Models for dense multilane vehicular traffic

TL;DR

This paper develops a mathematical model for dense, multi-lane vehicular traffic, analyzing its well-posedness, bounds, and the limit as the number of lanes becomes large, resulting in a continuum traffic flow model.

Contribution

It introduces a multi-lane traffic model with lane-changing dynamics, proves its well-posedness, and derives a continuum limit as the number of lanes increases.

Findings

The multi-lane model is well-posed with specific bounds.

Lane-changing behavior is proportional to velocity differences.

The model converges to a continuum limit as lanes increase.

Abstract

We study vehicular traffic on a road with multiple lanes and dense, unidirectional traffic following the traditional Lighthill-Whitham-Richards model where the velocity in each lane depends only on the density in the same lane. The model assumes that the tendency of drivers to change to a neighboring lane is proportional to the difference in velocity between the lanes. The model allows for an arbitrary number of lanes, each with its distinct velocity function. The resulting model is a well-posed weakly coupled system of hyperbolic conservation laws with a Lipschitz continuous source. We show several relevant bounds for solutions of this model that are not valid for general weakly coupled systems. Furthermore, by taking an appropriately scaled limit as the number of lanes increases, we derive a model describing a continuum of lanes, and show that the $N$-lane model converges to a weak solution of the continuum model.

Figures

Equations

$$u_t+(uv(u){)}_x=0,$$

$${\partial }_t{u}_1+{\partial }_x(u_1{v}_1(u_1))$$

$${\partial }_t{u}_2+{\partial }_x(u_2{v}_2(u_2))$$

$$S(u_1,u_2)=K(v_2(u_2)-v_1(u_1))\cdot \{\begin{array}{ll}{u}_1& v_2(u_2)\ge v_1(u_1),\\ u_2& v_2(u_2)<v_1(u_1).\end{array}$$

$$\sum _{i=1}^N{\Vert u_i(t)-{\bar{u}}_i(t)\Vert }_{L^1(\mathbb{R})}\le \sum _{i=1}^N{\Vert u_{i,0}-{\bar{u}}_{i,0}\Vert }_{L^1(\mathbb{R})},$$

$$\{\begin{array}{l}{u}_t+kf(u{)}_x+(k^{\prime }f(u){)}_y={\left(ku{g}_y\right)}_y,\\ g(u{)}_y{|}_{y=0,1}=0,\\ u{|}_{t=0}=u_0,\end{array}$$

$$S(u_1,u_2)$$

$$=K\left[{\left(v_2(u_2)-v_1(u_1)\right)}^{+}{u}_1-{\left(v_2(u_2)-v_1(u_1)\right)}^{-}{u}_2\right]$$

$${\partial }_t{u}_1+{\partial }_x(u_1{v}_1(u_1))$$

$${\partial }_t{u}_2+{\partial }_x(u_2{v}_2(u_2))$$

$$\displaystyle\int_{0}^{\infty}\int_{\mathbb{R}}\big{(}u_{1}\varphi_{t}+u_{1}v_{1}(u_{1})\varphi_{x}-S(u_{1},u_{2})\varphi\big{)}\,dxdt+\int_{\mathbb{R}}u_{1,0}\varphi|_{t=0}\,dx$$

$$\displaystyle\int_{0}^{\infty}\int_{\mathbb{R}}\big{(}u_{2}\varphi_{t}+u_{2}v_{2}(u_{2})\varphi_{x}+S(u_{1},u_{2})\varphi\big{)}\,dxdt+\int_{\mathbb{R}}u_{2,0}\varphi|_{t=0}\,dx$$

$$\displaystyle\int_{0}^{\infty}\int_{\mathbb{R}}\big{(}\eta(u_{1})\varphi_{t}+q_{1}(u_{1})\varphi_{x}\big{)}\,dxdt+$$

$$\ge {\int }_0^{\infty }{\int }_{\mathbb{R}}{\eta }^{\prime }(u_1)\phi S(u_1,u_2)dxdt,$$

$$\displaystyle\int_{0}^{\infty}\int_{\mathbb{R}}\big{(}\eta(u_{2})\varphi_{t}+q_{2}(u_{2})\varphi_{x}\big{)}\,dxdt+$$

$$\ge -{\int }_0^{\infty }{\int }_{\mathbb{R}}{\eta }^{\prime }(u_2)\phi S(u_1,u_2)dxdt,$$

