The Godunov Method for a 2-Phase Model

TL;DR

This paper applies the Godunov numerical method to a phase-transition traffic model, demonstrating its effectiveness through numerical tests and comparing it with a different existing model.

Contribution

It introduces the application of the Godunov method to a specific phase-transition traffic model and analyzes its differences with another established model.

Findings

Numerical tests confirm the validity of the Godunov method for this model.

The paper highlights key differences between the proposed model and an alternative model.

The method effectively captures phase transitions in traffic flow.

Abstract

We consider the Godunov numerical method to the phase-transition traffic model, proposed in [6], by Colombo, Marcellini, and Rascle. Numerical tests are shown to prove the validity of the method. Moreover we highlight the differences between such model and the one proposed in [1], by Blandin, Work, Goatin, Piccoli, and Bayen.

Figures

Equations

$$\left\{\begin{array}[]{l}\partial_{t}\rho+\partial_{x}\left(\rho\,v(\rho,\eta)\right)=0\vspace{.2cm}\\ \partial_{t}\eta+\partial_{x}\left(\eta\,v(\rho,\eta)\right)=0\end{array}\right.\quad\mbox{with}\quad v(\rho,\eta)=\min\left\{V_{\max},\frac{\eta}{\rho}\,\psi(\rho)\right\},$$

$${\partial }_t\rho +{\partial }_x\left(\rho V\right)=0,$$

$$\left\{\begin{array}[]{l}\partial_{t}\rho+\partial_{x}(\rho v)=0\vspace{.2cm}\\ \partial_{t}w+v\,\partial_{x}w=0\end{array}\right.\qquad\mbox{ with }\qquad v=\min\left\{V_{\max},\,w\,\psi(\rho)\right\}\,,$$

$$F$$

$$C$$

$$\begin{array}[]{@{}rcl@{\qquad}rcl@{}}\psi(0)\!\!&\!\!=\!\!&\!\!1,\!\!&\!\!\psi(R)\!\!&\!\!=\!\!&\!\!0,\\ \psi^{\prime}(\rho)\!\!&\!\!\leq\!\!&\!\!0,\!\!&\!\!\displaystyle\frac{d^{2}\ }{d\rho^{2}}\left(\rho\,\psi(\rho)\right)\!\!&\!\!\leq\!\!&\!\!0\quad\mbox{ for all }\rho\in[0,R]\,.\end{array}$$

$$\begin{array}[]{@{}rcl@{\quad}rcl@{}}\lambda_{1}(\rho,\eta)\!\!&\!\!=\!\!&\!\!\eta\,\psi^{\prime}(\rho)+v(\rho,\eta),&\lambda_{2}(\rho,\eta)\!\!&\!\!=\!\!&\!\!v(\rho,\eta),\\[5.0pt] r_{1}(\rho,\eta)\!\!&\!\!=\!\!&\!\!\left[\begin{array}[]{c}-\rho\\ -\eta\end{array}\right],&r_{2}(\rho,\eta)\!\!&\!\!=\!\!&\!\!\left[\begin{array}[]{c}1\\ \eta\left(\frac{1}{\rho}-\frac{\psi^{\prime}(\rho)}{\psi(\rho)}\right)\end{array}\right],\\ \nabla\lambda_{1}\cdot r_{1}\!\!&\!\!=\!\!&\!\!\displaystyle-\frac{d^{2}\ }{d\rho^{2}}\left[\rho\,\psi(\rho)\right],&\nabla\lambda_{2}\cdot r_{2}\!\!&\!\!=\!\!&\!\!0,\\ \mathcal{L}_{1}(\rho;\rho_{o},\eta_{o})\!\!&\!\!=\!\!&\!\!\displaystyle\eta_{o}\frac{\rho}{\rho_{o}},&\mathcal{L}_{2}(\rho;\rho_{o},\eta_{o})\!\!&\!\!=\!\!&\!\!\displaystyle\frac{\rho\,v(\rho_{o},\eta_{o})}{\psi(\rho)},\;\rho_{o}<R.\end{array}$$

