\ell$ is the Lagrangian coordinate of the i-th car. Note carefully that the weights $\Phi _{ij \alpha }$ are computed by averaging the kernel $\Phi (\cdot -z_i)$ (centred at car i) between the Lagrangian particles $z_{i+j}$ (car $i+j$) and $z_{i+1+j}$ (car $i+1+j$). The cars are here labelled in the driving direction,¹ so that the weights $\Phi _{ij \alpha }$ decrease with the car number (increasing $z_i$). Averaging between Lagrangian particles is different from more traditional approach (1.5). The contrast between the position $x_i$ of car i and the Lagrangian coordinate $z_i$ is that $x_i$ represents the actual physical position of the car in space, while $z_i$ is a mathematical construct (labelling) used to describe the car’s position relative to other cars.

The corresponding macroscopic equation becomes

$$\begin{aligned} \partial _t \left( \frac{1}{u(z,t)}\right) -\partial _z V\left( \,\left[ \, \overline{\frac{1}{u(z,t)}}\,\right] ^{-1} \, \right) =0, \quad z\in \mathbb {R}, \,\, t>0, \end{aligned}$$

(1.9)

where

$$\begin{aligned} \overline{\frac{1}{u(z,t)}}= \int \limits _z^\infty \Phi _{\alpha }(\zeta -z) \frac{1}{u(\zeta ,t)}\, d\zeta . \end{aligned}$$

(1.10)

In other words, in terms of the Lagrangian variable $y=y(z,t)=\frac{1}{u(z,t)}$ (“amount of road per car”, also known as “spacing” or “gap” between cars), we obtain a nonlocal conservation law of the form

$$\begin{aligned} \partial _t y-\partial _z W\left( \overline{y}\right) =0, \quad \overline{y}(z,t)= \int \limits _z^\infty \Phi _{\alpha }(\zeta -z)y(\zeta ,t)\, d\zeta , \quad W(y):=V\left( \frac{1}{y}\right) . \end{aligned}$$

(1.11)

Formally, as the filter size $\alpha$ approaches zero, the local Lagrangian PDE $\partial _t (1/u)-\partial _z V(u)=0$ is obtained. This PDE can be transformed into Eulerian PDE (1.1) through a change of variable [29]. Nonlocal LWR equations (1.1) are Eulerian models, while model (1.9) analysed in this paper is a Lagrangian model. The main difference between the two is the coordinate system used. In Eulerian coordinates, traffic is observed from a fixed point and the coordinates are fixed in space, while in Lagrangian coordinates, traffic is observed from a car travelling with the flow and coordinates move with the cars. In Eulerian coordinates, the main variable is density u as a function of space x and time t, while in the Lagrangian formulation, it is spacing y as a function of “car number” z and time t (the smaller the spacing, the higher the traffic density, and vice versa). Lagrangian traffic flow models have become increasingly important in recent times, as advancements in technology have allowed for the collection of data via GPS, on-board sensors, and smartphones. This provides more accurate Lagrangian traffic measurements.

We will see that the mathematical and numerical analysis of Lagrangian PDE (1.9) becomes fairly simple, whereas its Eulerian counterpart leads to a complicated PDE that appears much harder to analyse directly. Besides, we are able to rigourously justify the zero-filter limit of (1.9). More precisely, we show the existence, uniqueness, and $L^1$ stability of solutions to (1.9), for any fixed value of the filter size $\alpha >0$. To prove the existence of a weak solution, we use approximate solutions obtained from the FtL model and compactness arguments. The resulting solution is regular enough to make it easy to prove the uniqueness and stability of the weak solution. A key aspect of our approach is that we derive estimates and strong convergence for the filtered variable

$$\begin{aligned} w:=\overline{y}=\int \limits _z^\infty \Phi _{\alpha }(\zeta -z) y(\zeta ,t)\, d\zeta , \end{aligned}$$

(1.12)

rather than for the original variable $y=1/u$ itself. This allows for simple proofs of estimates that are independent of the filter size $\alpha$, which is at variance with the more traditional analyses of [3, 7, 15, 16]. As a result, we can consider a sequence $\left\{ w_{\alpha }=\overline{y_\alpha }\right\} _{\alpha >0}$ of filtered solutions of (1.9) and show that a subsequence converges strongly in $L^1_{{\text {loc}}}$ to a function w that is a solution of the (Lagrangian form) of LWR Eq. (1.1). Besides, we demonstrate that $w_{\alpha }$ dissipates any convex entropy function, which implies that the limit w is the unique Kružkov entropy solution of the LWR equation. We even provide an explicit rate of convergence, namely that $\left\| w_{\alpha }(t)-w(t)\right\| _{L^1(\mathbb {R})}\le C \sqrt{\alpha }$. It is worth noting that the zero-filter limit has only recently been successfully studied in [9, 12], but only for the first nonlocal conservation law in (1.2). Our work provides a different approach for studying the alternative nonlocal Lagrangian model (1.9), which is distinct from (1.2), and its zero-filter limit.

In this study, we also demonstrate that the variable $y_\alpha$ converges strongly through the estimation of $w_\alpha -y_\alpha$ in the $L^1$ norm for exponential kernels. Based on numerical experiments, the same appears to be true for Lipschitz kernels. However, the convergence is not expected for general discontinuous kernels. Our numerical experiments indicate that as $\alpha$ approaches zero, oscillations persist in the variable $y_\alpha$ for discontinuous kernels.

The paper is structured as follows: Sect. 2 analyses a fully discrete scheme for $w_\alpha$. Section 3 explores the connection between $y_\alpha =1/u_\alpha$ and $w_\alpha$. Section 4 provides an Eulerian formulation for the discussed Lagrangian PDE for easy comparison with existing literature. Section 5 examines the zero-filter limit. Finally, Sect. 6 showcases numerical examples.

2 Analysis of a fully discrete scheme

In this section, we will present and analyse a fully discrete numerical approach based on nonlocal FtL model (1.7). The numerical examples for this approach will be provided in Sect. 6. Before that, however, we will list some properties of the averaging kernel $\Phi _\alpha$ and the associated averaging operator.

Let $\Phi :\mathbb {R}_+\rightarrow \mathbb {R}_+$ be a nonincreasing function such that

$$\begin{aligned} \int \limits \limits _0^\infty \Phi (z)\,dz = 1\ \ \ \text {and}\ \ \ \int \limits _0^\infty z\Phi (z)\,dz < \infty . \end{aligned}$$

(2.1)

For $\alpha >0$ define

$$\begin{aligned} \Phi _\alpha (z)=\frac{1}{\alpha } \Phi \Bigl (\frac{z}{\alpha }\Bigr ), \end{aligned}$$

(2.2)

and for any suitable function $h:\mathbb {R}\rightarrow \mathbb {R}$ define

$$\begin{aligned} \overline{h}(z)=\int \limits _z^\infty \Phi _\alpha (\zeta -z)h(\zeta )\,d\zeta = \int \limits _0^\infty \Phi _\alpha (\zeta )h(z+\zeta )\,d\zeta . \end{aligned}$$

(2.3)

We have that

$$\begin{aligned}&\overline{h}'(z) =\overline{h'}(z)\ \ \text {if }h\text { is differentiable,}\\&\left\| \overline{h}\right\| _{L^p(\mathbb {R})} \le \left\| h\right\| _{L^p(\mathbb {R})},\ \ p\in [1,\infty ],\\&\overline{h}'(z) =\int \limits _0^\infty \Phi '_\alpha (\zeta ) \left[ h(z)-h(z+\zeta )\right] \,d\zeta =-\int \limits _z^\infty \Phi '_\alpha (\zeta -z) \left[ h(\zeta )-h(z)\right] \,d\zeta , \end{aligned}$$

if $\Phi$ is differentiable.

We shall consider a time-forward Euler discretization of the system of ODEs (1.7). We set ${\Delta z}=\ell >0$ and employ the usual notation $z_j=(j-1/2){\Delta z}$, $j \in \mathbb {Z}/2$, $z_{1/2}=0$, and $\lambda ={\Delta t}/{\Delta z}$, where ${\Delta t}>0$ is a sufficiently small (to be specified) number. Subtracting the equation for $x_i'$ in (1.7) from that for $x_{i+1}'$ and dividing the result by ${\Delta z}$, we get

$$\begin{aligned} \frac{d}{dt} \Bigl (\frac{1}{u_i(t)}\Bigr ) = \frac{1}{{\Delta z}} \Bigl ( V\Bigl (\, \overline{\frac{1}{u_{i+1}}}\, \Bigr ) - V\Bigl (\,\overline{\frac{1}{u_{i}}}\,\Bigr )\Bigr ), \end{aligned}$$

(2.4)

where

$$\begin{aligned} \overline{h}_i = \sum _{j\ge i} \Phi _{ij\alpha }h_j, \quad \Phi _{ij\alpha }=\int \limits _{z_{j-1/2}}^{z_{j+1/2}} \Phi _\alpha (\zeta -z_{i-1/2})\,d\zeta , \quad i\in \mathbb {Z}, \end{aligned}$$

and we have used (1.3). Semi-discrete scheme (2.4) represents an approximation of nonlocal Lagrangian PDE (1.9). Throughout the paper, $\Phi _{ij\alpha }$ and $\Phi _{i,j,\alpha }$ are used interchangeably, with either commas or no commas in their notation.

To greatly facilitate the analysis, we will shift our focus from the variable $y=1/u$ to its filtered counterpart by introducing

$$\begin{aligned} w_i=\overline{\frac{1}{u_{i}}}, \quad W(w)=V\Bigl (\frac{1}{w}\Bigr ), \quad V\in C^1([0,\infty ))\text { nonincreasing} \end{aligned}$$

(2.5)

as previously mentioned in the introduction, cf. (1.12).

Applying the $\overline{\;\cdot \;}$ operator to (2.4), we get

$$\begin{aligned} \frac{d}{dt} w_i = \frac{1}{{\Delta z}}\left( \overline{W(w_{i+1})} -\overline{W(w_{i})}\right) , \ \ \ i\in \mathbb {Z}. \end{aligned}$$

We shall analyse the following scheme for this system of ODEs:

$$\begin{aligned} \begin{aligned} w^{n+1}_i&= w^n_i + \lambda \left( \overline{W}\negthinspace ^{\;n}_{i+1/2}-\overline{W}\negthinspace ^{\;n}_{i-1/2}\right) ,\ \ n\ge 0,\\ w^0_i&= \sum _{j\ge i} \Phi _{ij\alpha } y_{0,j}, \end{aligned} \ \ i\in \mathbb {Z}, \end{aligned}$$

(2.6)

where $w^n_i\approx w_i(n{\Delta t})$ and

$$\begin{aligned} \overline{W}\negthinspace ^{\;n}_{i-1/2}=\sum _{j\ge i} \Phi _{ij\alpha } W(w^n_j),\ \ \ \ y_{0,i}=\frac{1}{u_{0,i}}=\frac{x_{i+1}(0)-x_i(0)}{\ell }. \end{aligned}$$

It is readily verified that the infinite matrix $\Phi _{ij\alpha }$ satisfies

$$\begin{aligned} \Phi _{i-1,j-1,\alpha }&=\int \limits _{z_{j-3/2}}^{z_{j-1/2}}\Phi _\alpha (\zeta -z_{i-3/2})\,d\zeta =\int \limits _{z_{j-1/2}}^{z_{j+1/2}} \Phi _\alpha (\zeta -z_{i-1/2})\,d\zeta =\Phi _{ij\alpha },\\ \sum _{j\ge i} \Phi _{ij\alpha }&= \sum _{j\ge 1} \Phi _{1j\alpha } = \sum _{j\ge 1} \int \limits _{z_{j-1/2}}^{z_{j+1/2}} \Phi _\alpha (\zeta )\,d\zeta =\int \limits _0^\infty \Phi (\zeta )\,d\zeta = 1,\\ \sum _{i\in \mathbb {Z}} \sum _{j\ge i} \Phi _{ij\alpha } \, \mu _j&=\sum _{i\in \mathbb {Z}} \sum _{k=1}^\infty \Phi _{i,i+k-1,\alpha } \, \mu _{i+k-1} =\sum _{i\in \mathbb {Z}} \sum _{k=1}^\infty \Phi _{1,k,\alpha } \, \mu _{i+k-1} =\sum _{j\in \mathbb {Z}} \mu _j \sum _{k=1}^\infty \Phi _{1,k,\alpha } \\&= \sum _{i\in \mathbb {Z}} \mu _i,\\ \sum _{i\in \mathbb {Z}} \Bigl |\sum _{j\ge i} \Phi _{ij\alpha } \, \mu _j\Bigr |^p&\le \sum _{i\in \mathbb {Z}} \left| \mu _j\right| ^p, \ \ 1\le p<\infty ,\\ \sup _{i\in \mathbb {Z}}\Bigl |\sum _{j\ge i} \Phi _{ij\alpha } \, \mu _j\Bigr |&\le \sup _{i\in \mathbb {Z}}\left| \mu _i\right| . \end{aligned}$$

The following lemma demonstrates that scheme (2.6) for the filtered variable $w=\overline{y}$ adheres to the classical monotonicity criteria of Harten, Hyman, and Lax. The monotonicity of the scheme ensures that the numerical solution does not create spurious oscillations or produce unphysical values outside of the set of initial conditions. Note that the (exact) solution operator for the original variable $y=1/u$ is not monotone.

Lemma 2.1

If ${\Delta t}$ and ${\Delta x}$ are chosen such that the CFL -condition

$$\begin{aligned} 0\le \lambda \sup _wW'(w) \le 1 \end{aligned}$$

(2.7)

holds, then scheme (2.6) is monotone in the sense that

$$\begin{aligned} w^n_i \ge \widetilde{w}^n_i \ \ \text {for all }i\in \mathbb {Z}\ \ \ \Longrightarrow \ \ \ w^{n+1}_i \ge \widetilde{w}^{n+1}_i \ \ \text {for all }i\in \mathbb {Z}, \end{aligned}$$

where $\widetilde{w}^{n+1}$ is a corresponding solution of (2.6).

