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

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

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

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

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:

where

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

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

Now the weights are determined by

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,^{Footnote 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

where

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

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

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.

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

For \(\alpha >0\) define

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

We have that

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

where

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

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

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

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

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

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

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

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

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

Proof

We compute

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

Proof

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

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

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

\(\square \)

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

and set

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

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

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

\(\square \)

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

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

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

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

and write (2.6) as

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

If we insert the definition of \(\varphi ^n_i\)

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

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

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

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

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

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

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

Subtracting these inequalities we get

\(\square \)

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

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

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

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].

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 \(y_i^n=1/u_i^n\):

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

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

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

or

with the bilinear function G defined by

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

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

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):

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

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

where c does not depend on \(\alpha \).

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

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 \(\ell \mapsto \bigl (\sigma (\ell , x,t),t\bigr )\):

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

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

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

Define

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),

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),

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

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

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

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

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

where

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

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

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

where

and \(\psi _t(z)\) satisfies

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

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

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

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 \((\eta ,Q)\)

where

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

and consequently

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

If we use the Kružkov entropy

we obtain

Thus for this choice

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

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

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

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

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

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

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

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:

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

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

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

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

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

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

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

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

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

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

and let

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

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

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

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

Proof

Using Theorem 5.1

where

Thus

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

Therefore we can proceed as follows:

where we have used (2.1). Hence

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

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

\(\square \)

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),

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

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 \(\alpha \rightarrow 0\).

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

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

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

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

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

We consider the (box) initial condition

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.

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

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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

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

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

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

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

cf. (1.3). Hence, we expect that

for sufficiently small \(\alpha \).

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: \(\alpha =1/128\)

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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

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 \(\left\{ 1/2,1/8,1/32,1/128\right\} \).

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, \(\Phi =\Phi _{\textrm{box}}\)

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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

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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:

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.