$$\rho(0,x)=\left\{\begin{array}[]{l@{\quad\mbox{ if }}rcl}\rho^{l}&x&<&0\\ \rho^{r}&x&>&0\end{array}\right.\qquad\eta(0,x)=\left\{\begin{array}[]{l@{\quad\mbox{ if }}rcl}\eta^{l}&x&<&0\\ \eta^{r}&x&>&0\end{array}\right.$$

$$\begin{array}[]{ll}\left\{\begin{array}[]{ll}\partial_{t}\rho+\partial_{x}(\rho V)=0,&\textrm{ if }(\rho,q)\in\Omega_{f},\vspace{.2cm}\\ \left\{\begin{array}[]{l}\partial_{t}\rho+\partial_{x}(\rho v(\rho,q))=0,\\ \partial_{t}q+\partial_{x}(qv(\rho,q))=0.\end{array}\right.&\textrm{ if }(\rho,q)\in\Omega_{c},\end{array}\right.\end{array}$$

$$\begin{array}[]{l}\Omega_{f}=\left\{(\rho,q)\in[0,R]\times[0,+\infty[:\,q=\frac{R(\rho-\sigma)}{\sigma(R-\rho)},\,\,0\leq\rho\leq\sigma_{+}\right\}\vspace{.2cm}\\ \Omega_{c}=\left\{(\rho,q)\in[0,R]\times[0,+\infty[:\,v(\rho,q)\leq V,\,\,\frac{q^{-}}{R}\leq\frac{q}{\rho}\leq\frac{q^{+}}{R}\right\}\,.\end{array}$$

$$v(\rho,q)=\left\{\begin{array}[]{l@{\quad}l}V,&\textrm{ if }(\rho,q)\in\Omega_{f}\vspace{.2cm}\\ v^{eq}(\rho)(1+q)=\frac{V\sigma}{R-\sigma}\left(\frac{R}{\rho}-1\right)\left(1+q\right),&\textrm{ if }(\rho,q)\in\Omega_{c}\,,\end{array}\right.$$

$$\left(\rho_{0},\eta_{0}\right)(x)=\left\{\begin{array}[]{ll}(\rho^{l},\eta^{l})&\textrm{ if }x<0\\ (\rho^{r},\eta^{r})&\textrm{ if }x>0\end{array}\right.$$

$$\left(\rho_{0},q_{0}\right)(x)=\left\{\begin{array}[]{ll}(\rho^{l},q^{l})&\textrm{ if }x<0\\ (\rho^{r},q^{r})&\textrm{ if }x>0\end{array}\right.$$

$$\left\{\begin{array}[]{l}\frac{q^{m}}{\rho^{m}}=\frac{q}{R}\\ v_{c}(\rho^{m},q^{m})=v_{c}(\rho^{r},q^{r})\,.\end{array}\right.$$

$$\Lambda \left(({\rho }^l,q^l),({\rho }^m,q^m)\right)=\frac{{\rho }^{m}v\left({\rho }^m,q^m\right)-{\rho }^{l}V}{{\rho }^m-{\rho }^l}.$$

$$\left\{\begin{array}[]{l}\frac{q^{p}}{\rho^{p}}=\frac{q^{-}}{R}\\ \Lambda\left((\rho^{l},q^{l}),(\rho^{p},q^{p})\right)=\lambda_{1}(\rho^{p},q^{p})\,.\end{array}\right.$$

$$\begin{array}{rl}{u}_j^{n+1}& =\left({\rho }_j^{n+1},{\eta }_j^{n+1}\right)=u_j^n-\frac{\Delta t}{\Delta x}\left({\mathcal{F}}_{j+1/2}^n-{\mathcal{F}}_{j-1/2}^n\right)\\ & =\left({\rho }_j^n,{\eta }_j^n\right)-\frac{\Delta t}{\Delta x}\left[\left(F_{j+1/2}^n,G_{j+1/2}^n\right)-\left(F_{j-1/2}^n,G_{j-1/2}^n\right)\right],\end{array}$$