Proof

We compute

$$\begin{aligned} \frac{\partial w^{n+1}_i}{\partial w^n_k}= {\left\{ \begin{array}{ll} 0, &{}k<i,\\ 1-\lambda \Phi _{ii\alpha } W'(w_i), &{}k=i,\\ \lambda \bigl (\Phi _{ik\alpha } -\Phi _{i,k+1,\alpha }\bigr ) W'(w_k), &{}k>i, \end{array}\right. }\ \ \ \ge 0, \end{aligned}$$

if (2.7) holds, since $\Phi _{ii\alpha }\le 1$ and $\Phi _{ik\alpha } -\Phi _{i,k+1,\alpha }\ge 0$. $\square$

As a direct result of the monotonicity, scheme (2.6) for the filtered variable w is also $L^1$ contractive (stable with respect to the initial data).

Corollary 2.2

Assume that CFL-condition (2.7) holds and let $\widetilde{w}^n_i$ be the result of applying scheme (2.6) to the initial data $\widetilde{y}_{0,i}$. Then

$$\begin{aligned} {\Delta z}\sum _i\left| w^n_i-\widetilde{w}^n_i\right| \le {\Delta z}\sum _i\left| w^0_i-\widetilde{w}^0_i\right| = {\Delta z}\sum _i\left| y_{0,i}-\tilde{y}_{0,i}\right| . \end{aligned}$$

Proof

Since the scheme is monotone, we can use the Crandall–Tartar lemma [17, Lemma 2.13] on the set

$$\begin{aligned} D_{a,b}=\Bigl \{ \left\{ w_i\right\} _{i\in \mathbb {Z}} \;\Bigm |\; 1\le w_i<\infty ,\ \ {\Delta z}\sum _{i\le 0} \left| w_i-a\right|<\infty , \ \ {\Delta z}\sum _{i\ge 0} \left| w_i-b\right| <\infty \Bigr \}, \end{aligned}$$

and conclude that the corollary holds. $\square$

The monotonicity of scheme (2.6) for the filtered variable implies several basic estimates that are independent of the filter size $\alpha$. This is a key feature of using the filtered variable, as it allows for the numerical scheme to be stable and well-balanced as $\alpha \rightarrow 0$. These estimates are not used to prove the convergence of the scheme to the filtered version of nonlocal Lagrangian PDE (1.9) (for fixed $\alpha$), but rather to address the behaviour of the scheme in the limit as $\alpha$ approaches zero. This is important because it helps to ensure consistency with the original LWR model. We will return to the zero-filter limit of (1.9) in Sect. 5.

Corollary 2.3

Assume that CFL-condition (2.7) holds. Then

$$\begin{aligned} 1\le \inf _{i} y_{0,i}&\le w^n_i\le \sup _{i} y_{0,i}, \end{aligned}$$

(2.8)

$$\begin{aligned} \sum _i\left| w^n_{i+1}-w^n_i\right|&\le \sum _i\left| y_{0,i+1}-y_{0,i}\right| , \end{aligned}$$

(2.9)

$$\begin{aligned} {\Delta z}\sum _i \left| w^{n+1}_i-w^n_i\right|&\le {\Delta t}\left\| W'\right\| _{L^\infty }\left| y_{0,\cdot }\right| _{BV}. \end{aligned}$$

(2.10)

Proof

To prove (2.8), observe that the constants $c=\inf _{i} y_{0,i}$ and $C=\sup _{i} y_{0,i}$ are solutions to scheme (2.6) and then apply monotonicity. To prove BV bound (2.9), set $\widetilde{w}^n_i=w^n_{i+1}$ in Corollary 2.2. To prove $L^1$-continuity (2.10), choose $\widetilde{w}=w^{n+1}_i$ in Corollary 2.2 and calculate

$$\begin{aligned} {\Delta z}\sum _i\left| w^{n+1}_i-w^n_i\right|&\le {\Delta z}\sum _i\left| w^1_i-w^0_i\right| = {\Delta t}\sum _i\left| \overline{W}\negthinspace ^{\;0}_{i+1/2}-\overline{W}\negthinspace ^{\;0}_{i-1/2}\right| \\&= {\Delta t}\sum _i \Bigl | \sum _{j\ge i+1} \Phi _{i+1,j,\alpha }W(w^0_{j})- \sum _{j\ge i} \Phi _{ij\alpha }W(w^0_{j})\Bigr |\\&={\Delta t}\sum _i \Bigl | \sum _{j\ge i} \Phi _{ij\alpha }W(w^0_{j+1})- \sum _{j\ge i} \Phi _{ij\alpha }W(w^0_{j})\Bigr |\\&\le {\Delta t}\sum _i \sum _{j\ge i}\Phi _{ij\alpha } \left| W(w^0_{j+1})-W(w^0_{j})\right| \\&\le {\Delta t}\left\| W'\right\| _\infty \sum _i \left| w^0_{j+1}-w^0_{j}\right| \le {\Delta t}\left\| W'\right\| _\infty \left| y_{0,\cdot }\right| _{BV}. \end{aligned}$$

$\square$

Next, we will estimate the variations in space and time of the solution $w_i^n$ of scheme (2.6) for the filtered variable $w=\overline{y}$. These estimates will be dependent on the filter size $\alpha$, but they will be sufficient to demonstrate uniform convergence to a Lipschitz continuous limit $w_\alpha (x,t)$ for a fixed value of $\alpha$. As we wish to bound the “derivatives” of $w^n_i$, let us define

$$\begin{aligned} \Delta w^n_{j+1/2}=w^n_{j+1}-w^n_j,\ \ \Delta W_j=W(w_{j+1})-W(w_j)\ \ \text {and}\ \ \Delta \overline{W}\negthinspace _j = \overline{W}\negthinspace _{j+1/2}-\overline{W}\negthinspace _{j-1/2}, \end{aligned}$$

and set

$$\begin{aligned} \left( \Delta \widehat{w}\right) ^n=\sup _i\left| \Delta w^n_{i+1/2}\right| . \end{aligned}$$

(2.11)

Note that $\Delta \overline{W}\negthinspace _i = \sum _{j\ge i} \Phi _{ij\alpha }\Delta W_j$.

Lemma 2.4

Assume that CFL-condition (2.7) holds. We have

$$\begin{aligned} (\Delta \widehat{w})^{n}&\le (\Delta \hat{w})^0 \exp \Bigl (\frac{C}{\alpha } t^n\Bigr ),\end{aligned}$$

(2.12)

$$\begin{aligned} \sup _i\left| w^{n+1}_i - w^n_i\right|&\le \lambda \left\| W'\right\| _\infty (\Delta \widehat{w})^0 \exp \Bigl (\frac{C}{\alpha } t^n\Bigr ), \end{aligned}$$

(2.13)

where $t^n=n{\Delta t}$, $(\Delta \widehat{w})^{n}$ is defined in (2.11), and the constant C is independent of n, ${\Delta z}$, and $\alpha$.

Proof

We calculate

$$\begin{aligned} \left| \Delta w^{n+1}_{i+1/2}\right|&=\left| \Delta w^n_{i+1/2} + \lambda \left( \Delta \overline{W}\negthinspace ^{\;n}_{j+1}-\Delta \overline{W}\negthinspace ^{\;n}_{j}\right) \right| \\&\le \left| \Delta w^n_{i+1/2}\right| + \lambda \Bigl | \sum _{j\ge i+1} \Phi _{i+1,j,\alpha } \Delta W^n_j - \sum _{j\ge i} \Phi _{ij\alpha }\Delta W^n_j\Bigr |\\&=\left| \Delta w^n_{i+1/2}\right| + \lambda \Bigl |\sum _{j\ge i+1} \left( \Phi _{i+1,j,\alpha }-\Phi _{ij\alpha }\right) \Delta W^n_j -\lambda \Phi _{1,1,\alpha }\Delta W^n_i\Bigr |\\&\le \left| \Delta w^n_{i+1/2}\right| -\lambda \sum _{j\ge 1} \left( \Phi _{1,j+1,\alpha }-\Phi _{1,j,\alpha }\right) \left| \Delta W^n_j\right| +\lambda \Phi _{1,1,\alpha }\left| \Delta W^n_i\right| \\&\le \left| \Delta w^n_{i+1/2}\right| + {\Delta t}\left\| W'\right\| _\infty \sum _{j\ge 1} \frac{\Phi _{1,j,\alpha }-\Phi _{1,j+1,\alpha }}{{\Delta z}}\left| \Delta w^n_{j+1/2}\right| + {\Delta t}\left\| W'\right\| _\infty \frac{\Phi _{1,1,\alpha }}{{\Delta z}}\left| \Delta w^n_{j+1/2}\right| \\&\le (\Delta \widehat{w})^n \biggl (1+{\Delta t}\left\| W'\right\| _\infty \Bigl (\, \sum _{j\ge 1} \frac{\Phi _{1,j,\alpha }-\Phi _{1,j+1,\alpha }}{{\Delta z}}+\frac{\Phi _{1,1,\alpha }}{{\Delta z}} \, \Bigr )\biggr )\\&= (\Delta \widehat{w})^n \Bigl (1+2{\Delta t}\left\| W'\right\| _\infty \frac{\Phi _{1,1,\alpha }}{{\Delta z}} \Bigr ), \end{aligned}$$

which implies (2.12). We can also use this to prove (2.13),

$$\begin{aligned} \left| w^{n+1}_i - w^n_i\right|&=\lambda \left| \Delta \overline{W}\negthinspace ^{\;n}_{i+1/2}\right| \le \lambda \sum _{j\ge i} \Phi _{ij\alpha } \left| W(w^n_{j+1})-W(w^n_j)\right| \\&\le \lambda \left\| W'\right\| _\infty (\Delta \widehat{w})^n \sum _{j\ge i} \Phi _{ij\alpha } \le \lambda \left\| W'\right\| _\infty (\Delta \widehat{w})^0 \exp \Bigl (\frac{C}{\alpha } t^n\Bigr ). \end{aligned}$$

$\square$

The main theorem of this section states that the solutions to scheme (2.6) for the filtered variable converge to a Lipschitz continuous weak solution of the filtered version of nonlocal Lagrangian PDE (1.9) (for a fixed $\alpha$). To assist the convergence proof, define $w_{{\Delta t},\alpha }(z,t)$ to be the bi-linear interpolation of the points $\left\{ (z_i,t^n,w^n_i)\right\}$ with $j\in \mathbb {Z}$ and $n\ge 0$.

Theorem 2.5

Let $0<T<\infty$ and assume that as ${\Delta t}\rightarrow 0$, ${\Delta z}\rightarrow 0$ in such a way that CFL condition (2.7) is always satisfied. Let $W(\cdot )$ be defined by (2.5) and consider an initial function $1\le y_0\in BV(\mathbb {R})$. Let $\alpha >0$ be fixed and assume furthermore that the sequence of initial functions $\left\{ w_{{\Delta t},\alpha }(z,0)\right\} _{{\Delta t}>0}$ is such that $\left| \partial _zw_{{\Delta t},\alpha }(z,0)\right| \le M$, where M does not depend on ${\Delta t}$ (but on $\alpha$). Suppose the averaging kernel $\Phi _\alpha$ satisfies (2.1), (2.2). Then there exists a Lipschitz continuous function $w_\alpha :\mathbb {R}\times [0,T]\mapsto \mathbb {R}$ such that

$$\begin{aligned} \lim _{{\Delta t}\rightarrow 0} w_{{\Delta t},\alpha }=w_\alpha \quad \text {in }C(K\times [0,T]), \,\, \forall K\subset \subset \mathbb {R}. \end{aligned}$$

Moreover, $w_\alpha$ is a weak (distributional) solution of

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _tw_\alpha = \partial _z\overline{W(w_\alpha )},&{}\ \ z\in \mathbb {R},\ \ 0<t\le T,\\ w_\alpha (z,0)=\overline{y_0},&{}\ \ z\in \mathbb {R}, \end{array}\right. } \end{aligned}$$

(2.14)

where the averaging (overline) operator is defined by (2.3), i.e.

$$\begin{aligned} \int \limits _0^T\int \limits _\mathbb {R}w_\alpha (z,t)\partial _t\varphi (z,t) -\overline{W(w_\alpha )}\partial _z\varphi (z,t)\,dzdt = \int \limits _\mathbb {R}w_\alpha (z,T)\varphi (z,T)- w_\alpha (z,0)\varphi (z,0)\,dz \end{aligned}$$

for all test functions $\varphi \in C^\infty _0(\mathbb {R}\times [0,T])$. The solution is uniquely determined by the initial data.

Proof

The uniform convergence $w_{{\Delta t},\alpha }\rightarrow w_\alpha$ follows by the Arzelà-Ascoli theorem and Lemma 2.4.

