# A Macroscopic Traffic Model based on Driver Reaction and Traffic Stimuli
## 3 Roe Decomposition  
To evaluate the performance, the proposed and PW models are discretized using the Roe decomposition technique [38]. This decomposition approximates discontinuities and has been shown to provide consistent and accurate results for vehicular traffic flow models [39]. In vector form, the conserved form of these models is given by  
\[G_{t}+f(G)_{x}=S(G), \tag{19}\]  
where the subscripts \(t\) and \(x\) denote temporal and spatial derivatives, respectively. \(G\) denotes the data variables, \(f(G)\) denotes the vector of functions of the data variables, and \(S(G)\) is the vector of source terms. The system in Equation (19) can be represented in quasilinear form as  
\[\frac{\partial G}{\partial t}+A(G)\frac{\partial G}{\partial x}=0, \tag{20}\]  
where \(A(G)\) is the Jacobian matrix of the gradients of the functions of variables \(\rho\) and \(\rho v\). This matrix is used to find the eigenvalues and eigenvectors. The eigenvalues are not only useful to obtain approximate solutions but also to analyze traffic system hyperbolicity. The conserved form of the PW model is obtained by multiplying Equation (2) by \(v\)  
\[v\rho_{t}+v(\rho v)_{x}=0. \tag{21}\]  
Now, substituting  
\[v\rho_{t}=(\rho v)_{t}-\rho v_{t}. \tag{22}\]  
into Equation (21) gives  
\[\rho v_{t}=(\rho v)_{t}+v(\rho v)_{x}. \tag{23}\]  
Multiplying Equation (3) by \(\rho\) gives  
\[\rho v_{t}+\rho vv_{x}+C_{0}^{2}\rho_{x}=\rho\frac{v_{e}(\rho)-v}{\tau}. \tag{24}\]  
Now, consider  
\[\rho vv_{x}=(\rho vv)_{x}-v(\rho v)_{x}, \tag{25}\]  
and substituting Equations (23) and (25) into Equation (24) gives  
\[(\rho v)_{t}+(\rho vv)_{x}+C_{0}^{2}\rho_{x}=\rho\frac{v_{e}(\rho)-v}{\tau}. \tag{26}\]  
Multiplying and dividing \((\rho vv)_{x}\) by \(\rho\), we have  
\[(\rho vv)_{x}=\left(\frac{(\rho v)^{2}}{\rho}\right)_{x}, \tag{27}\] \begin{table}
\begin{tabular}{c c c} \hline
**Term** & **PW Model** & **Proposed Model** \\ \hline Anticipation & \(\frac{C_{0}^{2}}{\rho}\frac{\partial\rho}{\partial x}\) & \(\frac{-\frac{\partial v_{x}-v}{\rho}v_{x}^{\prime}(\rho)}{\rho}\frac{\partial \rho}{\partial x}\) \\ \hline Relaxation & \(\frac{v_{e}(\rho)-v}{\tau}\) & \(\frac{v_{e}(\rho)-v}{\tau}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Payne–Whitham (PW) and proposed model parameters. so that Equation (26) can be written as  
\[(\rho v)_{t}+\left(\frac{(\rho v)^{2}}{\rho}+C_{0}^{2}\rho\right)_{x}=\rho\frac{v _{e}(\rho)-v}{\tau}, \tag{28}\]  
which is the conserved form of the PW model [40]. This can be expressed in vector form as  
\[G=\left( \begin{array}{c}\rho\\ \rho v\end{array} \right),f(G)=\left( \begin{array}{c}\rho v\\ \frac{(\rho v)^{2}}{\rho}+{C_{0}}^{2}\rho\end{array} \right),S(G)=\left( \begin{array}{c}0\\ \rho\frac{v_{e}(\rho)-v}{\tau}\end{array} \right). \tag{29}\]  
The second equation of the proposed model in Equation (18) is given by  
\[v_{t}+vv_{x}-\frac{v_{m}-v}{h}v_{e}^{{}^{\prime}}(\rho)\frac{1}{\rho}\rho_{x}= \frac{v_{e}(\rho)-v}{\tau}, \tag{30}\]  
and multiplying by \(\rho\) gives  
\[\rho v_{t}+\rho vv_{x}-\frac{v_{m}-v}{h}v_{e}^{{}^{\prime}}(\rho)\rho_{x}= \rho\frac{v_{e}(\rho)-v}{\tau} \tag{31}\]  
Substituting Equations (23) and (25) into Equation (31), we have  
\[(\rho v)_{t}+\left(\frac{(\rho v)^{2}}{\rho}\right)_{x}-\frac{v_{m}-v}{h}v_{e }^{{}^{\prime}}(\rho)\rho_{x}=\rho\frac{v_{e}(\rho)-v}{\tau}, \tag{32}\]  
and since \(v_{e}(\rho)_{x}=v_{e}^{{}^{\prime}}(\rho)\rho_{x}\)  
\[(\rho v)_{t}+\left(\frac{(\rho v)^{2}}{\rho}-\frac{v_{m}-v}{h}v_{e}(\rho) \right)_{x}=\rho\frac{v_{e}(\rho)-v}{\tau}. \tag{33}\]  
The proposed model in vector form is then  
\[G=\left( \begin{array}{c}\rho\\ \rho v\end{array} \right),f(G)=\left( \begin{array}{c}\rho v\\ \frac{(\rho v)^{2}}{\rho}-\frac{v_{m}-v}{h}v_{e}(\rho)\end{array} \right),S(G)= \left( \begin{array}{c}0\\ \rho\frac{v_{e}(\rho)-v}{\tau}\end{array} \right). \tag{34}\]  
The Jacobian matrix for the PW model is  
\[A(G)=\left( \begin{array}{cc}0&1\\ -v^{2}+C_{0}^{2}&2v\end{array}