$$F^{n}_{j+1/2}=\left\{\begin{array}[]{l@{\qquad}l}\rho_{l}V_{\max}&\mbox{ if }\quad u_{l},u_{r}\in F\\ \eta_{m}\psi(\rho_{m})&\mbox{ if }\quad u_{l},u_{r}\in C\\ \rho_{\circ}V_{\max}&\mbox{ if }\quad u_{l}\in C,u_{r}\in F\\ \eta_{m}\psi(\rho_{m})&\mbox{ if }\quad u_{l}\in F,\,u_{r}\in C,\,\rho_{l}V_{\max}>\eta_{m}\psi(\rho_{m})\\ \rho_{l}V_{\max}&\mbox{ if }\quad u_{l}\in F,\,u_{r}\in C,\,\rho_{l}V_{\max}<\eta_{m}\psi(\rho_{m})\end{array}\right.$$

$$G^{n}_{j+1/2}=\left\{\begin{array}[]{ll}\eta_{l}V_{\max}&\mbox{ if }\quad u_{l},u_{r}\in F\\ \frac{\eta_{m}^{2}}{\rho_{m}}\psi(\rho_{m})&\mbox{ if }\quad u_{l},u_{r}\in C\\ \eta_{\circ}V_{\max}&\mbox{ if }\quad u_{l}\in C,\,u_{r}\in F\\ \frac{\eta_{m}^{2}}{\rho_{m}}\psi(\rho_{m})&\mbox{ if }\quad u_{l}\in F,\,u_{r}\in C,\,\rho_{l}V_{\max}>\eta_{m}\psi(\rho_{m})\\ \eta_{l}V_{\max}&\mbox{ if }\quad u_{l}\in F,\,u_{r}\in C,\,\rho_{l}V_{\max}<\eta_{m}\psi(\rho_{m})\,.\end{array}\right.$$

$$R=1,V_{\max }=60\text{Km/h},\check{w}=120\text{Km/h},\hat{w}=140\text{Km/h},$$

$$\rho_{0}(x)=\left\{\begin{array}[]{ll}1,&x\leq 500\\ 0,&x>500\end{array}\right.\quad w_{0}(x)=\left\{\begin{array}[]{ll}\check{w}+\frac{\hat{w}-\check{w}}{500}x,&x\leq 500\\ 0,&x>500.\end{array}\right.$$

$$T=300\text{s},\Delta x=1\text{m},\Delta t=0.7\frac{\Delta x}{V_{\max }}.$$

$$\rho_{0}(x)=\left\{\begin{array}[]{ll}1,&x\leq 500\\ 0,&x>500\end{array}\right.\quad w_{0}(x)=\left\{\begin{array}[]{ll}\hat{w}+\frac{\check{w}-\hat{w}}{500}x,&x\leq 500\\ 0,&x>500.\end{array}\right.$$

$$\begin{split}\rho_{0}(x)&=\left\{\begin{array}[]{ll}0,&x\leq 500\\ 0.2+0.5\,\frac{x-500}{2000},&500<x<2500\\ 0,&x>2500\end{array}\right.\\ w_{0}(x)&=\left\{\begin{array}[]{ll}0,&x\leq 500\\ \hat{w}+\frac{\check{w}-\hat{w}}{2000}\left(x-500\right),&500<x<2500\\ 0,&x\geq 2500.\end{array}\right.\end{split}$$

$$T=200\text{s},\Delta x=1\text{m},\Delta t=0.7\frac{\Delta x}{V_{\max }}.$$

Full text

2000 Mathematics Subject Classification: 35L65, 90B20

Key words and phrases: $2$–Phase Model, Continuum Traffic Models, Godunov Scheme, Hyperbolic Systems of Conservation Laws.

1 Introduction

Aim of this paper is to present the Godunov method for approximating the solutions to the $2$–phase traffic model, introduced in [6] by Colombo, Marcellini, and Rascle. The model consists on the system of conservation laws

[TABLE]

where $\rho=\rho(t,x)$ is the car density at time $t>0$ and at position $x\in{\mathbb{R}}$, $\eta=\eta(t,x)$ is a generalized momentum, $v=v(\rho,\eta)$ is the avarage speed of cars, $V_{\max}$ is a positive constant describing the maximum speed of cars, and $\psi=\psi(\rho)$ is a given decreasing function. This model is an extension of the famous Lighthill-Whitham-Richards (LWR) model, see [15, 17], and it is obtained by assuming that different kinds of drivers have different maximal speeds $w=w\left(\rho,\eta\right)$, where $w=\eta/\rho\in\left[\check{w},\hat{w}\right]$, for suitable constants $0<\check{w}<\hat{w}$.