$$a^{\pm}=\frac{1}{2}\big{(}\left|a\right|\pm a\big{)},\quad H(a)=\mathbf{1}_{[0,\infty)}(a),$$

$$0$$

$$H(x)+H(-x)=1,(x^{+}{)}^{\prime }=H(x),(x^{-}{)}^{\prime }=-H(-x),x\ne 0.$$

$${\partial }_t(u_i{)}^{-}+{\partial }_x{q}_i^{-}(u_i)=(-1{)}^{i}H(-u_i)S(u_1,u_2)$$

$${\int }_{\mathbb{R}}(u_i(\tau ,x){)}^{-}dx\le {\int }_{\mathbb{R}}(u_{i,0}(x){)}^{-}dx+(-1{)}^{i+1}{\int }_0^{\tau }{\int }_{\mathbb{R}}H(-u_i)S(u_1,u_2)dxdt,$$

$${\int }_{\mathbb{R}}(u_1(\tau ,x){)}^{-}+(u_2(\tau ,x){)}^{-}dx\le {\int }_0^{\tau }{\int }_{\mathbb{R}}r(u_1,u_2)dxdt,$$

$$r(u_1,u_2)=S(u_1,u_2)(H(-u_1)-H(-u_2)).$$

$$r(u_1,u_2)$$

$$\le 0.$$

$${\partial }_t(u_i-1{)}^{+}+{\partial }_x{q}_i^{+}(u_i)\le (-1{)}^{i+1}H(u_i-1)S(u_1,u_2)$$

$${\int }_{\mathbb{R}}\left[(u_1(\tau ,x)-1{)}^{+}+(u_2(\tau ,x)-1{)}^{+}\right]dx\le {\int }_0^{\tau }{\int }_{\mathbb{R}}r(u_1,u_2)dxdt,$$

$$r(u_1,u_2)=S(u_1,u_2)(H(u_1-1)-H(u_2-1)).$$

$$r(u_1,u_2)$$

$$\le 0,$$

$$v_2\left(u_{2,0}(x)\right)\ge v_1\left(u_{1,0}(x)\right),$$

$$v_2\left(u_2(t,x)\right)\ge v_1\left(u_1(t,x)\right)$$

$$v_1(u)=1.5(1-u)\text{and}{v}_2(u)=2.5(1-u),$$

$$u_{1,0}(x)=u_{2,0}(x)={\sin }^2(\pi x/2).$$

$$S_i(u_i,u_{i+1})=\left[{\left(v_{i+1}(u_{i+1})-v_i(u_i)\right)}^{+}{u}_i-{\left(v_{i+1}(u_{i+1})-v_i(u_i)\right)}^{-}{u}_{i+1}\right],$$

Full text

The resulting model is a well-posed weakly coupled system of hyperbolic conservation laws with a Lipschitz continuous source. We show several relevant bounds for solutions of this model that are not valid for general weakly coupled systems.

Furthermore, by taking an appropriately scaled limit as the number of lanes increases, we derive a model describing a continuum of lanes, and show that the $N$-lane model converges to a weak solution of the continuum model.

Research was supported by the grant Waves and Nonlinear Phenomena (WaNP) (no 250070) from the Research Council of Norway.

1. Introduction

The Lighthill–Whitham–Richards (LWR) model for unidirectional traffic on a single road, see [13, 16], reads

[TABLE]

where $u=u(t,x)$ denotes the density of vehicles at the position $x$ and time $t$, and $v=v(u)$ is a given velocity function. The LWR-model expresses conservation of vehicles and is a well-established model for dense unidirectional single lane vehicular traffic on a homogeneous road without exits and entries. Furthermore, it serves as the standard textbook example to gain intuition regarding the behavior of solutions of scalar one-dimensional hyperbolic conservation laws, see, e.g., [10].

Given the importance of vehicular traffic modeling in modern society, it is no wonder that the LWR-model has been generalized to describe several important scenarios in dense traffic flow. Indeed, “traffic hydrodynamics” has become a research field in its own right, where the flow of vehicles is modeled by conservation laws or balance equations. In the general context, the LWR-model is the simplest model among the many hydrodynamic traffic models. Among the other models often used is the Aw–Rascle model [1], which is a system of conservation laws where the velocity $v$ is not a given function of $u$, but satisfies a second conservation law. It is thus considerably more complicated than the simple LWR-model. For a general introduction to how conservation laws are used in traffic modeling, see [9, 3] and the many references therein.