For a fixed test function $\varphi$ define

$$\begin{aligned} \varphi ^n_i = \int \limits _{z_{i-1/2}}^{z_{i+1/2}}\int \limits _{t_n}^{t_{n+1}} \varphi (z,t) \,dtdz, \end{aligned}$$

and write (2.6) as

$$\begin{aligned} \frac{1}{{\Delta t}}\left( w^{n+1}_i-w^n_i\right) - \frac{1}{{\Delta z}}\left( \overline{W}\negthinspace ^{\;n}_{i+1/2}-\overline{W}\negthinspace ^{\;n}_{i-1/2}\right) =0. \end{aligned}$$

Multiply this with $\varphi ^n_i$, sum over $n=0,1,\ldots ,N-1$, where $N{\Delta t}=T$, and over $i\in \mathbb {Z}$ and finally sum by parts to arrive at

$$\begin{aligned} \sum _{i\in \mathbb {Z}}\sum _{n=1}^{N-1} w^n_i \frac{1}{{\Delta t}}\left( \varphi ^n_i-\varphi ^{n-1}_i\right) -\sum _{i\in \mathbb {Z}}\sum _{n=0}^{N-1} \overline{W}\negthinspace ^{\;n}_{i-1/2} \frac{1}{{\Delta z}}\left( \varphi ^n_{i}-\varphi ^n_{i-1}\right) =\sum _{i\in \mathbb {Z}}\frac{1}{{\Delta t}} w^N_i\varphi ^{N-1}_i-\frac{1}{{\Delta t}} w^0_i\varphi ^{0}_i. \end{aligned}$$

If we insert the definition of $\varphi ^n_i$

$$\begin{aligned}&\sum _{i\in \mathbb {Z}}\sum _{n=1}^{N-1} w^n_i \int \limits _{z_{i-1/2}}^{z_{i+1/2}}\int \limits _{t_n}^{t_{n+1}} \frac{1}{{\Delta t}}\left( \varphi (z,t)-\varphi (z,t-{\Delta t})\right) \,dtdz \nonumber \\&\qquad -\sum _{i\in \mathbb {Z}}\sum _{n=0}^{N-1} \overline{W}\negthinspace ^{\;n}_{i-1/2} \int \limits _{z_{i-1/2}}^{z_{i+1/2}}\int \limits _{t_n}^{t_{n+1}} \frac{1}{{\Delta z}}\left( \varphi (z,t)-\varphi (z-{\Delta z},t)\right) \,dtdz\nonumber \\&\quad =\sum _{i\in \mathbb {Z}} w^N_i \int \limits _{z_{i-1/2}}^{z_{i+1/2}}\frac{1}{{\Delta t}}\int \limits _{t_{N-1}}^{t_{N}} \varphi (z,t)\,dtdz - w^0_i \int \limits _{z_{i-1/2}}^{z_{i+1/2}}\frac{1}{{\Delta t}}\int \limits _{0}^{{\Delta t}} \varphi (z,t)\,dtdz. \end{aligned}$$

(2.15)

Now define the piecewise constant function (this is “omega”, not “double-u”)

$$\begin{aligned} \omega _{{\Delta t},\alpha }(z,t)=w^n_i\ \ \text {for }(z,t)\in [z_{i-1/2},z_{i+1/2})\times [t^n,t^{n+1}). \end{aligned}$$

(2.16)

Since $w_{{\Delta t},\alpha }$ is uniformly Lipschitz continuous with a Lipschitz constant L not depending on ${\Delta t}$ we have that $\left| \omega _{{\Delta t},\alpha }(z,t)-w_{{\Delta t},\alpha }(z,t)\right| \le L{\Delta t}$. Furthermore

$$\begin{aligned} \overline{W}\negthinspace ^{\;n}_{i-1/2}&=\sum _{j\ge i} \Phi _{ij\alpha } W(w^n_j) =\sum _{j\ge i} \int \limits _{z_{j-1/2}}^{z_{j+1/2}} \Phi _\alpha (\zeta -z_{i-1/2})\,d\zeta W(w^n_j)\\&=\int \limits _{z_{i-1/2}}^\infty \Phi _\alpha (\zeta -z_{i-1/2}) W(\omega _{{\Delta t},\alpha }(\zeta ,t))\,d\zeta =\overline{W(\omega _{{\Delta t},\alpha })}(z_{i-1/2},t). \end{aligned}$$

Since W is Lipschitz, it follows that $W(\omega _{{\Delta t},\alpha })$ converges a.e. and in $L^1_{{\text {loc}}}$ to $W(w_\alpha )$. Additionally, as the $\overline{\;\cdot \;}$ operator is continuous in $L^\infty$, we also have that $\overline{W(\omega _{{\Delta t},\alpha })}$ converges a.e. and in $L^1_{{\text {loc}}}$ to $\overline{W(w_\alpha )}$. Hence also the piecewise constant function $\mathcal {W}$ defined by

$$\begin{aligned} \mathcal {W}_{\Delta t}(z,t) =\overline{W(\omega _{{\Delta t},\alpha })}(z_{i-1/2},t) \ \ \ \text {for }z\in [z_{i-1/2},z_{i+1/2}), \end{aligned}$$

will converge in $L^1_{{\text {loc}}}$ to $\overline{W(w_\alpha )}$ as ${\Delta t}\rightarrow 0$. With this notation, (2.15) can be rewritten

$$\begin{aligned}&\int \limits _\mathbb {R}\int \limits _{{\Delta t}}^T \omega _{{\Delta t},\alpha }(z,t) \frac{1}{{\Delta t}}\left( \varphi (z,t)-\varphi (z,t-{\Delta t})\right) \,dtdz \nonumber \\&\qquad - \int \limits _\mathbb {R}\int \limits _{0}^{T} \mathcal {W}_{\Delta t}(z,t) \frac{1}{{\Delta z}}\left( \varphi (z,t)-\varphi (z-{\Delta z},t)\right) \,dtdz\nonumber \\&\quad =\int \limits _\mathbb {R}\omega _{{\Delta t},\alpha }(z,T) \frac{1}{{\Delta t}}\int \limits _{t_N-{\Delta t}}^{t_{N}} \varphi (z,t)\,dtdz - \int \limits _\mathbb {R}\omega _{{\Delta t},\alpha }(z,0) \frac{1}{{\Delta t}}\int \limits _{0}^{{\Delta t}} \varphi (z,t)\,dtdz. \end{aligned}$$

(2.17)

Now we can send ${\Delta t}$ to 0 in (2.17) and conclude that $w_\alpha$ is a (Lipschitz continuous) distributional solution of (2.14).

Finally, the assertion of uniqueness follows directly from the $L^1$ contraction principle stated in upcoming Theorem 5.3. $\square$

Finally, we will demonstrate a discrete entropy inequality for the filtered scheme. Although this inequality will not be used directly in our analysis, it serves as an important validation of the numerical scheme (see also Corollary 2.3). The inequality shows that as the filter size becomes increasingly small, the numerical scheme accurately captures the correct solution. This is a crucial aspect, as it ensures the accuracy and well-balanced nature of the scheme used.

Lemma 2.6

If CFL-condition (2.7) holds, then for any constant c

$$\begin{aligned} \left| w^{n+1}_i-c\right| \le \left| w^n_i-c\right| + \lambda \sum _{j\ge i}\Phi _{ij\alpha }\left( Q_c(w^n_{j+1})-Q_c(w^n_i)\right) , \end{aligned}$$

where $Q_c(w)=\mathop {\textrm{sign}}\limits \left( w-c\right) (W(w)-W(c))$.

Proof

For $\varvec{w}=\left\{ w_i\right\} _{i\in \mathbb {Z}}$ we define

$$\begin{aligned} G(\varvec{w})_i = w_i+\lambda \sum _{j\ge i}\Phi _{ij\alpha } \left( W(w_{j+1})-W(w_j)\right) , \end{aligned}$$

and observe that the mapping $\varvec{w}\mapsto G(\varvec{w})$ is monotone in the sense that if $v_i\le w_i$ for all i, then $G(\varvec{v})_i\le G(\varvec{w})_i$ for all i. Using G the scheme reads $w^{n+1}_i = G\left( \varvec{w}^n\right) _i$. Let $\varvec{c}$ denote the constant vector with all entries equal to the number c, $\max \left\{ \varvec{a},\varvec{b}\right\} _i=\max \left\{ a_i,b_i\right\}$, and $\min \left\{ \varvec{a},\varvec{b}\right\} _i=\min \left\{ a_i,b_i\right\}$. Then we have

$$\begin{aligned} G\left( \max \left\{ \varvec{w}^n,\varvec{c}\right\} \right) _i&=\max \left\{ \varvec{w}^n_i,c\right\} + \lambda \sum _{j\ge i}\Phi _{ij\alpha } \left( W\left( \max \left\{ w^n_{j+1},c\right\} \right) -W\left( \max \left\{ w^n_{j},c\right\} \right) \right) \\&\le \max \left\{ G(\varvec{w}^n)_i,c\right\} ,\\ G\left( \min \left\{ \varvec{w}^n,\varvec{c}\right\} \right) _i&=\min \left\{ w^n_i,c\right\} + \lambda \sum _{j\ge i}\Phi _{ij\alpha } \left( W\left( \min \left\{ w^n_{j+1},c\right\} \right) -W\left( \min \left\{ w^n_{j},c\right\} \right) \right) \\&\ge \min \left\{ G(\varvec{w}^n)_i,c\right\} . \end{aligned}$$

Subtracting these inequalities we get

$$\begin{aligned} \left| w^{n+1}_i-c\right|&=\max \left\{ G(\varvec{w}^n)_i,c\right\} -\min \left\{ G(\varvec{w}^n)_i,c\right\} \\&\le \max \left\{ w^n_i,c\right\} -\min \left\{ w^n_i,c\right\} \\&\qquad + \lambda \sum _{j\ge i} \Phi _{ij\alpha } \bigl [\left( W\left( \max \left\{ w^n_{j+1},c\right\} \right) -W\left( \min \left\{ w^n_{j+1},c\right\} \right) \right) \\&\qquad -\left( W\left( \max \left\{ w^n_{j},c\right\} \right) -W\left( \min \left\{ w^n_{j},c\right\} \right) \right) \bigr ]\\&\quad =\left| w^n_i-c\right| + \lambda \sum _{j\ge i} \Phi _{ij\alpha }\left( Q_c\left( w^n_{j+1}\right) -Q_c\left( w^n_i\right) \right) . \end{aligned}$$

$\square$

Recall that $w_\alpha$ is the Lipschitz continuous weak solution of (2.14), which is the filtered version of nonlocal Lagrangian PDE model (1.9). Using similar reasoning as in the proof of Theorem 2.5, it can be demonstrated that $w_\alpha$ satisfies the Kružkov entropy inequalities $\partial _t\left| w_\alpha -c\right| \le \partial _z\overline{Q_c(w_\alpha )}$, for $c\in \mathbb {R}$. In Sect. 5 we will show that a refined version of this entropy inequality is satisfied by any Lipschitz continuous weak solution of (2.14).

Remark 2.7

The unique form of the “filtered equation”, i.e. nonlocal PDE (2.14), suggests it can be interpreted as a fractional conservation law, where the spatial derivative is a fractional derivative operator. Recent studies, such as those referenced in [1, 2, 19] and many other others, have explored perturbations of conservation laws through the use of fractional diffusion or more general Lévy operators. This connection will be further clarified in the following.

Recall that the transport part of nonlocal PDE (2.14) can be written in the form

$$\begin{aligned} \partial _z\overline{W(w_\alpha )}(z,t) = \int \limits _0^\infty \bigl (-\Phi '_\alpha \bigr ) (\zeta ) \bigl [W(w_\alpha (z+\zeta ,t)) -W(w_\alpha (z,t))\bigr ]\,d\zeta . \end{aligned}$$

For motivational reasons, let us specify the kernel as $\Phi _\alpha (z)=e^{-z/\alpha }/\alpha$. Then it follows that $\bigl (-\Phi _\alpha '\bigr )(z)= \Phi _\alpha (z)/\alpha$ and $\int \limits _0^\infty \bigl (-\Phi _\alpha '\bigr )(z)\, dz=1/\alpha$, but note that $\int \limits _0^\infty z\bigl (-\Phi _\alpha '\bigr )(z)\, dz=1$.

Introducing the measure $\pi (dz)$ on $\mathbb {R}$ defined by

$$\begin{aligned} \pi (dz)= \bigl (-\Phi _\alpha '\bigr )(z) \chi _{(-\infty ,0]}(z)\,dz, \end{aligned}$$

which satisfies first moment condition $\int \limits _{\mathbb {R}} \left| z\right| \, \pi (dz)<\infty$, we may express the term $\partial _z\overline{W(w_\alpha )}(z,t)$ as $\int \limits _{\mathbb {R}}\bigl [W(w_\alpha (z+\zeta ,t)) -W(w_\alpha (z,t))\bigr ]\,\pi (d\zeta )$. Dropping the $\alpha$-subscript, nonlocal PDE (2.14) now becomes

$$\begin{aligned} \partial _tw=\int \limits _{\mathbb {R}}\bigl [W(w(z+\zeta ,t)) -W(w(z,t))\bigr ]\,\pi (d\zeta ). \end{aligned}$$

The measure $\pi (dz)$ depends discontinuously on the position z, which contrasts with studies such as [1, 2, 19]. Aiming for a generalised traffic flow model, we may treat $\pi (d\zeta )$ as a general Lévy measure, which describes the distribution of jumps in a Lévy process. In particular, one-sided Lévy processes (subordinators) may be relevant. A Lévy process is a stochastic process with independent and stationary increments and can be thought of as an extension of Brownian motion. Lévy processes and fractional derivatives can be used to model various types of anomalous diffusion phenomena, including the spread of information in complex transportation systems impacted by factors such as network structure, individual behaviour, and external disruptions. Fractional derivatives are nonlocal operators that account for long-range interactions and memory effects. A famous example of a Lévy measure is provided by $\pi (dz)=|z|^{-(1+\gamma )}\, \chi _{|z|<1} \, dz$, for $\gamma \in (0,2)$. This example is related to the fractional Laplacian $\Delta _\alpha :=-(-\Delta )^{\frac{\gamma }{2}}$on $\mathbb {R}$. For more information on Lévy processes, including one-sided processes (subordinators), see [25].

3 The nonlocal Lagrangian PDE for $y=1/u$

Let us discuss the relationship between the scheme for the filtered variable $w=\overline{y}$ and a (fully discrete) scheme for the original variable $y=1/u$. Assuming that the nonlocal operator $\overline{\;\cdot \;}$ is invertible (which is true for certain averaging kernels, such as $\Phi _\alpha (z)=e^{-z/\alpha }/\alpha$), then we can directly recover the values $\left\{ y_i^n\right\}$ from the values $\left\{ w_i^n\right\}$ computed via scheme (2.6). Alternatively, we can start from a fully discrete version of (2.4) for $y_i^n=1/u_i^n$:

$$\begin{aligned} y^{n+1}_i = y^n_i+\lambda \left( W(w_{i+1}^n)-W(w_i^n)\right) , \quad i\in \mathbb {Z},\,\, n\in \mathbb {N}, \end{aligned}$$

(3.1)

where, for $n=0$, $\left\{ y^0_i\right\}$ is an approximation of the initial function $y_0=1/u_0$, and $w_i^n= \sum _{j\ge i} \Phi _{ij\alpha }y_j^n$, $\Phi _{ij\alpha }=\int \limits _{z_{j-1/2}}^{z_{j+1/2}} \Phi _\alpha (\zeta -z_{i-1/2})\,d\zeta$, $i\in \mathbb {Z}$. This is an explicit upwind (Godunov-type) scheme for approximating solutions $y=1/u$ to nonlocal Lagrangian PDE (1.9). Applying the averaging operator $\overline{\;\cdot \;}$ to (3.1) leads to scheme (2.6) for the filtered variable $w_i^n=\overline{y}_i^n=\overline{\frac{1}{u_i^n}}$.