The model described by (1.1) belongs to the class of hyperbolic phase transition models for traffic, whose aim is to describe the different traffic regimes; see [11]. The pioneer work in this class is the model proposed in 2002 by Colombo; see [3, 4, 5]. Its distinctive feature is that the two phases are disconnected. Subsequently other phase transition models have been introduced in the literature; see [1, 6, 9, 12, 16]. In particular, the models in [1, 6], although very similar in the fundamental diagram, are indeed different, since solutions to Riemann problems have different structures and contain a different number of waves. Also the derivation of the two models is completely different: the construction in [1] is done imposing a priori two phases, the free and the congested one, while in [6] the two phases are obtained as a consequence of the speed limit $V_{\max}$.

In the present paper, we describe the Godunov numerical scheme for system (1.1). The Godunov method is a finite volume method for conservation laws; it is based on the solution to the Riemann problem in order to approximate the flux between two contiguous cells; see [10, 14]. In our setting, the fundamental diagram in the conserved variable is convex; hence the application of this method is straightforward. Instead the nonconvexity of the fundamental diagram for the model in [1] is a source of various problems for the Godunov method; see [2, 18].

The paper is organized as follow. In Section 2 we describe the 2-Phase Traffic Model (1.1). In Section 3 we make a comparison between the $2$–Phase model (1.1) and that in [1]. Finally in Section 4, we describe the Godunov method to the model (1.1). Some numerical integrations conclude the paper.

2 Notations and Description of the Phase Transition Model

In this section we fix notations and we recall some properties concerning the $2$–Phase traffic model introduced in [6]. As already said, the model (1.1) is an extension of the classical LWR model, given by the following scalar conservation law

[TABLE]

where $\rho$ is the traffic density and $V=V(t,x,\rho)$ is the speed.

We consider the folllowing two assumptions on the speed:

• •

We assume that, at a given density, different drivers may choose different velocities, that is, we assume that $V=w\,\psi(\rho)$, where $\psi=\psi(\rho)$ is a $\mathbf{C^{2}}$ function and $w=w(t,x)$ is the maximal speed of a driver, located at position $x$ at time $t$.

• •

We impose an overall bound on the speed $V_{\max}$.

We get the following $2\times 2$ system

[TABLE]

and then the model in (1.1), with the change of variables $\eta=\rho w$.

As a consequence of the introduction of the speed bound $V_{\max}$, system (1.1) produces two phases, the free and the congested one, described by the sets

[TABLE]

In Figure 1 it is represented the fundalmental digram in the coordinates $(\rho,\eta)$ and $(\rho,\rho v)$. Note that $F$ and $C$ are closed sets and $F\cap C\neq\emptyset$. Note also that $F$ is one-dimensional in the $(\rho,\rho v)$ plane of the fundamental diagram, while it is two-dimensional in the $(\rho,\eta)$ coordinates.

We recall the following assumptions:

(H-1)

$R,\check{w},\hat{w},V_{\max}$ are positive constants, with $V_{\max}<\check{w}<\hat{w}$.

(H-2)

$\psi\in\mathbf{C^{2}}\left([0,R];[0,1]\right)$ is such that

[TABLE]

(H-3)

Waves of the first family in the congested phase $C$ have negative speed.

We recall the eigenvalues, right eigenvectors, and Lax curves $\eta=\mathcal{L}_{i}(\rho;\rho_{o},\eta_{o})$ in $C$:

[TABLE]

When $\rho_{o}=R$, the 2-Lax curve through $(\rho_{o},\eta_{o})$ is the segment $\rho=R$, $\eta\in[R\check{w},R\hat{w}]$.

In order to apply th Godunov’s method in Section 4, we recall the description of the solutions of the Riemann problem for the model (1.1). First, we recall all the possible waves in the solution.

• •

A Linear wave is a wave connecting two states in the free phase.

• •

A Phase Transition Wave is a wave connecting a left state $\left(\rho_{l},\eta_{l}\right)\in F$ with a right state $\left(\rho_{r},\eta_{r}\right)\in C$ satisfying $\frac{\eta_{l}}{\rho_{l}}=\frac{\eta_{r}}{\rho_{r}}$.