In this paper we introduce a new model for multilane dense vehicular traffic where the underlying model for each lane remains the LWR-model. Our basic assumption is that drivers prefer to drive faster, and that the tendency of a vehicle to change lane is proportional to the difference in velocity between neighboring lanes. If (1.1) describes the density of vehicles in a particular lane, the multilane behavior is described by a source term, accounting for lane changes. The result is thus a system of weakly coupled scalar conservation laws.

More precisely, consider two lanes denoted $1$ and $2$, the model we study, reads

[TABLE]

where the change of lanes is codified in

[TABLE]

Here $u_{i}$ denotes the density in lane $i$. The system constitutes a weakly coupled $2\times 2$ system of one-dimensional hyperbolic conservation laws, and there is ample theory available for systems of this type, see Section 2. The system readily generalizes to an arbitrary number of lanes, see Section 3. We show that the general system with $N$ lanes has a unique entropy solution, and that the solution is well-posed in the sense that one has a surprising $L^{1}$ stability

[TABLE]

for two solutions $u_{i}$ and $\bar{u}_{i}$, see Theorems 3.2 and 3.3. Note that the $L^{1}$ stability does not hold in general for systems of balance laws, that is, hyperbolic conservation laws with source.

The models invites for considering the continuum limit where the number of lanes increases to infinity. We organize the parallel lanes along the $x$-axis, and measure the distance between the lanes along the $y$-axis. The distance between the lanes is scaled as $\Delta y=1/N$, where $N$ denotes the number of lanes. For simplicity we assume that the velocity function is given by $v_{i}(u)=-k(y_{i})g(u)$ where $y_{i}=i\Delta y$, and $-g(u)$ is the velocity function. We scale the function such that $g(0)=-1$ and $g(1)=0$. We need to scale the constant $K$ as $K=1/\Delta y^{2}$. We consider given initial data $u_{0}\colon\mathbb{R}\times[0,1]\to[0,1]$, where the initial data for lane $i$ is $u_{i,0}$ is given by (4.20) and with solution $u_{i}$. We interpolate this function to $u_{\Delta y}$ where $u_{\Delta y}\colon[0,\infty)\times\mathbb{R}\times[0,1]\to[0,1]$. We assume that $k$ is smooth and positive with $k^{\prime}(0)=k^{\prime}(1)=0$. In Theorem 4.2 we show that $u_{\Delta y}\to u$ where $u$ is a weak solution of

[TABLE]

where the flux function $f$ is defined as $f(u)=uv(u)$. This equation is an interesting anisotropic and degenerate parabolic equation with non-trivial boundary conditions in the $y$-direction.

There is a plethora of approaches to the modeling of multilane dense traffic, and the most relevant to our approach here can be found in [5, 11, 12, 14], using either kinetic models or the Aw–Rascle model or variations thereof, or [2, 4] where more involved source terms modeling the change of lanes, are employed. See [15] for a survey of various models for lane changing.

The rest of this paper is organized as follows: In Section 2 we detail the two-lane case, and show that $u_{i}\in[0,1]$ is an invariant region. In Section 3 we state the $N$-lane model, and prove a number of estimates on the solution. Finally, in Section 4, we study the limit as $N\to\infty$. See [6] for a model for two-dimensional traffic flow on highways. Analogously to the analysis of numerical schemes for degenerate parabolic equations, we establish enough estimates on the solution, enabling us to conclude that a limit exists, and that this limit is a weak solution of a degenerate convection-diffusion equation. All sections are illustrated by numerical examples.

2. A continuum model for two-lane vehicular traffic

Consider a road with two lanes, each with its own velocity function. The lanes are homogeneous, and traffic on the road is unidirectional. We assume that the vehicular traffic is dense, allowing for a continuum formulation. Let $u_{i}$ and $v_{i}=v_{i}(u_{i})$ denote the density and velocity, respectively, in lane $i$.

In this paper we focus on the interaction between the two lanes. We assume that drivers prefer to drive in the faster lane, and the tendency of a vehicle to change lane is proportional to the difference in velocity. Thus the flow from lane $1$ to lane $2$ equals

[TABLE]

where $K$ is a constant, $(a)^{+}=\max\left\{a,0\right\}$ and $(a)^{-}=-\min\left\{a,0\right\}$. The flow from lane $2$ to lane $1$ equals $-S(u_{1},u_{2})$. The classical Lighthill–Whitham–Richards model implies the following model describing the two-lane traffic

[TABLE]

where $x$ is the position along the road and $t$ denotes time. This $2\times 2$ system of hyperbolic conservation laws is weakly coupled with a Lipschitz continuous source term.