The ($\alpha$-independent) bound of the subsequent lemma implies that scheme (3.1) converges weakly to a limit $y_\alpha$, which will be proven later to be a solution of nonlocal PDE (1.11).

Lemma 3.1

Let $1\le y_0\in BV(\mathbb {R})$ be given. If CFL-condition (2.7) holds, then

$$\begin{aligned} \inf _{z\in \mathbb {R}} y_0(z)\le y^n_i\le \sup _{z\in \mathbb {R}} y_0(z), \end{aligned}$$

(3.2)

for every $\alpha >0$ and $i\in \mathbb {Z}$, $n\ge 0$, where $\left\{ y^n_i\right\} _{i,n}$ solves (3.1).

Proof

Introduce the notation

$$\begin{aligned} I_j=\int \limits _{z_j}^{z_{j+1}} \Phi _\alpha (\zeta )\,d\zeta \ \ \ \text {and} \ \ \ A^n_i = \frac{W(w^n_{i+1})-W(w^n_{i})}{w^n_{i+1}-w^n_i}\ge 0. \end{aligned}$$

By a summation by parts, the scheme for $y^n_i$ (3.1) can be written

$$\begin{aligned} y^{n+1}_i-y^n_i&=\lambda \left( W\left( w^n_{i+1}\right) -W\left( w^n_i\right) \right) =\lambda A^n_i \left( w^n_{i+1}-w^n_i\right) \\&=\lambda A^n_i \sum _{j=1}^\infty I_{j-1}\left( y^n_{i+j} - y^n_{i+j-1}\right) \\&=\lambda A^n_i\Bigl ( \sum _{j=1}^\infty \left( I_{j-1}-I_j\right) y^n_{i+j} - I_0y^n_i\Bigr )\hspace{3cm} \Bigl ( I_0=\sum _{j=1}^\infty \left( I_{j-1}-I_j\right) \Bigr ) \\&=\lambda A^n_i \sum _{j=1}^\infty \left( I_{j-1}-I_j\right) \left( y^n_{i+j}-y^n_i\right) , \end{aligned}$$

or

$$\begin{aligned} y^{n+1}_i=G(A^n_i,y_i^n,y_{i+1}^n,y_{i+2}^n,\ldots ), \end{aligned}$$

with the bilinear function G defined by

$$\begin{aligned} G(A,\varvec{y})=\left( 1-\lambda A I_0\right) y_1 + \lambda A \sum _{j=1}^\infty \left( I_{j-1}-I_j\right) y_{j+1}=y_1+ \lambda A \sum _{j=1}^\infty \left( I_{j-1}-I_j\right) \left( y_{j+1}-y_1\right) , \end{aligned}$$

for a number A and a vector $\varvec{y}=\left\{ y_i\right\} _{i=1}^\infty$. Observe that $G(A,y,y,y,\ldots )=y$ and that for fixed $A\ge 0$, the map $\left\{ y_i\right\} \mapsto G(A,\left\{ y_i\right\} )$ (by the CFL-condition and the fact that $I_{j-1}\ge I_j$) is monotone increasing in each argument $y_1,y_2,y_3,\ldots$. Set

$$\begin{aligned} \check{y}=\inf _{i\in \mathbb {Z}} y^n_i\ \ \ \text {and}\ \ \ \hat{y}=\sup _{i\in \mathbb {Z}} y^n_i. \end{aligned}$$

For any $i\in \mathbb {Z}$ and any $n\ge 0$

$$\begin{aligned} \check{y}&=G\left( A^n_i,\check{y},\check{y},\check{y},\ldots \right) \le G\left( A^n_i,y^n_i,y^n_{i+1},y^n_{i+2},\ldots \right) \\ {}&=y^{n+1}_i= G\left( A^n_i,y^n_i,y^n_{i+1},y^n_{i+2},\ldots \right) \le G\left( A^n_i,\hat{y},\hat{y},\hat{y},\ldots \right) =\hat{y}. \end{aligned}$$

Hence $\inf _{i\in \mathbb {Z}} y^n_i\le \inf _{i\in \mathbb {Z}} y^{n+1}_i\le \sup _{i\in \mathbb {Z}} y^{n+1}_i \le \sup _{i\in \mathbb {Z}} y^{n}_i$, and the lemma follows by induction. $\square$

We denote by $w_{{\Delta t},\alpha }(z,t)$ the bi-linear interpolation of the points $\left\{ (z_i,t^n,w^n_i)\right\}$ with $j\in \mathbb {Z}$, $n\ge 0$, and $t^n=n{\Delta t}$, recalling (3.1). Based on Theorem 2.5, we conclude that $w_{{\Delta t},\alpha }(z,t)$ converges uniformly on compacts to a Lipschitz continuous limit $w_\alpha (z,t)$ as ${\Delta t}\rightarrow 0$. The piecewise constant interpolation of the points $\left\{ (z_i,t^n,w^n_i)\right\}$ is denoted by $\omega _{{\Delta t},\alpha }(z,t)$, and it converges a.e. and thus in $L^1(K\times [0,T])$, $\forall K\subset \subset \mathbb {R}$. The piecewise constant interpolation of the points $\left\{ (z_i,t^n,y^n_i)\right\}$ is denoted by $y_{{\Delta t},\alpha }(z,t)$. Due to estimate (3.2), $y_{{\Delta t},\alpha }$ is bounded in $L^\infty (\mathbb {R}\times \mathbb {R}_+)$ uniformly in ${\Delta t}$ (and $\alpha$). Hence, there exists a subsequence $\left\{ y_{{\Delta t}_m,\alpha }\right\} _{m\in \mathbb {N}}$ that converges weak-$\star$ in $L^\infty (\mathbb {R}\times \mathbb {R}_+)$ to some limit $y_\alpha$. This implies that the functions $y_\alpha$, $w_\alpha$ satisfy (weakly) nonlocal Lagrangian PDE (1.11) with $w_\alpha =\overline{y_\alpha }$. By the uniqueness of solutions (from Remark 3.3), the entire sequence $\left\{ y_{{\Delta t},\alpha }\right\}$ converges. In summary, we have proved the following proposition:

Proposition 3.2

Suppose the assumptions of Theorem 2.5 hold. There exists a pair $\bigl (y_\alpha ,w_\alpha \bigr )$, with $1\le y_\alpha \in L^\infty (\mathbb {R}\times \mathbb {R}_+)$ and $w_\alpha \in \bigl ({\text {Lip}}_{{\text {loc}}} \cap L^\infty \bigr )(\mathbb {R}\times \mathbb {R}_+)$, such that the following convergences hold as ${\Delta t}\rightarrow 0$ (with $\alpha >0$ fixed):

$$\begin{aligned}&y_{{\Delta t},\alpha }\rightarrow y_\alpha \quad \text {weak}-\star \text { in }L^\infty (\mathbb {R}\times \mathbb {R}_+), \\ {}&w_{{\Delta t},\alpha }\rightarrow w_\alpha \text { uniformly on compacts of }\mathbb {R}\times \mathbb {R}_+. \end{aligned}$$

Besides, $\bigl (y_\alpha ,w_\alpha \bigr )$ is a weak solution of

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _ty_{\alpha } = \partial _zW(w_{\alpha }),&{}\quad z\in \mathbb {R},\,t>0, \\ \displaystyle w_{\alpha }(z,t)=\int \limits _{z}^{\infty } \Phi _\alpha \left( {\zeta -z}\right) y_{\alpha }(\zeta ,t) \,d\zeta , &{} \quad z\in \mathbb {R},\,t>0, \\ y_{\alpha }(z,0)=y_0(z), &{} \quad z\in \mathbb {R}. \end{array}\right. } \end{aligned}$$

(3.3)

Weak solutions from the class $L^\infty (\mathbb {R}\times \mathbb {R}_+)\times \bigl ({\text {Lip}}_{{\text {loc}}} \cap L^\infty \bigr )(\mathbb {R}\times \mathbb {R}_+)$ are uniquely determined by their initial data.

Remark 3.3

To conclude this section, we examine the stability of nonlocal Lagrangian PDE (3.3) in response to perturbations in the averaging kernel $\Phi$. Suppose $\Phi _1$ and $\Phi _2$ both adhere to the same assumptions outlined in (2.1) as $\Phi$. Consider the solutions $y_{1,\alpha }$ and $y_{2,\alpha }$ to (3.3) with $\Phi _{1,\alpha }$ and $\Phi _{2,\alpha }$ as the averaging kernels, see (2.2), and $y_{1,0}$, $y_{1,0}$ as the initial data. A simple calculation yields the stability estimate

$$\begin{aligned} \left\| y_{1,\alpha }(\cdot ,t)-y_{2,\alpha }(\cdot ,t)\right\| _{L^1(\mathbb {R})} \le&e^{ct/\alpha } \left\| y_{1,0}-y_{2,0}\right\| _{L^1(\mathbb {R})} \\ {}&+c\alpha \bigl (e^{ct/\alpha }-1\bigr ) \left\| \Phi _1-\Phi _2\right\| _{L^1(\mathbb {R})} +c\bigr (e^{ct/\alpha }-1\bigr ) \left\| \Phi _1'-\Phi _2'\right\| _{L^1(\mathbb {R})}. \end{aligned}$$

where c does not depend on $\alpha$.

4 Eulerian formulation

One can transform nonlocal Lagrangian PDE (3.3)—or (1.9)—into an Eulerian PDE via a change of variable, assuming that smooth solutions exist. However, this results in a complex and difficult-to-analyse Eulerian PDE. We only display this PDE here to highlight differences from other nonlocal Eulerian traffic flow equations, like (1.2). Wagner’s result [29] provides a rigourous framework for converting Lagrangian PDEs to Eulerian PDEs for weak solutions.

The Eulerian form of (1.9) reads

$$\begin{aligned}&\partial _t{{\widetilde{u}}} +\partial _x\left( {{\widetilde{u}}}\, V\left( \,\left[ \,\overline{\frac{1}{{{\widetilde{u}}}(x,t)}} \,\right] ^{-1}\,\right) \right) =0, \end{aligned}$$

(4.1)

$$\begin{aligned}&\overline{\frac{1}{{{\widetilde{u}}}(x,t)}} =\int \limits _{x}^{\infty }\Phi _\alpha \left( \, \int \limits _x^\sigma \widetilde{u}(\theta ,t) \,d\theta \,\right) \,d\sigma . \end{aligned}$$

(4.2)

We may rewrite (4.2) in a slightly clearer form. Since $0<u_*\le {{\widetilde{u}}}\le 1$, the function $\sigma \mapsto \int \limits _x^\sigma {{{\widetilde{u}}}(\theta ,t)} \,d\theta$ is invertible and $\int \limits _x^\infty \widetilde{u}(\theta ,t) \,d\theta =\infty$. Therefore, we may express $\overline{\frac{1}{\widetilde{u}}}$ at the point (x, t) as a weighted harmonic mean of $\widetilde{u}$ around different points $\ell \mapsto \bigl (\sigma (\ell , x,t),t\bigr )$:

$$\begin{aligned} \overline{\frac{1}{{{\widetilde{u}}}(x,t)}} =\int \limits _{0}^{\infty }\Phi _\alpha \left( \ell \right) \frac{1}{\widetilde{u}(\sigma (\ell , x,t),t)}\,d\ell , \end{aligned}$$

(4.3)

where $\sigma (\ell ,x,t)$ satisfies $\ell =\int \limits _x^{\sigma (\ell , x,t)}\widetilde{u}(\theta ,t) \,d\theta$; the new variable $\ell$ should not be confused with the $\ell$ appearing in (1.3).

Remark 4.1

Formally, by sending $\alpha \rightarrow 0$ in (4.1) and (4.2), we arrive at local LWR equation (1.1). To see this, note that the relation $\ell =\int \limits _x^\sigma \widetilde{u}(\theta ,t)\,d\theta$ implies $0=\int \limits _x^{\sigma (0, x,t)}{{{\widetilde{u}}}(\theta ,t)}\,d\theta$, from which we conclude that $\sigma (0,x,t)=x$. As a result, sending $\alpha \rightarrow 0$ in (4.3) yields

$$\begin{aligned} \int \limits _{0}^{\infty }\Phi _\alpha \left( \ell \right) \frac{1}{{\widetilde{u}}(\sigma (\ell , x,t),t)} \,d\ell \longrightarrow \frac{1}{\widetilde{u}(\sigma (0,x,t),t)} =\frac{1}{{\widetilde{u}}(x,t)}, \end{aligned}$$

and then (4.1) becomes (1.1): $\partial _t\widetilde{u}+\partial _x\bigl (\widetilde{u} V(\widetilde{u})\bigr )=0$.