• •

A Wave of the First Family is a wave connecting a left state $\left(\rho_{l},\eta_{l}\right)\in C$ with a right state $\left(\rho_{r},\eta_{r}\right)\in C$ such that $\frac{\eta_{l}}{\rho_{l}}=\frac{\eta_{r}}{\rho_{r}}$.

• •

A Wave of the Second Family is a wave connecting a left state $\left(\rho_{l},\eta_{l}\right)\in C$ with a right state $\left(\rho_{r},\eta_{r}\right)\in C$ such that $v\left(\rho_{l},\eta_{l}\right)=v\left(\rho_{r},\eta_{r}\right)$.

2.1 The Riemann Problem

Under the assumptions (H-1), (H-2) and (H-3), for all states $(\rho^{l},\eta^{l})$, $(\rho^{r},\eta^{r})\in F\cup C$, the Riemann problem consisting of (1.1) with initial data

[TABLE]

admits a unique self similar weak solution $(\rho,\eta)=(\rho,\eta)(t,x)$ constructed as follows:

1. (1)

If $(\rho^{l},\eta^{l}),(\rho^{r},\eta^{r})\in F$, then the solution consists of a linear wave separating $(\rho^{l},\eta^{l})$ from $(\rho^{r},\eta^{r})$. 2. (2)

If $(\rho^{l},\eta^{l}),(\rho^{r},\eta^{r})\in C$, then the solution consists of a wave of the first family (shock or rarefaction) between $(\rho^{l},\eta^{l})$ and a middle state $(\rho^{m},\eta^{m})$, followed by a wave of the second family between $(\rho^{m},\eta^{m})$ and $(\rho^{r},\eta^{r})$. 3. (3)

If $(\rho^{l},\eta^{l})\in C$ and $(\rho^{r},\eta^{r})\in F$, then the solution consists of a wave of the first family separating $(\rho^{l},\eta^{l})$ from a middle state $(\rho_{\circ},\eta_{\circ})$ (which belongs to the intersection between $F$ and $C$) and by a linear wave separating $(\rho_{\circ},\eta_{\circ})$ from $(\rho^{r},\eta^{r})$. 4. (4)

If $(\rho^{l},\eta^{l})\in F$ and $(\rho^{r},\eta^{r})\in C$, then the solution consists of a phase transition wave between $(\rho^{l},\eta^{l})$ and a middle state $(\rho^{m},\eta^{m})$ (which is in $C$), followed by a wave of the second family between $(\rho^{m},\eta^{m})$ and $(\rho^{r},\eta^{r})$.

3 Comparison with the Model in [1]

In this section we briefly describe the $2$–Phase traffic model, introduced by Blandin, Work, Goatin, Piccoli and Bayen in [1], and we show the main differences with (1.1).

The phase transition traffic model in [1] has been developed as an extension of the Colombo phase transition model [3] and can be described by the following system

[TABLE]

where $\rho$ and $q$, represent respectively the car traffic density and the linearized momentum, $v=v\left(\rho,q\right)$ is the average speed of cars, while the sets $\Omega_{f}$ and $\Omega_{c}$, respectively the free and the congested phases, are given by

[TABLE]

In (3.2), $R>0$ denotes the maximal density, $V>0$ is the maximal velocity attained by the vehicles in the free phase, $q^{-}$ and $q^{+}$ are respectively the minimum and the maximum value of the momentum $q$, and the constants $\sigma$ and $\sigma_{+}$ are suitable constants in $[0,R]$. The velocity $v:\Omega_{f}\cup\Omega_{c}\to[0,+\infty[$ is defined by

[TABLE]

where the equilibrium velocity $v^{eq}$ represents the desired speed of cars when the density is $\rho$. In the congested phase $\Omega_{c}$, when $q=0$, the velocity of cars is the equilibrium velocity $v^{eq}$.

The models (1.1) and (3.1) have some similarities. Both models are described by two intersecting phases: the free phase and the two-dimensional congested phase. In Figures 1 and 2 the fundamental diagrams, respectively for (1.1) and for (3.1), are drawn. Note however that the free phase in (3.1) is one-dimensional both in the conserved quantity coordinates $\left(\rho,q\right)$ and in the coordinates $\left(\rho,\rho v\right)$. In the model (1.1) the free phase is two-dimensional in the conserved quantity coordinates $\left(\rho,\eta\right)$.