The velocities $v_{i}=v_{i}(u_{i})$ are strictly decreasing positive functions, and we assume that they are scaled such that $v_{1}(1)=v_{2}(1)=0$. For simplicity, we scale space and time such that $K=1$.

It is well-known that this system in general only allows for weak solutions $u_{i}\in L^{1}(\mathbb{R})\cap BV(\mathbb{R})$, the set of integrable functions of finite total variation, see, e.g., [10]. Furthermore, the issue of uniqueness of the solution is non-trivial and one needs to require that the solution satisfies an entropy condition.

Definition 2.1**.**

Let $v_{i}=v_{i}(u_{i})$ be strictly decreasing positive functions such that $v_{1}(1)=v_{2}(1)=0$. Assume that $u_{i,0}\in L^{1}([0,1])\cap BV([0,1])$ for $i=1,2$. We say that $u_{i}\in C([0,\infty);L^{1}(\mathbb{R}))$ with $u_{i}(t,\,\cdot\,)\in BV(\mathbb{R})$ for $t\in[0,\infty)$ is a weak solution of (2.2) with initial data $u_{i,0}$ if

[TABLE]

for all compactly supported test functions $\varphi\in C^{\infty}_{0}([0,\infty)\times\mathbb{R})$.

The solution is called an entropy solution if

[TABLE]

for all convex functions $\eta$ where $q_{i}$ satisfies $q_{i}^{\prime}(u)=\eta^{\prime}(u)f_{i}^{\prime}(u)$ with $f_{i}(u)=uv_{i}(u)$, and for all compactly supported non-negative test functions $\varphi\in C^{\infty}_{0}([0,\infty)\times\mathbb{R})$, $\varphi\geq 0$.

Remark 2.2**.**

It suffices that (2.3) holds for $\eta$ of the form $\eta(u)=\left|u-k\right|$ for all constants $k\in\mathbb{R}$, see [10, Remark 2.1]. In that case $q_{i}(u)=\operatorname{sign}\left(u-k\right)(f_{i}(u)-f_{i}(k))$.

Remark 2.3**.**

The existence and uniqueness of entropy solutions to (2.2) follows by Theorem 3.2 below.

We will throughout the paper use the following notation:

[TABLE]

where $\mathbf{1}_{M}$ is the indicator (characteristic) function of a set $M$. Note that

[TABLE]

We shall also employ the convention that $C$ denotes a “generic” finite positive constant, independent of critical parameters, whose actual value may change from one occurrence to the next. Similarly, we use $C_{\alpha}$ to denote a positive function $c(\alpha)<\infty$ for $\alpha<\infty$.

This model (2.2) has the natural invariant region $u\in[0,1]$. This is the content of the following lemma.

Lemma 2.4**.**

Let $u_{1}$ and $u_{2}$ be entropy solutions in the sense of Definition 2.1, with initial data $u_{i,0}$ for $i=1,2$. If $u_{i,0}(x)\in[0,1]$ for all $x$ and $i=1,2$, then $u_{i}(t,x)\in[0,1]$ for all $x$ and for $t>0$.

Proof.

To show that $u_{i}\geq 0$ if $u_{i,0}\geq 0$ we use the entropy $\eta(u)=u^{-}$. Then

[TABLE]

in $\mathcal{D}^{\prime}$ for $i=1,\,2$. We use a non-negative test function $\varphi(x,t)\approx\mathbf{1}_{[0,\tau]}$ to find that

[TABLE]

Adding these two equations and using that $(u_{i,0})^{-}=0$, we get

[TABLE]

with

[TABLE]

We have that

[TABLE]

Hence $u_{i}(\tau,x)\geq 0$ for almost all $x$.

Similarly, by using the convex entropy $\eta(u)=(u-1)^{+}$ we get

[TABLE]

in $\mathcal{D}^{\prime}$, the set of distributions. By the same argument as before, we arrive at

[TABLE]

with

[TABLE]

We have that

[TABLE]

if $u_{1}$ and $u_{2}$ are non-negative. ∎

Remark 2.5**.**

There are also other invariant regions for this equation. If

[TABLE]

then

[TABLE]

for $t>0$.