Under the assumption of smooth solutions, we will outline a derivation of (4.1) and (4.2). For a derivation that works for weak solutions, see [29]. Let $\psi _t(z)$ satisfy

$$\begin{aligned} \partial _z\psi _t(z)=\frac{1}{u(z,t)},\qquad \partial _t\psi _t(z)=V\left( \,\left[ \,\overline{\frac{1}{u(z,t)}} \,\right] ^{-1}\, \right) . \end{aligned}$$

(4.4)

Denote by $\psi _t^{-1}(\cdot )$ the inverse of $\psi _t(\cdot )$, so that

$$\begin{aligned} \psi _t(\psi _t^{-1}(x))=x. \end{aligned}$$

(4.5)

Define

$$\begin{aligned} \widetilde{u}(x,t)=u(\psi _t^{-1}(x),t). \end{aligned}$$

(4.6)

Differentiating (4.5) with respect to x yields $\partial _z\psi _t(\psi _t^{-1}(x))\partial _x\psi _t^{-1}(x)=1$. Thus, by (4.4), $\partial _x\psi _t^{-1}(x)$ equals $1/\partial _z\psi _t(\psi _t^{-1}(x))=u(\psi _t^{-1}(x),t)$, and, thanks to (4.6),

$$\begin{aligned} \partial _x\psi _t^{-1}(x)=\widetilde{u}(x,t). \end{aligned}$$

(4.7)

Differentiating (4.5) with respect to t yields $\partial _z\psi _t(\psi _t^{-1}(x))\partial _t\psi _t^{-1}(x) +\partial _t\psi _t(\psi _t^{-1}(x))=0$. Hence, using (4.4) and (4.6),

$$\begin{aligned} \partial _t\psi _t^{-1}(x) =-u(\psi _t^{-1}(x),t) V\left( \,\left[ \,\overline{\frac{1}{u(\psi _t^{-1}(x),t)}} \,\right] ^{-1}\, \right) =-\widetilde{u}(x,t) V\left( \,\left[ \,\overline{\frac{1}{{{\widetilde{u}}}(x,t)}}\, \right] ^{-1}\,\right) . \end{aligned}$$

(4.8)

Using (4.6), (4.8), and (1.9) to express $\partial _t u(z,t)$ as $-u^2(z,t)\partial _z V\Bigl ( \,\left[ \, \overline{\frac{1}{u(z,t)}}\,\right] ^{-1}\, \Bigr )$, we obtain

$$\begin{aligned} \partial _t\widetilde{u}(x,t)&=\partial _zu(\psi _t^{-1}(x),t)\partial _t\psi _t^{-1}(x) +\partial _tu(\psi _t^{-1}(x),t) \\ {}&= -\partial _zu(\psi _t^{-1}(x),t) u(\psi _t^{-1}(x),t) \partial _zV\left( \,\left[ \,\overline{\frac{1}{u(\psi _t^{-1}(x),t)}} \,\right] ^{-1}\, \right) \\ {}&\qquad -u^2(\psi _t^{-1}(x),t) V\left( \,\left[ \,\overline{\frac{1}{u(\psi _t^{-1}(x),t)}} \,\right] ^{-1}\, \right) \\ {}&= -u(\psi _t^{-1}(x),t)\partial _z\left( u(\psi _t^{-1}(x),t) V\left( \,\left[ \,\overline{\frac{1}{u(\psi _t^{-1}(x),t)}} \,\right] ^{-1}\,\right) \right) . \end{aligned}$$

In view of (4.7) and (4.6), this yields

$$\begin{aligned} \partial _t\widetilde{u}(x,t)&= -\partial _x\psi _t^{-1}(x)\partial _z\left( u(\psi _t^{-1}(x),t) V\left( \,\left[ \,\overline{\frac{1}{u(\psi _t^{-1}(x),t)}} \,\right] ^{-1}\, \right) \right) \\ {}&=-\partial _x\left( u(\psi _t^{-1}(x),t) V\left[ \,\overline{\frac{1}{u(\psi _t^{-1}(x),t)}} \,\right] ^{-1}\,\right) =-\partial _x\left( \widetilde{u}(x,t) V\left( \,\left[ \,\overline{\frac{1}{\widetilde{u}(x,t)}} \,\right] ^{-1}\,\right) \right) , \end{aligned}$$

which is (4.1). Furthermore, using (4.6) and (1.10),

$$\begin{aligned} \overline{\frac{1}{\widetilde{u}(x,t)}}&=\overline{\frac{1}{u(\psi _t^{-1}(x),t)}} = \int \limits _{\psi _t^{-1}(x)}^{\infty } \Phi _\alpha \left( {\zeta -\psi _t^{-1}(x)}\right) \frac{1}{u(\zeta ,t)}\,d\zeta . \end{aligned}$$

Introduce the change of variable $\zeta =\psi _t^{-1}(\sigma )$ for $\sigma \in [x,\infty )$, so that $d\zeta =\partial _x\psi _t^{-1}(\sigma ) \, d\sigma =\widetilde{u}(\sigma ,t)\, d\sigma$, cf. (4.7) and (4.6). Then

$$\begin{aligned} \overline{\frac{1}{\widetilde{u}(x,t)}}&= \int \limits _{x}^{\infty } \Phi _\alpha \left( {\psi _t^{-1}(\sigma ) -\psi _t^{-1}(x)}\right) \,d\sigma =\int \limits _{x}^{\infty }\Phi _\alpha \left( \int \limits _x^\sigma {\partial _x\psi _t^{-1}(\theta )}\,d\theta \right) \,d\sigma \\&=\int \limits _{x}^{\infty }\Phi _\alpha \left( \int \limits _x^\sigma {\widetilde{u}(\theta ,t)} \,d\theta \right) \,d\sigma , \end{aligned}$$

which is (4.2).

Remark 4.2

For comparative purposes, let us discuss the relationship between Lagrangian and Eulerian variables in the “standard” nonlocal traffic flow equations (1.2), starting with the first equation. The macroscopic Lagrangian model corresponding to nonlocal FtL model (1.4) is

$$\begin{aligned} \partial _t \left( \frac{1}{u(z,t)}\right) -\partial _z V\left( \overline{u}(z,t) \right) =0, \quad z\in \mathbb {R}, \,\, t>0, \end{aligned}$$

where

$$\begin{aligned} \overline{u}(z,t)=\int \limits _{\psi _t(z)}^\infty \Phi _{\alpha }(\zeta -\psi _t(z))u(\psi _t^{-1}(\zeta ),t)\, d\zeta , \end{aligned}$$

(4.9)

and $\psi _t(z)$ satisfies the equations

$$\begin{aligned} \partial _z\psi _t(z)=\frac{1}{u(z,t)}, \qquad \partial _t\psi _t(z)=V\left( \overline{u}(z,t) \right) . \end{aligned}$$

By repeating the steps that led to (4.1) and (4.2), with necessary adjustments to account for the differences between (1.10) and (4.9), we derive the first Eulerian PDE in (1.2) for the function $\widetilde{u}(x,t)=u(\psi _t^{-1}(x),t)$. These adjustments include expressing (4.9) as

$$\begin{aligned} \overline{\widetilde{u}}(x,t) =\overline{u(\psi _t^{-1}(x),t)}&=\int \limits _{\psi _t(z)}^\infty \Phi _{\alpha }(\zeta -\psi _t(z)) u(\psi _t^{-1}(\zeta ),t)\, d\zeta =\int \limits _x^\infty \Phi _{\alpha }(\zeta -x) \widetilde{u}(\zeta ,t)\, d\zeta . \end{aligned}$$

Similarly, the macroscopic Lagrangian model corresponding to (1.6) takes the form

$$\begin{aligned} \partial _t \left( \frac{1}{u(z,t)}\right) -\partial _z \overline{V\left( u(z,t)\right) }=0, \quad z\in \mathbb {R}, \,\, t>0, \end{aligned}$$

where

$$\begin{aligned} \overline{V\left( u(z,t) \right) } =\int \limits _{\psi _t(z)}^\infty \Phi _{\alpha }(\zeta -\psi _t(z)) V(u(\psi _t^{-1}(\zeta ),t))\, d\zeta , \end{aligned}$$

and $\psi _t(z)$ satisfies

$$\begin{aligned} \partial _z\psi _t(z)=\frac{1}{u(z,t)}, \qquad \partial _t\psi _t(z)=\overline{V\left( u(z,t) \right) }. \end{aligned}$$

Using the same reasoning, the second Eulerian PDE in (1.2) is derived.

5 Zero-filter limit of the nonlocal model

In this section, we will examine a sequence of Lipschitz continuous weak solutions $w_\alpha$, indexed by the filter size $\alpha >0$, of the filtered version of nonlocal Lagrangian PDE (1.9), see (2.14) and Theorem 2.5. We will prove that these solutions have $\alpha$-independent estimates, precise entropy equalities, and converge to the unique entropy solution of original LWR equation (1.1) in Lagrangian coordinates.

Let $(\eta ,Q)$ be an entropy/entropy-flux pair, i.e. $\eta$ is a convex, twice continuously differentiable function and Q is a function satisfying $Q'(w)=\eta '(w)W'(w)$. Multiply (2.14) with $\eta '(w(z,t))$ to get

$$\begin{aligned} \partial _t\eta (w_\alpha )&=\partial _z\overline{Q(w_\alpha )} +\eta '(w_\alpha )\partial _z\overline{W(w_\alpha )} -\partial _z\overline{Q(w_\alpha )}\\ {}&=\partial _z\overline{Q(w_\alpha )}\\ {}&\quad + \int \limits _0^\infty \Phi '_\alpha (\zeta ) \bigl [ \left( \eta '(w_\alpha (z,t)) W(w_\alpha (z,t)) -Q(w_\alpha (z,t))\right) \\ {}&\quad \; - \left( \eta '(w_\alpha (z,t))W(w_\alpha (z+\zeta ,t)) -Q(w_\alpha (z+\zeta ,t))\right) \bigr ]\,d\zeta \\ {}&=\partial _z\overline{Q(w_\alpha )} + \int \limits _0^\infty \Phi '_\alpha (\zeta ) H(w_\alpha (z,t),w_\alpha (z+\zeta ,t))\,d\zeta , \end{aligned}$$

where, recalling that $W'(\cdot )\ge 0$,

$$\begin{aligned} H(a,b)&=\left[ \bigl (\eta '(a) W(a)-Q(a)\bigr ) - \bigl (\eta '(a)W(b)-Q(b)\bigr )\right] \\ {}&= \int \limits ^a \bigl (\eta '(a)-\eta '(\sigma )\bigr )W'(\sigma )\,d\sigma - \int \limits ^b \bigl (\eta '(a)-\eta '(\sigma )\bigr )W'(\sigma )\,d\sigma \\ {}&=\int \limits _a^b \bigl (\eta '(\sigma )-\eta '(a)\bigr )W'(\sigma )\,d\sigma =\int \limits _a^b \int \limits _a^\sigma \eta ''(\mu )\,d\mu \, W'(\sigma )\,d\sigma \ge 0. \end{aligned}$$

Since $\Phi '_a\le 0$, we have proved that a solution $w_\alpha$ of (2.14) satisfies an entropy (in)equality.

Theorem 5.1

Let $w_\alpha$ be a Lipschitz continuous distributional solution of (2.14), see Theorem 2.5. Then for any entropy/entropy-flux pair $(\eta ,Q)$

$$\begin{aligned} \partial _t\eta (w_\alpha (z,t))+D(z,t) =\partial _z\overline{Q(w_\alpha )}(z,t), \end{aligned}$$

(5.1)

where

$$\begin{aligned} D(z,t)= \int \limits _0^\infty \bigl (-\Phi _\alpha '\bigr )(\zeta ) \int \limits _{w_\alpha (z,t)}^{w_\alpha (z+\zeta ,t)}\int \limits _{w_\alpha (z,t)}^{\sigma } \eta ''(\mu ) W'(\sigma )\,d\mu \,d\sigma \,d\zeta \ge 0. \end{aligned}$$

Remark 5.2

For concrete choices of the entropy $\eta$ we obtain more precise estimates. If we suppose $\inf _{\mu ,\sigma } [\eta ''(\mu )W'(\sigma )] \ge 2c > 0$ for some constant c, then

$$\begin{aligned} \int \limits _a^b \int \limits _a^\sigma \eta ''(\mu )\,d\mu W'(\sigma )\,d\sigma \ge c(b-a)^2, \end{aligned}$$

and consequently

$$\begin{aligned} D(z,t)\ge c\int \limits _0^\infty \bigl (-\Phi '_\alpha \bigr )(\zeta ) \bigl (w_\alpha (z+\zeta ,t)-w_\alpha (z,t)\bigr )^2\,d\zeta \,\, (\ge 0). \end{aligned}$$

For example, specifying $\eta (w)=w^2/2$ and integrating (5.1) over $[-R,R]\times [0,T]$, we obtain the additional a priori estimate

$$\begin{aligned} \int \limits _0^T\int \limits _{-R}^R \int \limits _0^\infty \bigl (-\Phi '_\alpha \bigr )(\zeta ) \bigl (w_\alpha (z+\zeta ,t)-w_\alpha (z,t)\bigr )^2\,d\zeta \, dz\,dt\le C_R. \end{aligned}$$

If we use the Kružkov entropy

$$\begin{aligned} \eta (w)=\left| w-k\right| , \ \ \eta '(w)=\mathop {\textrm{sign}}\limits \left( w-k\right) , \ \ \eta ''(w)=2\delta _k(w), \end{aligned}$$

we obtain

$$\begin{aligned} H(w_\alpha (z,t),w_\alpha (z+\zeta ,t))= {\left\{ \begin{array}{ll} 2\left| w_\alpha (z+\zeta ,t)-w_\alpha (z,t)\right| &{} \text {if }k\text { is between }w_\alpha (z,t) \text { and }w_\alpha (z+\zeta ,t),\\ 0 &{} \text {otherwise.} \end{array}\right. } \end{aligned}$$

Thus for this choice

$$\begin{aligned} D(z,t)=2\int \limits _0^\infty \bigl (-\Phi _\alpha '\bigr )(\zeta ) \left| w_\alpha (z+\zeta ,t)-w_\alpha (z,t)\right| \chi _{[m(z,\zeta ),M(z,\zeta )]} (\zeta )\,d\zeta , \end{aligned}$$

where $\chi _I$ denotes the indicator function of the interval I and

$$\begin{aligned} m(z,\zeta )=\min \left\{ w_\alpha (z+\zeta ,t),w_\alpha (z,t)\right\} , \ \ \ M(z,\zeta )=\max \left\{ w_\alpha (z+\zeta ,t),w_\alpha (z,t)\right\} . \end{aligned}$$

Next we demonstrate that the Lipschitz continuous weak solutions of filtered PDE (1.9) exhibit continuity with respect to the initial data in the $L^1$ norm. Specifically, we show that the solution operator is $L^1$ contractive. It is important to note that solutions of (2.14) cannot be integrated over $\mathbb {R}$. However, the theorem below demonstrates that the difference between two solutions, if they are initially integrable, will be integrable over $\mathbb {R}$ at later times.

Theorem 5.3

Let $w_\alpha$ be a solution of (2.14) and let $v_\alpha$ be another solution with initial data $r_0$, see Theorem 2.5. If $y_0-r_0\in L^1(\mathbb {R})$, then $w_\alpha (\cdot ,t)-v_\alpha (\cdot ,t) \in L^1(\mathbb {R})$ for $t>0$, and

$$\begin{aligned} \left\| w_\alpha (\cdot ,t)-v_\alpha (\cdot ,t)\right\| _{L^1(\mathbb {R})}\le \left\| y_0-r_0\right\| _{L^1(\mathbb {R})}. \end{aligned}$$

In particular, Lipschitz continuous weak solutions are uniquely determined by their initial data.