The main difference between (1.1) and (3.1) lies in the solution of the Riemann problem, in particular when the left state belongs to the free phase and the right state to the congested one.

The following result for (1.1) holds.

Proposition 3.1

Consider the system (1.1) and fix the states $(\rho^{l},\eta^{l})\in F$, $(\rho^{r},\eta^{r})\in C$. The Riemann problem for (1.1) with the initial condition

[TABLE]

is solved with at most two waves.

Proof.  Since $(\rho^{l},\eta^{l})\in F$ and $(\rho^{r},\eta^{r})\in C$, the solution to the Riemann problem consists of a shock wave connecting $(\rho^{l},\eta^{l})$ to a middle state $(\rho^{m},\eta^{m})$, possibly followed by a wave of the second family connecting $(\rho^{m},\eta^{m})$ to $(\rho^{r},\eta^{r})$; see Subsection 2.1 and Figure 3. This concludes the proof.   $\square$

The following result for (3.1) holds.

Proposition 3.2

Consider the system (3.1) and fix the states $(\rho^{l},q^{l})\in\Omega_{f}$, $(\rho^{r},q^{r})\in\Omega_{c}$. The Riemann problem for (3.1) with the initial condition

[TABLE]

is solved with at most three waves.

Proof.  We use here the notations in [1]. First define the intermediate state $(\rho^{m},q^{m})\in\Omega_{c}$ as he solution to the system

[TABLE]

Let $\Lambda\left((\rho^{l},q^{l}),(\rho^{m},q^{m})\right)$ be the speed of the phase transition wave connecting $(\rho^{l},q^{l})$ to $(\rho^{m},q^{m})$. Its value is given by the Rankine-Hugoniot condition

[TABLE]

Denoting with $\lambda_{1}(\rho^{m},q^{m})$ the first eigenvalue of the Jacobian matrix of the flux at $(\rho^{m},q^{m})$, the following possibilities hold.

$\Lambda\left((\rho^{l},q^{l}),(\rho^{m},q^{m})\right)\geq\lambda_{1}(\rho^{m},q^{m})$. In this case, the solution to the Riemann problem consists of a phase transition wave connecting $(\rho^{l},q^{l})$ to $(\rho^{m},q^{m})$, possibly followed by a contact discontinuity wave connecting $(\rho^{m},q^{m})$ to $(\rho^{r},q^{r})$; see Figure 3. 2. 2.

$\Lambda\left((\rho^{l},q^{l}),(\rho^{m},q^{m})\right)<\lambda_{1}(\rho^{m},q^{m})$. Define $(\rho^{p},q^{p})\in\Omega_{c}$ the solution to

[TABLE]

The solution to the Riemann problem consists of a phase transition wave connecting $(\rho^{l},q^{l})$ to $(\rho^{p},q^{p})$, of a rarefaction wave of the first family connecting $(\rho^{p},q^{p})$ to $(\rho^{m},q^{m})$, and of contact discontinuity wave connecting $(\rho^{m},q^{m})$ to $(\rho^{r},q^{r})$; see Figure 3.

The proof is so concluded.

$\square$

4 The Godunov Method

In this section we describe the Godunov method for the model (1.1) under the assumptions (H-1), (H-2), and (H-3); see [10, 13, 14]. We recall that such numerical scheme is a finite volume method, based on Riemann problems within computational cells in order to obtain the numerical fluxes.

Consider the system (1.1), with the conserved variables $(\rho,\eta)$; see Figure 1. Introduce the space discretization $\Delta x$ and the time dicretization $\Delta t$ satisfying the Courant-Friedrichs-Lewy conditions (see [7, 8]) as in [13]. Define $\nu=\Delta t/\Delta x$, and, for all $j\in{\mathbb{Z}}$ and all $n\in{\mathbb{N}}$, the points $x_{j+1/2}=j\Delta x$, $x_{j}=\left(j-1/2\right)\Delta x$, the time $t^{n}=n\Delta t$, and the cell $C_{j}^{n}=\left\{t^{n}\right\}\times\left[x_{j-1/2},x_{j+1/2}\right]$ of length $\Delta x$.