2.1. An example

We finish our discussion of the two-lane case by exhibiting an example. The velocities on the two roads are

[TABLE]

and the initial data

[TABLE]

Of course, we do not have entropy solutions in closed form, so instead we use a numerical approximation generated by the Engquist–Osher scheme with $800$ grid points in the interval $[0,2]$. Figure 1 shows the computed solution at $t=0.375$, $t=0.75$, $t=1.125$ and $t=1.5$. For comparison, we have also included the single lane model with the (average of $v_{1}$ and $v_{2}$) speed $v(u)=2(1-u)$. We see that there is the expected change of lane to the faster lane, and that a shock builds up in the fast lane to the left of the shock in the slow lane.

3. Multilane model

The model (2.2) can be generalized to an arbitrary number of lanes. Consider a road with $N$ lanes. Traffic is unidirectional and dense. Each lane has its specific velocity function $v_{i}$ depending only on the density in that lane, thus $v_{i}=v_{i}(u_{i})$, where $u_{i}$ is the density in lane $i$.

Assume that drivers prefer to drive in the faster lane, and this tendency increases with the velocity difference with adjacent lanes. Thus the flow from lane $i$ to lane $i+1$ equals

[TABLE]

where we have scaled time such that the constant of proportionality is one. We then get, in the analogous manner to the derivation of (2.2), that

[TABLE]

coupled with the boundary conditions

[TABLE]

Definition 3.1**.**

Let $v_{i}=v_{i}(u_{i})$ be Lipschitz continuous functions, and assume that $u_{i,0}\in L^{1}(\mathbb{R})\cap L^{\infty}(\mathbb{R})$, for $i=1,\dots,N$. We say that $u_{i}\in C([0,\infty);L^{1}(\mathbb{R}))$ is a weak solution of (3.1) with initial data $u_{i,0}$ if

[TABLE]

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

It is an entropy solution if

[TABLE]

for all convex functions $\eta$, and for all non-negative test functions $\varphi\in C^{\infty}_{0}([0,\infty)\times\mathbb{R})$. Here $q_{i}$ is defined by $q_{i}^{\prime}(u)=\eta^{\prime}(u)f_{i}^{\prime}(u)$ with $f_{i}(u)=uv_{i}(u)$.

The wellposedness of the system of equations (3.1) is ensured by the following general theorem from [8], see also [7].

Theorem 3.2** ([7], Theorem 3.13).**

Assume that $v_{i}$ and $u_{i,0}$ are as in Definition 3.1. The there exists a unique entropy solution $u=\left\{u_{i}\right\}_{i=1}^{N}$. Furthermore, if $\bar{u}=\left\{\bar{u}_{i}\right\}_{i=1}^{N}$ is another entropy solution with initial data $\left\{\bar{u}_{i,0}\right\}_{i=1}^{N}$, then

[TABLE]

A fundamental property of hyperbolic conservation law is the $L^{1}$ contractivity of solutions in the sense that the spatial $L^{1}$-norm of the difference between two entropy solutions at a specific time does not increase in time. This property is in general lost for weakly coupled systems, or for scalar conservation laws with a source. The general bound (3.4) does not imply $L^{1}$ contractivity. However, for the system (3.1), the special form of the source yields $L^{1}$ contractivity for the whole solution, as the next theorem shows.

Theorem 3.3**.**

Consider two entropy solutions $u=\left\{u_{i}\right\}_{i=1}^{N}$ and $\bar{u}=\left\{\bar{u}_{i}\right\}_{i=1}^{N}$ of (3.1) with initial data $u_{0}=\left\{u_{i,0}\right\}$ and $\bar{u}_{0}=\left\{\bar{u}_{i}\right\}$, respectively. Then we have

[TABLE]

Proof.

By using Kružkov’s doubling of variables technique we get

[TABLE]

in $\mathcal{D}^{\prime}$. Subtracting the equation for $u_{i}$ and adding the equation for $\bar{u}_{i}$ we arrive at

[TABLE]

in $\mathcal{D}^{\prime}$. Choosing $\varphi\approx\mathbf{1}_{[0,\tau]}$ we infer that

[TABLE]

Recall that

[TABLE]

Now

[TABLE]

So if $u_{i}>\bar{u}_{i}$,

[TABLE]

where $0<c<\left|v_{i}^{\prime}\right|$. Therefore

[TABLE]

Define

[TABLE]

then (3.6) and the above inequality imply that

[TABLE]

Gronwall’s inequality then implies that

[TABLE]

Thus if $\Theta(0)=0$, i.e., $u_{i,0}(x)\leq\bar{u}_{i,0}(x)\;\text{a.e.}\;x$, then $\Theta(T)=0$ for $T>0$, i.e., $u_{i}(x,T)\leq\bar{u}_{i}(x,T)\;\text{a.e.}\;x$.