Proof

Subtracting the equation for $v_\alpha$ from that of $w_\alpha$ we get

$$\begin{aligned} \partial _t\left( w_\alpha -v_\alpha \right) =\partial _z\left( \overline{W(w_\alpha )-W(v_\alpha )}\right) . \end{aligned}$$

Using the notation $\Delta W(z,t)=W(w_\alpha (z,t))-W(v_\alpha (z,t))$, we multiply this with $\mathop {\textrm{sign}}\limits \left( w_\alpha (z,t)-v_\alpha (z,t)\right) =\mathop {\textrm{sign}}\limits \left( \Delta W(z,t)\right)$ and get

$$\begin{aligned} \partial _t\left| w_\alpha -v_\alpha \right|&= \mathop {\textrm{sign}}\limits \left( w_\alpha -v_\alpha \right) \partial _z\left( \overline{W(w_\alpha )-W(v_\alpha )}\right) \nonumber \\&=\int \limits _0^\infty \Phi _\alpha '(\zeta ) \mathop {\textrm{sign}}\limits \left( \Delta W(z,t)\right) \left( \Delta W(z,t)-\Delta W(z+\zeta ,t)\right) \,d\zeta \nonumber \\ {}&\le \int \limits _0^\infty \Phi _\alpha '(\zeta ) \left( \left| \Delta W(z,t)\right| -\left| \Delta W(z+\zeta ),t\right| \right) \,d\zeta \nonumber \\ {}&=\partial _z\int \limits _0^\infty \Phi _\alpha (\zeta ) \left| \Delta W(z+\zeta ,t)\right| \,d\zeta =\partial _z\overline{\left| W(w_\alpha )-W(v_\alpha )\right| }. \end{aligned}$$

(5.2)

Let $\delta >0$ be a constant, define $f_\delta (z)=e^{-\delta \left| z\right| }$, and observe that

$$\begin{aligned} f'_\delta (z)=-\delta \mathop {\textrm{sign}}\limits \left( z\right) f_\delta (z),\ \ \left| f'_\delta (z)\right| \le \delta f_\delta (z). \end{aligned}$$

Multiply (5.2) with $f_\delta (z)$ and integrate in z to get

$$\begin{aligned} \frac{d}{dt} \int \limits _\mathbb {R}f_\delta (z)\left| w_\alpha (z,t)-v_\alpha (z,t)\right| \,dz&\le -\int \limits _\mathbb {R}f_\delta '(z) \overline{\left| \Delta W\right| }(z,t)\,dz =\delta \int \limits _\mathbb {R}\mathop {\textrm{sign}}\limits \left( z\right) f_\delta (z) \overline{\left| \Delta W\right| }(z,t)\,dz\\ {}&\le \delta \int \limits _0^\infty f_\delta (z) \overline{\left| \Delta W\right| }(z,t)\,dz\\ {}&= \delta \int \limits _0^\infty \overline{f_\delta \left| \Delta W\right| }(z,t)\,dz +\delta \int \limits _0^\infty f_\delta (z)\overline{\left| \Delta W\right| }(z,t)- \overline{f_\delta \left| \Delta W\right| }(z,t)\,dz \\ {}&\le \delta \left\| W'\right\| _\infty \int \limits _\mathbb {R}f_\delta (z) \left| w_\alpha (z,t)-v_\alpha (z,t)\right| \,dz\\ {}&\qquad +\delta \int \limits _0^\infty \int \limits _0^\infty \Phi _\alpha (\zeta ) \left( f_\delta (z)-f_\delta (z+\zeta )\right) \left| \Delta W(z+\zeta ,t)\right| \,d\zeta dz\\ {}&= \delta \left\| W'\right\| _\infty \int \limits _\mathbb {R}f_\delta (z) \left| w_\alpha (z,t)-v_\alpha (z,t)\right| \,dz\\ {}&\qquad +\delta \int \limits _0^\infty \int \limits _0^\infty \Phi _\alpha (\zeta ) e^{-\delta z}\left( 1-e^{-\delta \zeta }\right) \left| \Delta W(z+\zeta ,t)\right| \,d\zeta dz\\ {}&\le \delta \left\| W'\right\| _\infty \int \limits _\mathbb {R}f_\delta (z) \left| w_\alpha (z,t)-v_\alpha (z,t)\right| \,dz\\ {}&\qquad +M \int \limits _0^\infty \Phi _\alpha (\zeta ) \left( 1-e^{-\delta \zeta }\right) \,d\zeta \\ {}&\le \delta \left\| W'\right\| _\infty \int \limits _\mathbb {R}f_\delta (z) \left| w_\alpha (z,t)-v_\alpha (z,t)\right| \,dz +M \delta \int \limits _0^\infty \Phi _\alpha (\zeta )\zeta \,d\zeta \\ {}&=\delta \left\| W'\right\| _\infty \int \limits _\mathbb {R}f_\delta (z) \left| w_\alpha (z,t)-v_\alpha (z,t)\right| \,dz +M \delta c \alpha , \end{aligned}$$

where M is a bound on $\left| \Delta W\right|$ and $c=\int \limits _0^\infty \Phi (\zeta )\zeta \,d\zeta <\infty$, see (2.1). We invoke Gronwall’s inequality and obtain

$$\begin{aligned} \int \limits _\mathbb {R}f_\delta (z) \left| w_\alpha (z,t)-v_\alpha (z,t)\right| \,dz&\le e^{\delta \left\| W'\right\| _\infty t}\int \limits _\mathbb {R}f_\delta (z) \left| w_\alpha (z,0)-v_\alpha (z,0)\right| \,dz\\ {}&\quad + \frac{M c\alpha }{\left\| W'\right\| _\infty } \left( e^{\delta \left\| W'\right\| _\infty t} -1\right) . \end{aligned}$$

Since $w_\alpha (\cdot ,0)-v_\alpha (\cdot ,0)\in L^1(\mathbb {R})$, we can use the monotone convergence theorem to take the limit as $\delta \rightarrow 0$, and this concludes the proof. $\square$

The following lemma presents three estimates that do not depend on the parameter $\alpha$, and when taken together, they imply the local $L^1$ precompactness of the sequence $\left\{ w_\alpha \right\} _{\alpha >0}$. These estimates are modelled on the discrete estimates from Corollary 2.3.

Lemma 5.4

Let $w_\alpha$ be the unique Lipschitz continuous solution of (2.14), see Theorem 2.5. Then the following $\alpha$-independent estimates hold:

$$\begin{aligned}&\inf _x y_0(z)\le w_\alpha (z,t)\le \sup _z y_0(z,t), \end{aligned}$$

(5.3)

$$\begin{aligned}&\left| w_\alpha (\cdot ,t)\right| _{BV(\mathbb {R})} \le \left| y_0\right| _{BV(\mathbb {R})}, \end{aligned}$$

(5.4)

$$\begin{aligned}&\left\| w_\alpha (\cdot ,t)-w_\alpha (\cdot ,s)\right\| _{L^1(\mathbb {R})}\le \left| t-s\right| \,\left\| W'\right\| _\infty \left| y_0\right| _{BV(\mathbb {R})}. \end{aligned}$$

(5.5)

Proof

Note the translation invariance of $\Phi _\alpha$ in $\overline{\;\cdot \;}$, see the second part of (2.3). Consequently, choosing $v_\alpha (z,0)=w_\alpha (z+\zeta ,0)$ in Theorem 5.3, we conclude that $\left| w_\alpha (\cdot ,t)\right| _{BV(\mathbb {R})} \le \left| w_\alpha (\cdot ,0)\right| _{BV(\mathbb {R})}\le \left| y_0\right| _{BV(\mathbb {R})}$. This proves (5.4).

To prove (5.5), for $t>s$ we calculate

$$\begin{aligned}&\left\| w_\alpha (\cdot ,t)-w_\alpha (\cdot ,s)\right\| _{L^1(\mathbb {R})} \le \int \limits _\mathbb {R}\int \limits _s^t \left| \overline{\partial _zW(w_\alpha )}(z,\tau )\right| \,d\tau \,dz \\ {}&\qquad \le \int \limits _s^t \int \limits _\mathbb {R}\left| \partial _zW(w_\alpha (z,\tau ))\right| \,dz\,d\tau \\ {}&\qquad \le \left\| W'\right\| _\infty \int \limits _s^t \left| w_\alpha (\cdot ,\tau )\right| _{BV(\mathbb {R})}\,d\tau \le (t-s)\left\| W'\right\| _\infty \left| y_0\right| _{BV(\mathbb {R})}. \end{aligned}$$

It remains to prove (5.3). Let $a^+=\max \left\{ a,0\right\}$ and H(a) be the Heaviside function. By an approximation argument, the functions

$$\begin{aligned} \eta (w)=\left( w-k\right) ^+,\quad Q(w)=H(w-k)(W(w)-W(k)), \quad k\in \mathbb {R}, \end{aligned}$$

are admissible entropy/entropy-flux pairs. Since W is nondecreasing, $Q(w)=(W(w)-W(k))^+$. Using the notation of, and arguments similar to, the proof of Theorem 5.3 we find

$$\begin{aligned} \frac{d}{dt}\int \limits _\mathbb {R}f_\delta (z) \eta (w_\alpha (z,t))\,dz&\le - \int \limits _{0}^\infty f_\delta '(z) \overline{Q(w_\alpha )}(z,t)\,dz\\ {}&= \delta \int \limits _0^\infty \overline{f_\delta Q(w_\alpha )}(z,t)\,dz +\delta \int \limits _0^\infty f_\delta (z) \overline{Q(w_\alpha )}(z,t)-\overline{f_\delta Q(w_\alpha )}(z,t)\,dz\\ {}&\le \delta \int \limits _\mathbb {R}f_\delta (z) Q(w_\alpha (z,t))\,dz\\ {}&\qquad + \delta \int \limits _0^\infty \int \limits _0^\infty \Phi _\alpha (\zeta )\left( f_\delta (z)-f_\delta (z+\zeta )\right) Q(w_\alpha (z+\zeta ,t))\,d\zeta \,dz\\ {}&\le \delta \left\| W'\right\| _\infty \int \limits _\mathbb {R}f_\delta (z) \eta (w_\alpha (z,t))\,dz + M \delta c \alpha , \end{aligned}$$

where now M is a bound on Q. Next, Gronwall’s inequality yields

$$\begin{aligned} \int \limits _\mathbb {R}f_\delta (z) \eta (w_\alpha (z,t))\,dz&\le e^{\delta \left\| W'\right\| _\infty t}\int \limits _\mathbb {R}f_\delta (z) \eta (w_\alpha (z,0))\,dz + \frac{M c\alpha }{\left\| W'\right\| _\infty } \left( e^{\delta \left\| W'\right\| _\infty t} -1\right) . \end{aligned}$$

Thus if $w_\alpha (z,0)<k$ for almost all z then

$$\begin{aligned} \int \limits _\mathbb {R}f_\delta (z) \eta (w_\alpha (z,t))\,dz \le \frac{M c\alpha }{\left\| W'\right\| _\infty } \left( e^{\delta \left\| W'\right\| _\infty t} -1\right) , \end{aligned}$$

for all $\delta >0$. We send $\delta \rightarrow 0$ and conclude that if $w_\alpha (z,0)<k$ for almost all z, then $w_\alpha (z,t)<k$ for almost all z. The other inequality is proved using $\eta (w)=(w-k)^-$ and analogous arguments. $\square$

Consider now the scalar conservation law

$$\begin{aligned} \partial _tw = \partial _zW(w), \quad w(\cdot ,0)=y_0, \quad z\in \mathbb {R},\; t>0, \end{aligned}$$

(5.6)

which coincides with original LWR Eq. (1.1) written in Lagrangian coordinates, where $W(\cdot )=V(1/w)$, see (2.5), and V is the local speed function. By a solution of (5.6) we mean a distributional solution, i.e. a function $w=w(z,t)$ such that $w\in C([0,T];L^1_{\textrm{loc}}(\mathbb {R})) \cap L^\infty (\mathbb {R}\times [0,T])$, $T>0$, and

$$\begin{aligned} \int \limits _0^T \int \limits _\mathbb {R}w\partial _t\varphi - W(w)\partial _z\varphi \, dzdt = \int \limits _\mathbb {R}w(z,T)\varphi (z,T) - y_0(z)\varphi (z,0)\,dz, \end{aligned}$$

for all test functions $\varphi \in C^\infty _0(\mathbb {R}\times [0,T])$.

By an entropy solution of (5.6) we mean a weak solution which also satisfies

$$\begin{aligned} \int \limits _0^T \int \limits _\mathbb {R}\eta (w)\partial _t\varphi - Q(w)\partial _z\varphi \, dzdt \ge \int \limits _\mathbb {R}\eta (w(z,T))\varphi (z,T) - \eta (y_0(z))\varphi (z,0)\,dz, \end{aligned}$$

(5.7)

for all entropy/entropy-flux pairs $(\eta ,Q)$ and all non-negative test functions in $\varphi \in C^\infty _0(\mathbb {R}\times [0,T])$. If $y_0\in BV(\mathbb {R})$ (for example), there exists such unique entropy solution w of (5.6) [21].

By Lemma 5.4 the set $\left\{ w_\alpha \right\} _{\alpha >0}$ is precompact in $C([0,T];L^1_{{\text {loc}}}(\mathbb {R}))$, see e.g. [17, Theorem A.11]. Let $\left\{ \alpha \right\}$ be some subsequence such that $w=\lim _{\alpha \rightarrow 0} w_\alpha$ exists.

The following theorem demonstrates that the limit w satisfies the entropy inequalities, which identify the unique weak solution of (5.6). The fact that there is only one solution means that the entire sequence $\left\{ w_\alpha \right\}$ converges to w, rather than just a subsequence of it.

Theorem 5.5

Consider $W(\cdot )$ defined by (2.5) and an initial function $y_0\in BV(\mathbb {R})$ such that $1\le y_0$. Suppose the averaging kernel $\Phi _\alpha$ satisfies the conditions in (2.1) and (2.2). Then the limit $w=\lim _{\alpha \rightarrow 0} w_\alpha$ coincides with the unique entropy solution to (5.6).