The Godunov method consists in the following steps.

Approximate an initial condition $\left(\rho_{0},\eta_{0}\right)$ by a function $\left(\rho_{\nu}(0,\cdot),\eta_{\nu}(0,\cdot)\right)$, constant in each cell $C_{j}^{0}$. Define $u_{j}^{0}=\left(\rho_{j}^{0},\eta_{j}^{0}\right)=\left(\rho_{\nu}(0,x_{j}),\eta_{\nu}(0,x_{j})\right)$ for every $j\in{\mathbb{Z}}$. 2. 2.

Construct, for every $n\in{\mathbb{N}}$ and $j\in{\mathbb{Z}}$, the values $u^{n+1}_{j}$ by the recurrence formula

[TABLE]

where $\mathcal{F}^{n}_{j+1/2}=\left({F}^{n}_{j+1/2},{G}^{n}_{j+1/2}\right)$ is a numerical flux at the interface $x_{j+1/2}$, obtained through the solution of corresponding Riemann problems.

If $j\in{\mathbb{Z}}$, the numerical flux $\mathcal{F}^{n}_{j+1/2}=\left(F^{n}_{j+1/2},G^{n}_{j+1/2}\right)$ depends on the values $u^{n}_{j}$ and $u^{n}_{j+1}$. Namely, denoting for simplicity with $u_{l}=\left(\rho_{l},\eta_{l}\right)$ and $u_{r}=\left(\rho_{r},\eta_{r}\right)$ respectively $u^{n}_{j}$ and $u^{n}_{j+1}$, we have

[TABLE]

and

[TABLE]

Above $(\rho_{m},\eta_{m})\in C$ is the middle state in the solution to the Riemann problem (2.4); see Subsection 2.1. Moreover the point $(\rho_{\circ},\eta_{\circ})\in F\cap C$ is the middle point in the solution to the Riemann problem (2.4) when $u_{l}\in C$ and $u_{r}\in F$; see point (3) of Subsection 2.1. Note that the condition $\rho_{l}V_{\max}$ respectively greater or less than $\eta_{m}\psi(\rho_{m})$ means that, by the Rankine-Hugoniot condition, the phase transition wave connecting $(\rho_{l},\eta_{l})\in F$ and $(\rho_{m},\eta_{m})\in C$ has strictly negative or positive speed.

4.1 Example: the case of a traffic light with increasing $w$

We present here a simple situation of a road with a traffic light, which turns green at the initial time $t=0$. At the beginning all the vehicles are stopped at the maximal density and $w$ is linearly increasing between $\check{w}$ and $\hat{w}$. We assume that the road is $3$Km long, modeled by the real interval $(0,3000)$ and the traffic light is located at position $x=500$.

We choose, for the model (1.1), the parameters

[TABLE]

with $\psi(\rho)=1-\rho$ and initial conditions

[TABLE]

In Figure 5 we show the contour plot for the numerical solution, obtained with the Godunov scheme, using the following numerical parameters:

[TABLE]

4.2 Example: the case of a traffic light with decreasing $w$

We consider here a situation very similar to that of Subsection 4.1. The only difference is that the initial $w$ is decreasing between $\check{w}$ and $\hat{w}$. As before, we assume that the road is $3$Km long, modeled by the real interval $(0,3000)$ and the traffic light is located at position $x=500$.

We choose, for the model (1.1), the parameters (4.4), the function $\psi(\rho)=1-\rho$, initial conditions

[TABLE]

In Figure 6 we show the contour plot for the numerical solution, obtained with the Godunov scheme, using the numerical parameters (4.6).

4.3 Example: increasing $\rho$ and decreasing $w$

In this part we consider an example where the initial conditions are monotone functions: the density $\rho$ is increasing, while the maximal speed $w$ is decreasing. We assume that the road is $3$Km long, modeled by the real interval $(0,3000)$.

We choose the parameters (4.4), the function $\psi(\rho)=1-\rho$, and the initial conditions

[TABLE]

In Figure 7 we show the contour plot for the numerical solution, obtained with the Godunov scheme, using the following numerical parameters:

[TABLE]

Acknowledgments

The authors were partial supported by the INdAM-GNAMPA 2016 project “Balance Laws: Theory and Applications”.