By the Crandall–Tartar lemma [10, Lemma 2.13], this implies $L^{1}$ contractivity, i.e., if $u$ and $\bar{u}$ are entropy solutions to (3.1) with initial data $u_{0}$ and $\bar{u}_{0}$, then (3.5) holds for $t>0$. ∎

One way to enforce the boundary conditions (3.2), is to define $u_{0}(x,t)=u_{1}(t,x)$, $v_{0}(u)=v_{1}(u)$, $u_{N+1}(x,t)=u_{N}(x,t)$ and $v_{N+1}(u)=v_{N}(u)$. Henceforth we will use this convention.

Corollary 3.4**.**

Consider two solutions $u_{i}$ and $\bar{u}_{i}$ of (3.1) with initial data $u_{i,0}$ and $\bar{u}_{i,0}$, respectively, in the sense of Definition 3.1. Then we have

[TABLE]

Furthermore, we have

[TABLE]

In addition

[TABLE]

Proof.

Setting $\bar{u}_{i,0}=u_{i+1,0}$ for $i=1,\ldots,N$ yields (3.7). Similarly, defining $\bar{u}_{i,0}(x)=u_{i,0}(x+h)$, using (3.5), and sending $h$ to zero gives (3.8). To obtain time continuity we define $\bar{u}_{i.0}(x)=u_{i}(x,h)$, to get (3.9). ∎

We also note the following useful estimates. Define $f_{i}(u)=uv_{i}(u)$ and $\Delta^{-}_{i}a_{i}=a_{i}-a_{i-1}$, divide by $h$ and let $h\downarrow 0$ to find that

[TABLE]

If we assume that the quantity on the left is bounded by $C$, then we get

[TABLE]

Furthermore, we have the useful observation

[TABLE]

3.1. An example

We also here include an example. For $i=1,\ldots,8$ we set $u_{i,0}(x)=\sin^{2}(\pi x/2)$, and define

[TABLE]

Also in this case the depicted solutions were calculated with the Engquist–Osher scheme with $800$ grid points in the interval $[0,2]$. Figure 2 shows the computed solutions at $t=0.375$, $t=0.75$, $t=1.125$ and $t=1.5$. We see the expected change of lanes to the faster lanes, and that a shock builds up in the faster lanes to the left of the slower lanes.

4. Infinitely many lanes — the continuum limit

It is natural, at least mathematically, to consider the case where the lanes increase in number while at the same time get closer. Our aim in this section is therefore to investigate limit as $N\to\infty$ in the system in the previous section.

To this end we let (the number of lanes) $N$ be a positive integer and set $\Delta y=1/N$. Let $y_{i}=(i-1/2)\Delta y$ for $i=1,\ldots,N$. We shall also use the “divided difference” notation

[TABLE]

For simplicity, we restrict our presentation to the case where $v_{i}(u)=-k(y_{i})g(u)$ where $g$ is a differentiable function with $g^{\prime}(u)>0$, $g(0)=-1$ and $g(1)=0$. Define $f(u)=-ug(u)$. Throughout we will use the notation $f_{i}=f(u_{i})$, $g_{i}=g(u_{i})$ and $k_{i}=k(y_{i})$. Now we reintroduce the scaling constant $K$ in (2.1), and set $K=\kappa/\Delta y^{2}$. For the convenience of the reader we set $\kappa=1$. Thus, for $i=1,\ldots,N$, $u_{i}$ is the unique entropy (in the sense of Definition 3.1) solution of the balance equation

[TABLE]

with the boundary conditions

[TABLE]

It is also useful to define the function $u_{\Delta y}(t,x,y)$ by

[TABLE]

We shall investigate whether the family $\left\{u_{\Delta y}\right\}_{\Delta y=1/N}$, $N\in\mathbb{N}$ is compact and characterize the limit $\lim_{\Delta y\to 0}u_{\Delta y}$. To this end we must show a number of estimates.

The right-hand side of (4.1) equals

[TABLE]

Thus (4.1) reads

[TABLE]

for $i=1,\ldots,N$, and we have the boundary values

[TABLE]

Remark**.**

Observe that the above term $b_{i}$ is an upwind discretization of the transport term corresponding to $au_{y}$, with $a=(kg)_{y}$.