Proof

Let $\varphi$ be a non-negative test function and define

$$\begin{aligned} \Upsilon (w)&=\int \limits _0^T \int \limits _\mathbb {R}\eta (w)\partial _t\varphi - Q(w)\partial _z\varphi \, dzdt - \int \limits _\mathbb {R}\eta (w(z,T))\varphi (z,T) - \eta (y_0(z))\varphi (z,0)\,dz,\\ \Upsilon _\alpha (w)&=\int \limits _0^T \int \limits _\mathbb {R}\eta (w)\partial _t\varphi - \overline{Q(w)}\partial _z\varphi \, dzdt - \int \limits _\mathbb {R}\eta (w(z,T))\varphi (z,T) - \eta (y_0(z))\varphi (z,0)\,dz. \end{aligned}$$

By Theorem 5.1$\Upsilon _\alpha (w_\alpha )\ge 0$. We write $\Upsilon (w)\ge \Upsilon _\alpha (w_\alpha )-\left| \Upsilon _\alpha (w_\alpha )-\Upsilon (w)\right| \ge -\left| \Upsilon _\alpha (w_\alpha )-\Upsilon (w)\right|$. Since $w_\alpha \rightarrow w$ in $C([0,T];L^1(\mathbb {R}))$, it is easily shown that $\left| \Upsilon _\alpha (w_\alpha )-\Upsilon (w)\right| \rightarrow 0$ as $\alpha \rightarrow 0$. Hence the limit w satisfies entropy inequality (5.7) which implies that w is a weak solution. $\square$

We have shown that $w_\alpha (\cdot ,t) \rightarrow w(\cdot ,t)$ in $L^1_{{\text {loc}}}$ as $\alpha \rightarrow 0$. By employing Kuznetsov’s lemma [17, Theorem 3.14] we can demonstrate that $w_\alpha \rightarrow w$ at a rate. For simplicity, we assume that $\lim _{\left| z\right| \rightarrow \infty } y_0(z)=c$ for some constant c. Since $v_\alpha =c$ is a solution of (2.14), Theorem 5.3 ensures that $w_\alpha (\cdot ,t)-c\in L^1(\mathbb {R})$. Since w solves scalar conservation law (5.6), by finite speed of propagation, $w(\cdot ,t)-c \in L^1(\mathbb {R})$ and thus $w_\alpha (\cdot ,t)-w(\cdot ,t)\in L^1(\mathbb {R})$. To state Kuznetsov’s lemma, we need some notation. Let $(\eta ,Q)$ be the Kružkov entropy/entropy-flux pair

$$\begin{aligned} \eta (w)=\left| w-k\right| , \quad Q(w,k)=\left| W(w)-W(k)\right| , \end{aligned}$$

and let

$$\begin{aligned} \begin{aligned} \Lambda _T(w,\varphi ,k)&=\int \limits _0^T \int \limits _\mathbb {R}\eta (w(z,t)) \partial _t\varphi (z,t)- Q(w(z,t),k)\partial _z\varphi (z,t)\, dzdt \\ {}&\qquad - \int \limits _\mathbb {R}\eta (w(z,T))\varphi (z,T) - \eta (y_0(z))\varphi (z,0)\,dz. \end{aligned} \end{aligned}$$

Let $\omega _\varepsilon$ be a standard mollifier and define the test function

$$\begin{aligned} \Omega _{\varepsilon _0,\varepsilon }(z,z',t,t')=\omega _{\varepsilon _0,\varepsilon }(t-t')\omega _\varepsilon (z-z'). \end{aligned}$$

Let $w_\alpha$ be the unique solution of (2.14) and let w be the entropy solution of (5.6). Observe that w and $w_\alpha$ share the same initial data. Finally define

$$\begin{aligned} \Lambda _{\varepsilon _0,\varepsilon }(w_\alpha ,w)= \int \limits _0^T\int \limits _\mathbb {R}\Lambda _T \left( w_\alpha ,\Omega \left( \cdot ,t',\cdot ,z'\right) ,w\left( z',t'\right) \right) \,dz'\,dt'. \end{aligned}$$

Since we know that $\left| w(\cdot ,t)\right| _{BV(\mathbb {R})}\le \left| y_0\right| _{BV(\mathbb {R})}$ and $\left| w_\alpha (\cdot ,t)\right| _{BV(\mathbb {R})}\le \left| y_0\right| _{BV(\mathbb {R})}$, in this context Kuznetsov’s lemma reads

$$\begin{aligned} \left\| w_\alpha (\cdot ,t)-w(\cdot ,t)\right\| _{L^1(\mathbb {R})} \le 2\left( \varepsilon +\left\| W'\right\| _\infty \varepsilon _0\right) \left| y_0\right| _{BV(\mathbb {R})}- \Lambda _{\varepsilon _0,\varepsilon }(w_\alpha ,w). \end{aligned}$$

This can be used to prove the following result quantifying the convergence $w_\alpha \rightarrow w$.

Theorem 5.6

Suppose the assumptions of Theorem 5.5 hold. Let $w_\alpha$ and w be solutions, respectively, of (2.14) and (5.6). Then

$$\begin{aligned} \left\| w_\alpha (\cdot ,t)-w(\cdot ,t)\right\| _{L^1(\mathbb {R})} \le 2\sqrt{2 T \left\| W'\right\| _\infty \left| y_0\right| _{BV(\mathbb {R})}\alpha }, \ \ \ \text {for }t\le T. \end{aligned}$$

Proof

Using Theorem 5.1

$$\begin{aligned} -\Lambda _{\varepsilon _0,\varepsilon }(w_\alpha ,w) =&-\overline{\Lambda }_{\varepsilon _0,\varepsilon }(w,w_\alpha ) + \overline{\Lambda }_{\varepsilon _0,\varepsilon }(w,w_\alpha ) - \Lambda _{\varepsilon _0,\varepsilon }(w_\alpha ,w)\\ {}&\le \left| \overline{\Lambda }_{\varepsilon _0,\varepsilon }(w,w_\alpha ) - \Lambda _{\varepsilon _0,\varepsilon }(w_\alpha ,w)\right| , \end{aligned}$$

where

$$\begin{aligned} \overline{\Lambda }_{\varepsilon _0,\varepsilon }(w_\alpha ,w)&= \int \limits _0^T\int \limits _\mathbb {R}\int \limits _0^T\int \limits _\mathbb {R}\left| w_\alpha (z,t)-w(z',t')\right| \partial _t\Omega (z,z',t,t')\\&\quad -\overline{Q(w_\alpha ,w(z',t'))}(z,t)\partial _z\Omega (z,z',t,t') \,dzdt\,dz'dt'\\ {}&\qquad - \int \limits _0^T\int \limits _\mathbb {R}\int \limits _\mathbb {R}\left| w_\alpha (z,T)-w(z',t')\right| \Omega (z,z',T,t')\\ {}&\qquad -\left| w_\alpha (z,0)-w(z',t')\right| \Omega (z,z',0,t')\,dz\,dz'dt'. \end{aligned}$$

Thus

$$\begin{aligned} \overline{\Lambda }_{\varepsilon _0,\varepsilon }(w,w_\alpha ) - \Lambda _{\varepsilon _0,\varepsilon }(w_\alpha ,w)&=\int \limits _0^T\int \limits _\mathbb {R}\int \limits _0^T\int \limits _\mathbb {R}\left( Q(w_\alpha (z,t),w(z',t'))-\overline{Q(w_\alpha ,w(z',t'))}(z,t)\right) \\ {}&\quad \times \partial _z\Omega (z,z',t,t') \,dzdt\,dz'dt'. \end{aligned}$$

Regarding the difference $Q(\ )-\overline{Q}(\ )$,

$$\begin{aligned} \left| Q(w_\alpha (z,t),w(z',t'))-\overline{Q(w_\alpha ,w(z',t'))}(z,t)\right|&=\Bigl | \int \limits _0^\infty \Phi _\alpha (\zeta ) \bigl (Q(w_\alpha (z,t),w(z',t'))\\ {}&\quad -Q(w_\alpha (z+\zeta ,t),w(z',t'))\bigr )\,d\zeta \Bigr |\\ {}&\le \left\| W'\right\| _\infty \int \limits _0^\infty \Phi _\alpha (\zeta ) \left| w_\alpha (z+\zeta ,t)-w_\alpha (z,t)\right| \,d\zeta . \end{aligned}$$

Therefore we can proceed as follows:

$$\begin{aligned} -\Lambda _{\varepsilon _0,\varepsilon }(w_\alpha ,w)&\le \left\| W'\right\| _\infty \int \limits _0^T\int \limits _\mathbb {R}\int \limits _0^T\int \limits _\mathbb {R}\int \limits _0^\infty \Phi _\alpha (\zeta ) \left| w_\alpha (z+\zeta ,t)-w_\alpha (z,t)\right| \\&\times \omega _{\varepsilon _0}(t-t')\omega _{\varepsilon }'(z-z') \,d\zeta \,dzdt\,dz'dt\\ {}&\le \left\| W'\right\| _\infty \int \limits _0^T \int \limits _0^\infty \Phi _\alpha (\zeta )\zeta \left| y_0\right| _{BV(\mathbb {R})} \frac{1}{\varepsilon } \,d\zeta \,dt\\ {}&\le T \left\| W'\right\| _\infty \left| y_0\right| _{BV(\mathbb {R})} \frac{\alpha }{\varepsilon }, \end{aligned}$$

where we have used (2.1). Hence

$$\begin{aligned} \left\| w_\alpha (\cdot ,t)-w(\cdot ,t)\right\| _{L^1(\mathbb {R})} \le 2\varepsilon + T \left\| W'\right\| _\infty \left| y_0\right| _{BV(\mathbb {R})} \frac{\alpha }{\varepsilon }, \end{aligned}$$

for $\varepsilon >0$. Minimising the right-hand side over $\varepsilon$ concludes the proof. $\square$

Theorems 5.5 and 5.6 state that as the filter size $\alpha$ approaches 0, the filtered variables $w_\alpha$, which are equal to $\overline{y_\alpha }$, converge strongly in $L^1_{{\text {loc}}}$ to the entropy solution of LWR conservation law (5.6). By Proposition 3.2, we know only that $y_\alpha$ converges weakly. The question of whether the Lagrangian variables $y_\alpha$ (spacing between cars) also converge strongly is a natural one, and our next result shows that this is true when using the exponential kernel.

Corollary 5.7

Suppose the assumptions of Theorem 5.5 hold, and specify $\Phi (\zeta )=e^{-\zeta }$. Let $y_\alpha$ and w be solutions, respectively, of (3.3) and (5.6). Then

$$\begin{aligned} \left\| y_\alpha (\cdot ,t)-w(\cdot ,t)\right\| _{L^1(\mathbb {R})} \le \alpha \left| y_0\right| _{BV(\mathbb {R})} +2\sqrt{2 T \left\| W'\right\| _\infty \left| y_0\right| _{BV(\mathbb {R})}\alpha }, \ \quad \text {for }t\in [0,T]. \end{aligned}$$

Proof

Due to the special choice of the function $\Phi$ we have the identity $-\alpha \partial _zw_\alpha +w_\alpha =y_\alpha$. Thus, using (5.4) and Theorem 5.6, we get

$$\begin{aligned} \left\| y_\alpha (\cdot ,t)-w(\cdot ,t)\right\| _{L^1(\mathbb {R})}&\le \left\| y_\alpha (\cdot ,t)-w_\alpha (\cdot ,t)\right\| _{L^1(\mathbb {R})} +\left\| w_\alpha (\cdot ,t)-w(\cdot ,t)\right\| _{L^1(\mathbb {R})} \\ {}&\le \alpha \left| w_\alpha (\cdot ,t)\right| _{BV(\mathbb {R})} +2\sqrt{2 T \left\| W'\right\| _\infty \left| y_0\right| _{BV(\mathbb {R})}\alpha } \\ {}&\le \alpha \left| y_0\right| _{BV(\mathbb {R})} +2\sqrt{2 T \left\| W'\right\| _\infty \left| y_0\right| _{BV(\mathbb {R})}\alpha }. \end{aligned}$$

$\square$

Remark 5.8

Let us examine conditions on the kernel $\Phi$ that enhance the weak convergence of ${y_\alpha }$ from Proposition 3.2 to strong convergence (to the limit w of $w_\alpha$). It appears that the only scenario is the one described in Corollary 5.7. Using (3.3),

$$\begin{aligned} \partial _zw_{\alpha }(z,t)&=-\Phi _\alpha (0)y_{\alpha }(z,t) -\int \limits _{0}^{\infty }\Phi _\alpha '\left( {\zeta }\right) y_{\alpha }(\zeta +z,t)\,d\zeta \\ {}&=\Phi _\alpha (0)(w_\alpha (z,t)-y_{\alpha }(z,t)) -\int \limits _{0}^{\infty }\left( \Phi _\alpha (0)\Phi _\alpha (\zeta ) +\Phi _\alpha '\left( {\zeta }\right) \right) y_{\alpha }(\zeta +z,t)\,d\zeta \\ {}&= \frac{\Phi (0)}{\alpha }(w_\alpha (z,t)-y_{\alpha }(z,t)) -\frac{1}{\alpha ^2}\int \limits _{0}^{\infty } \left( \Phi (0)\Phi \left( \frac{\zeta }{\alpha }\right) +\Phi '\left( \frac{\zeta }{\alpha }\right) \right) y_{\alpha }(\zeta +z,t)\,d\zeta . \end{aligned}$$

For every $R>0$, using (3.2) and (5.4),

$$\begin{aligned} \int \limits _{-R}^R&\left| w_\alpha (z,t)-y_{\alpha }(z,t)\right| \, dz \\ {}&\le \frac{\alpha }{\Phi (0)}\int \limits _{-R}^R \left| \partial _zw_{\alpha }(z,t)\right| \, dz +\frac{1}{\alpha \Phi (0)}\int \limits _{-R}^R\int \limits _{0}^{\infty } \left| \Phi (0)\Phi \left( \frac{\zeta }{\alpha }\right) +\Phi '\left( \frac{\zeta }{\alpha }\right) \right| y_{\alpha }(\zeta +z,t) \,d\zeta \,dz \\ {}&\le \frac{\alpha }{\Phi (0)}\left| w_\alpha (\cdot ,t)\right| _{BV(\mathbb {R})} +\frac{2R\left\| y_\alpha (\cdot ,t)\right\| _{L^\infty (\mathbb {R})}}{\Phi (0)} \int \limits _{0}^{\infty }\left| \Phi (0)\Phi \left( {\zeta }\right) +\Phi '\left( {\zeta }\right) \right| \,d\zeta \\ {}&\le \frac{\alpha }{\Phi (0)}\left| y_0\right| _{BV(\mathbb {R})} +\frac{2R\left\| y_0\right\| _{L^\infty (\mathbb {R})}}{\Phi (0)} \int \limits _{0}^{\infty }\left| \Phi (0)\Phi \left( {\zeta }\right) +\Phi '\left( {\zeta }\right) \right| \,d\zeta . \end{aligned}$$

Strong convergence is achieved only when the last term is zero, meaning $\Phi (0)\Phi \left( {\zeta }\right) +\Phi '\left( {\zeta }\right) =0$, which only holds when $\Phi (\zeta )=e^{-\zeta }$. Although numerical evidence suggests that strong convergence of $y_\alpha$ occurs for Lipschitz continuous kernels different from $e^{-\zeta }$, weak convergence (oscillations persist) is observed for BV (discontinuous) kernels in the limit as $\alpha \rightarrow 0$.