Similarly to (4.3), we also get the expression

[TABLE]

Recall (3.3) with $\eta(u)=u^{2}/2$ and $\varphi$ an approximation to $\mathbf{1}_{[0,T]}$. That gives

[TABLE]

where $\Pi_{T}=[0,T]\times\mathbb{R}$. We can sum this for $i=1,\ldots,N$, multiply with $\Delta y$ and do a summation by parts to get

[TABLE]

It will be useful to lower bound the last two terms on the left-hand side.

Recall first that

[TABLE]

for some constant $C$ independent of $\Delta y$. Using this and the fact that $\max_{u\in[0,1]}\left|g(u)\right|$ is bounded, as well as

[TABLE]

we have that

[TABLE]

Furthermore, note that the same argument yields

[TABLE]

Observe that

[TABLE]

and then use the inequality $(a+b)^{-}\geq a^{-}-\left|b\right|$. Thus, since $g^{\prime}>0$,

[TABLE]

where $0<c\leq\min_{i}k_{i}\min_{u}g^{\prime}(u)$. Similarly,

[TABLE]

and therefore

[TABLE]

Note that due to the monotonicity of $g$ we have for some $\tilde{u}$ between $u_{i}$ and $u_{1-1}$,

[TABLE]

We can now estimate the last two terms of the left-hand side of (4.7) from below. More precisely,

[TABLE]

which we can rewrite as

[TABLE]

using (4.10) and (4.11).

This implies that

[TABLE]

and

[TABLE]

Observe that by (4.9), (4.12) follows from (4.13), viz.

[TABLE]

By the same procedure, starting with (4.4) but using the alternate form (4.6) of the right-hand side, we arrive at the bounds

[TABLE]

and

[TABLE]

Combining the two bounds (4.12) and (4.14) we get

[TABLE]

In a similar manner, we find

[TABLE]

The other two bounds, (4.13) and (4.15) can be used for a continuity estimate. Write $u_{i-1/2}=(u_{i}+u_{i-1})/2$ and compute for $\ell\geq m$

[TABLE]

Squaring and integrating over $[0,T]\times\mathbb{R}$ gives

[TABLE]

By direct computations we have that

[TABLE]

which gives

[TABLE]

Multiplying with $\Delta y$ summing over $i$ and integrating in $x$, $t$, gives the bound, using (4.18) with $m=i-1$, $\ell=i$ and (4.17),

[TABLE]

Note that this also follows from (4.13), using that $u_{i}\in[0,1]$.

Convergence

We assume that $u_{0}\colon\mathbb{R}\times[0,1]\to\mathbb{R}$ is such that $0\leq u_{0}(x,y)\leq 1$ and that $u_{0}\in L^{1}\cap BV$. Now we assume that the initial data $u_{i,0}$ are such that there is a function $u_{0}(x,y)$ such that

[TABLE]

where $\Delta y=1/N$ and $y_{i-1/2}=(i-1)\Delta y$. Furthermore $0\leq u_{0}(x,y)\leq 1$. Since $u_{0}\in BV(\mathbb{R}\times[0,1])$,

[TABLE]

for some constant $C$ which is independent of $\Delta y$. For convenience, we have set $u_{0,0}=u_{0,1}$ and $u_{0,N+1}=u_{0,N}$.

We assume that $k\in C^{1}([0,1])$ is given, such that $k^{\prime}(0)=k^{\prime}(1)=0$, and $k(y)>0$ for $y\in[0,1]$. Define $k_{i}=k(y_{i})$. Let $u_{i}(t,x)$ be the entropy solutions to (4.4) with the boundary conditions

[TABLE]

which actually is a special case of (4.5). Then we define

[TABLE]

for $i=1,\ldots,N-1$ and $u_{\Delta y}(t,x,y)=u_{N}(t,x)$ if $y\in[y_{N-1/2},1]$. We have that $0\leq u_{\Delta y}(t,x,y)\leq 1$, $\left\|u_{\Delta y}(t,\,\cdot\,,\,\cdot\,)\right\|_{L^{1}(\mathbb{R}\times[0,1])}=\left\|u_{0}\right\|_{L^{1}(\mathbb{R}\times[0,1])}$, and, using the bounds (3.7) and (3.8), $\left\|u_{\Delta y}(t,\,\cdot\,,\,\cdot\,)\right\|_{BV(\mathbb{R}\times[0,1])}\leq C$, where $C$ is independent of $\Delta y$. Furthermore, using (4.18),