6 Numerical examples

This section presents three numerical experiments that showcase the features of our proposed model and compare it with established models in the field, giving a deeper understanding and valuable insights for future improvement.

6.1 Comparing different models

We compare solutions of the standard (local) LWR FtL model, the more sophisticated nonlocal FtL model given by (1.4), (1.5), and nonlocal FtL model (1.7), (1.8) proposed in this work.

Concretely, let the initial values (initial positions of vehicles) $x_i(0)=\tilde{x}_i(0)=\bar{x}_i(0)$ be specified as follows: Let $\ell$ be a small parameter (the length of a vehicle) and $\rho _0$ be a function such that $0<\rho _0(x)\le 1$ and that $\rho _0(x)$ is constant for x outside the interval (a, b). Then we set $x_1(0)=a$ and define $x_{i+1}(0)$, $u_i(0)$ by

$$\begin{aligned} \int \limits _{x_i(0)}^{x_{i+1}(0)} \rho _0(x)\,dx=\ell , \quad u_i(0)=\frac{\ell }{x_{i+1}(0)-x_i(0)}, \quad i=1,\ldots ,N, \end{aligned}$$

where N is the smallest integer such that $x_{N+1}(0)>b$. Finally, we set $u_{N+1}(0)=\rho _0(x_{N+1})$, $x_{N+1}=\infty$ and $u_0(0)=\rho _0(a-1)$. Given $\left\{ \tilde{x}_i\right\} _{i=1}^{N+1}$ with $x_{N+1}=\infty$ and $z_i=i\ell$ for $i=1,\ldots ,N$, $z_{N+1}=\infty$, define the $N\times N$ upper triangular matrices $\tilde{\Phi }_\alpha$ and $\overline{\Phi }_\alpha$ with entries

$$\begin{aligned} \tilde{\Phi }_{i,j,\alpha }=\int \limits _{\tilde{x}_j}^{\tilde{x}_{j+1}} \Phi _\alpha \left( \xi -\tilde{x}_i\right) \,d\xi , \ \ \ \ \overline{\Phi }_{i,j,\alpha }=\int \limits _{z_j}^{z_{j+1}} \Phi _\alpha \left( \zeta -z_i\right) \,d\zeta , \end{aligned}$$

respectively. Observe that $\tilde{\Phi }_{N,N,\alpha }=\overline{\Phi }_{N,N,\alpha }=1$. For $t>0$, $i=1,\ldots ,N-1$, let $x_i(t)$, $\tilde{x}_i(t)$, and $\bar{x}_i(t)$ solve

$$\begin{aligned} \text {(local FtL)} \qquad x_i'&=V(u_i),\ \ \ u_i = \frac{\ell }{x_{i+1}-x_i}, \end{aligned}$$

(6.1)

$$\begin{aligned} \text {(standard nonlocal FtL)} \qquad \tilde{x}_i'&=V(\tilde{u}_i),\ \ \ \tilde{u}_i = \sum _{j=i}^N\tilde{\Phi }_{i,j,\alpha } \frac{\ell }{\tilde{x}_{j+1}-\tilde{x}_j}, \end{aligned}$$

(6.2)

$$\begin{aligned} \text {(our nonlocal FtL)} \qquad \bar{x}_i'&=V(\bar{u}_i),\ \ \ \bar{u}_i = \Bigl (\, \sum _{j=i}^N \overline{\Phi }_{i,j,\alpha } \frac{\bar{x}_{i+1}-\bar{x}_i}{\ell }\, \Bigr )^{-1}, \end{aligned}$$

(6.3)

and $x_N'=\tilde{x}_N'=\bar{x}_N'=V(u_N)$, where V is a nonincreasing Lipschitz continuous function $V:[0,1]\mapsto [0,1]$ with $V(1)=0$. We define the piecewise constant function

$$\begin{aligned} u_\ell (x,t)= {\left\{ \begin{array}{ll} u_0 &{} x\le a,\\ u_i(t)&{} x_{i}(t)< x\le x_{i+1}(t), \ \ i=1,\ldots ,N-1,\\ u_{N}&{} x_N(t) < x. \end{array}\right. } \end{aligned}$$

The piecewise constant functions $\tilde{u}_\ell$ and $\bar{u}_\ell$ are defined analogously. To solve (6.1)–(6.3) numerically we utilise the explicit Euler scheme with ${\Delta t}=\ell$. In all our computations we use

$$\begin{aligned} V(v)=1-v\ \ \ \text {and}\ \ \ \Phi (\xi )=e^{-\xi }. \end{aligned}$$

We consider the (box) initial condition

$$\begin{aligned} \rho _0(x)= {\left\{ \begin{array}{ll} 1 &{} \left| x\right| <0.75,\\ 0.05 &{}\text {otherwise.} \end{array}\right. } \end{aligned}$$

(6.4)

If Fig. 1 we show a numerical solution to (6.1)–(6.3) computed with the explicit Euler scheme and $\alpha =0.5$ at $t=1.4$ for $\ell =0.06$ (left) and $\ell =0.005$ (right). It appears that the limits as $\ell \rightarrow 0$ of $\tilde{u}_\ell$ and $\bar{u}_\ell$ are different, and that both of these differ from the limit of $u_\ell$—the entropy solution of conservation law (1.1). We also observe that the limits of $\tilde{u}_\ell$ and $\bar{u}_\ell$ (as $\ell \rightarrow 0$) seem to have both positive and negative jumps and thus cannot satisfy an Oleinik-type entropy condition.

Fig. 1

Numerical solutions of (6.1)–(6.3) computed by the explicit Euler scheme. Left: $\ell =0.06$, right: $\ell =0.005$

The simulations show that when the speed is determined using weighted Lagrangian coordinates (6.3), vehicles drive faster compared to when the speed is determined by local FtL model (6.1) or Eulerian coordinates (6.2). This is because the Lagrangian distance between vehicles remains constant even if the Eulerian distance increases. The Lagrangian distance is always less than or equal to the Eulerian distance, giving the Lagrangian model more weight to spacings further ahead. As a result, in a decreasing density or thinly occupied road, the speed determined by the Lagrangian model is greater than or equal to that determined by the Eulerian model.

6.2 The zero-filter ($\alpha \rightarrow 0$) limit

We now study scheme (2.6) for $\alpha =1/2$, $\alpha =1/8$, $\alpha =1/32$, and $\alpha =1/128$ in order to compare $1/w_\alpha$ and $1/y_\alpha$ with $\rho$, where $\rho$ is the unique entropy solution of the local LWR model

$$\begin{aligned} \partial _t\rho +\partial _x(\rho V(\rho ))=0,\ \ \ \rho (x,0)=\rho _0(x). \end{aligned}$$

(6.5)

In this setting ($\rho _0=\textrm{const}$ outside an interval (a, b)), we define $u_i(0)$ and the matrix $\overline{\Phi }_\alpha$ as in the previous section and then define the initial data

$$\begin{aligned} y_i^0=\frac{1}{u_i(0)}\ \ \ \text {and}\ \ \ w^0_i = \sum _{j=i}^N \overline{\Phi }_{i,j,\alpha } y^0_j, \end{aligned}$$

(6.6)

for $i=1,\ldots ,N$. Set ${\Delta z}=\ell$, $\lambda ={\Delta t}/{\Delta x}$ where ${\Delta t}$ is chosen such that CFL-condition (2.7) holds. Let $w^n_i$ satisfy (2.6), which in this context reads

$$\begin{aligned} w_i^{n+1}=w^n_i+\lambda \Bigl (\sum _{j=i+1}^N \overline{\Phi }_{i+1,j,\alpha } W\left( w^n_j\right) - \sum _{j=i+1}^N \overline{\Phi }_{i,j,\alpha } W\left( w^n_j\right) \Bigr ), \end{aligned}$$

(6.7)

for $i=1,\ldots ,N$. The scheme for $y^n_i$ then reads

$$\begin{aligned} y_i^{n+1}=y^n_i+\lambda \left( W\left( w^n_{i+1}\right) -W\left( w^n_{i}\right) \right) , \end{aligned}$$

for $i=1,\ldots ,N$. It is not very elucidating to compare 1/y and 1/w with $\rho$ in Lagrangian coordinates, let therefore the “discrete Eulerian coordinates” $\xi _i^n$ be defined by

$$\begin{aligned} \xi ^n_1=x_1(0)+{\Delta t}\sum _{m=1}^n V\Bigl (\,\frac{1}{y^m_1}\,\Bigr ), \ \ \ \xi ^n_{i+1}=\xi ^n_i+y^n_i{\Delta z},\ \ i=1,\ldots ,N-1, \end{aligned}$$

cf. (1.3). Hence, we expect that

$$\begin{aligned} \frac{1}{w^n_i}\approx \rho (\xi ^n_i,t^n)\ \ \ \text {and}\ \ \ \frac{1}{y^n_i}\approx \rho (\xi ^n_i,t^n) \end{aligned}$$

for sufficiently small $\alpha$.

Fig. 2

Solutions of (6.7), (6.3), with initial data given in (6.4), (6.6). For all computations $t=1.2$ and $\ell =1/2000$. For comparisons we also show a numerical solution of (6.5). Upper left: $\alpha =1/2$, upper right: $\alpha =1/8$, lower left: $\alpha =1/32$, lower right: $\alpha =1/128$

Figure 2 shows 1/w, 1/y, and $\rho$ for different values of $\alpha$. In these plots the x axis is the Eulerian coordinates, i.e. we plot the points

$$\begin{aligned} \left( \xi ^n_i,1/w^n_i\right) \ \ \ \text {and}\ \ \ \left( \xi ^n_i,1/y^n_i\right) , \end{aligned}$$

for all relevant i, and n is such that $t^n=1.2$. The approximation to conservation law (6.5) is computed with the Engquist-Osher scheme on a fine grid. From this figure, we see that $\left\{ 1/w^n_i\right\} _{i=1}^N$ and $\left\{ 1/y^n_i\right\} _{i=1}^N$ approach $\left\{ \rho (\xi ^n_i,t^n)\right\} _{i=1}^N$ in $L^1$ as $\alpha$ traverses the sequence $\left\{ 1/2,1/8,1/32,1/128\right\}$.

Fig. 3

Solutions of (6.7), (6.3),with initial data given in (6.4), (6.6). For all computations $t=1.2$ and $\ell =1/5000$. In the left column, $\Phi =\Phi _{\textrm{tri}}$, in the right column, $\Phi =\Phi _{\textrm{box}}$

Fig. 4

Solutions of (6.7), (6.3),with initial data given in (6.4), (6.6), using the discontinuous filter $\Phi _{\textrm{box}}$ with $\alpha =1/256$ and $\ell =1/10000$. The figure to the right is just an enlargement of a region of the left figure

6.3 Convergence of $y_\alpha$ and the effect of different filters.

We proved that the filter $\Phi =\Phi _{\textrm{exp}}(z)=e^{-z}$ results in strong convergence of $y_\alpha$ to $1/\rho$, the entropy solution of local LWR conservation law (1.1). This convergence, which followed from $\left\| y_\alpha (\cdot ,t)-w_\alpha (\cdot ,t)\right\| _{L^1(\mathbb {R})} \lesssim \mathcal {O}(\alpha )$, was also seen in previous experiments. However, this strong convergence has only been proven for this specific filter and may not hold for others. To test this we experimented with other Lipschitz continuous filters:

$$\begin{aligned} \Phi _1(z)=\frac{4}{\pi }\frac{1}{(1+z^2)^2},\ \ \Phi _{\textrm{tri}}(z)=2\max \left\{ 1-z,0\right\} \ \ \text {and even}\ \ \Phi _2(z)=\frac{2}{\pi }\frac{1}{1+z^2}, \end{aligned}$$

although the last filter is not covered by the theory in this paper. Our numerical experiments show that $y_\alpha$ converges strongly for all filters. However, for the discontinuous filter $\Phi _{\textrm{box}}(z)=\chi _{(0,1)}(z)$, we observe weak convergence oscillations that persist as $\alpha \rightarrow 0$.

Oscillatory solutions can be attributed to stop-and-go traffic patterns [28]. Recall that stop-and-go traffic refers to a situation where cars frequently start and stop, resulting in waves of congestion that can propagate through a traffic flow and cause oscillations.

In Fig. 3 we compare computations using initial data (6.4), $\ell =1/5000$, and the filters $\Phi _{\textrm{tri}}$ (left column) and $\Phi _{\textrm{box}}$ (right column). In the first row $\alpha =1/32$ and in the second row $\alpha =1/128$.

From these computations, it is tempting to infer that (at least for these initial data) $y_\ell$ converges strongly to $1/\rho$ for the filter $\Phi _{\textrm{tri}}$ and only weakly to $1/\rho$ for the discontinuous filter $\Phi _{\textrm{box}}$. To substantiate our suspicion that $y_\ell$ only converges weakly, we did one final experiment in which we used the same initial data, but $\ell =1/10000$ and $\alpha =1/256$.

The result is depicted in Fig. 4. The right figure is a magnification of the region $x\in [0.2,0.3]$, $\rho \in [0.69,0.72]$ in the left figure.

Our experiment leads us to propose the conjecture that if a filter $\Phi$ is continuous, then the convergence of $y_\alpha$ to $1/\rho$ is strong. However, a proof has yet to be provided, except in the case of the exponential filter.