[TABLE]

where $C$ is independent of $\Delta y$. This is sufficient to conclude that there is a function $u\in C([0,\infty);L^{1}(\mathbb{R}\times[0,1]))$ and a sequence $\left\{\Delta y_{j}\right\}_{j=0}^{\infty}$, $\Delta y_{j}\to 0$ as $j\to\infty$, such that

[TABLE]

Furthermore, we have that $D^{-}u_{\Delta y_{j}}\rightharpoonup u_{y}$, therefore $u_{\Delta y_{j}}D^{-}u_{\Delta y_{j}}\rightharpoonup uu_{y}$. The bound (4.19) ensures that $uu_{y}\in L^{2}([0,T]\times\mathbb{R}\times[0,1])$.

Definition 4.1**.**

Set $\Omega=\mathbb{R}\times[0,1]$ and $\Omega_{T}=[0,T]\times\Omega$. Let $k=k(y)$ be as above, in particular $k^{\prime}(0)=k^{\prime}(1)=0$. We say that $u\in C([0,\infty);L^{1}(\Omega))$, such that $uu_{y}\in L^{2}(\Omega_{T})$, is a weak solution to

[TABLE]

if for all test functions $\varphi\in C^{\infty}_{0}(\Omega_{T})$,

[TABLE]

The aim is now to show that the limit $u$ is a weak solution in the above sense. Since $u_{i}$ is a weak solution of (4.4), we have

[TABLE]

for $i=1,\ldots,N$. We use $\varphi=\varphi^{i}$ where

[TABLE]

for a suitable test function $\varphi$. Next we multiply with $\Delta y$ and sum over $i=1,\ldots,N$ and do a summation by parts on the terms which have $D^{\pm}(\cdots)$. This will give us the weak formulation for $u_{\Delta y}$. For simplicity we assume that $\Delta y=\Delta y_{j}$, so that the whole sequence converges. Term by term we get

[TABLE]

as $\Delta y\to 0$.

Turning to (4.22), we have that

[TABLE]

where $\tilde{u}_{i-1/2}$ is between $u_{i}$ and $u_{i-1}$ and $\xi_{i-1/2}$ is between $u_{i}$ and $\tilde{u}_{i-1/2}$. Therefore

[TABLE]

The last term here vanishes as $\Delta y\to 0$ since

[TABLE]

where we used (4.13). Hence

[TABLE]

as $\Delta y\to 0$.

Now for (4.23), we have

[TABLE]

using (4.8), (4.16), and interpolation between $L^{1}$ and $L^{3}$. Thus \Delta y\bigl{|}\sum_{i=1}^{N}\text{\eqref{eq:term3}}\bigr{|}\rightarrow 0 as $\Delta y\to 0$.

It is straightforward to show that

[TABLE]

Hence, the limit $u$ is a weak solution.

We can sum up the result of our arguments in the following theorem.

Theorem 4.2**.**

Let $k\in C^{2}([0,1])$ such that $k^{\prime}(0)=k^{\prime}(1)=0$, and $k(y)>0$ for all $y\in[0,1]$, and assume that $g=g(u)$ is a strictly increasing differentiable function such that $g(0)=-1$ and $g(1)=0$.

Assume that $u_{0}\in L^{1}(\Omega)\cap BV(\Omega)$ and let $u_{\Delta y}$ be defined in (4.2) where $u_{i}$ solves (4.1) for $i=1,\ldots,N$.

Then there exists a sequence $N_{j}\to\infty$ and correspondingly $\Delta y_{j}=1/N_{j}\to 0$ such that the sequence of solutions $\left\{u_{\Delta y_{j}}\right\}_{j=1}^{\infty}$ has a limit, i.e.,

[TABLE]

The limit $u$ is a weak solution according to Definition 4.1.

We also have the regularity estimate

[TABLE]

4.1. An example

To illustrate the continuum limit, we have tested the “same” initial value problem as in Section 2.1 and Section 3.1. The relevant data are

[TABLE]

and

[TABLE]

We have used $\Delta y=1/60$ (i.e., 60 lanes) and solved (4.1) using the Engquist–Osher scheme with $800$ grid points in the interval $[0,2]$. Figure 3 shows the computed density $u$ at four different times. It is illuminating to compare this figure with Figures 1 and 2.