## 5 ROE Discretization of Models
From Equation (34), Equation (35) gives the quadratic relation of eigenvalues as  
\[\lambda^{2}-2\
u\lambda+\
u^{2}-\left[\left(\frac{-0.03599+0.000314\rho}{-0.5 979+0.0134\
u}\right)\frac{1}{\left(h_{s}+\tau v\right)}\right]^{2}=0. \tag{35}\]  
From Equation (35), the eigenvalues are given in Equation (36)  
\[\lambda_{1,2}=\
u\pm\left(\frac{-0.03599+0.000314\rho}{-0.5979+0.0134\
u(\rho) }\right)\frac{1}{\left(h_{s}+\tau v\right)}, \tag{36}\]which gives the velocity changes. That is, a change in traffic behavior is based on traffic emissions. To optimize the emissions, the coefficients of density and velocity can be controlled. To find the eigenvectors, we use  
\[|A(u)-\lambda I|x=0, \tag{37}\]  
and \(x=\left( \begin{matrix}1\\ x_{2}\end{matrix} \right)\), which gives eigenvector \(e_{1}\) as  
\[e_{1}=\left[ \begin{matrix}1\\ \
u+\frac{1}{h_{s}+\tau v}\left(\frac{-0.03599+0.000314\rho}{-0.5979+0.0134 \
u(\rho)}\right)\end{matrix} \right].\]  
Then, \(e_{2}\) is  
\[e_{2}=\left[ \begin{matrix}1\\ \
u-\frac{1}{h_{s}+\tau v}\left(\frac{-0.03599+0.000314\rho}{-0.5979+0.0134 \
u(\rho)}\right)\end{matrix} \right].\]  
To obtain the average velocity, \(\Delta\)f is  
\[\Delta\text{f}=\left[ \begin{matrix}\Delta\text{f}_{1}\\ \Delta\text{f}_{2}\end{matrix} \right]=A(u)\Delta u \tag{38}\]  
\[\Delta\text{f}=\left[ \begin{matrix}0&1\\ \left[\frac{1}{h_{s}+\tau v}\left(\frac{-0.03599+0.000314\rho}{-0.5979+0.0134 \
u}\right)\right]^{2}-\
u^{2}&2\
u\end{matrix} \right]\left[ \begin{matrix} \Delta\rho\\ \Delta\rho v\end{matrix} \right]=0 \tag{39}\]  
From Equation (39),  
\[\Delta\text{f}_{2}=\left(-\
u^{2}+\left[\frac{1}{h_{s}+\tau v}\left(\frac{-0. 03599+0.000314\rho}{-0.5979+0.0134\
u(\rho)}\right)\right]^{2}\right)\Delta \rho+2\
u\Delta\rho v. \tag{40}\]  
where from Equation (31),  
\[\text{f}(u)=\left[ \begin{matrix}\text{f}_{1}\\ \text{f}_{2}\end{matrix} \right]=\left[ \begin{matrix}\rho v\\ \left(\frac{\rho v}{\rho}\right)^{2}+\left[\frac{1}{h_{s}+\tau v}\left(\frac{-0.03599+0.000314\rho}{-0.5979+0.0134\
u(\rho)}\right)\right]^{2}\rho\end{matrix}  \right], \tag{41}\]  
and \(\text{f}\left(u\right)=\left[ \begin{matrix}\Delta\text{f}_{1}\\ \Delta\text{f}_{2}\end{matrix} \right]\) from Equation (41) is  
\[\Delta\text{f}\left(u\right)\!=\left[ \begin{matrix}\Delta\text{f}_{1}\\ \Delta\text{f}_{2}\end{matrix} \right]=\left[ \begin{matrix}\Delta\rho v\\ \Delta\left(\frac{\left(\rho v\right)^{2}}{\rho}+\left[\frac{1}{h_{s}+\tau v} \left(\frac{-0.03599+0.000314\rho}{-0.5979+0.0134\
u(\rho)}\right)\right]^{2} \rho\right)\end{matrix} \right]. \tag{42}\]  
Then, we obtain  
\[\Delta\text{f}_{2}=\Delta\left(\frac{\left(\rho v\right)^{2}}{\rho}+\left[ \frac{1}{h_{s}+\tau v}\left(\frac{-0.03599+0.000314\rho}{-0.5979+0.0134\
u( \rho)}\right)\right]^{2}\rho\right). \tag{43}\]  
Comparing Equation (40) with (43), the quadratic relation of velocity is  
\[\
u^{2}\Delta\rho-2\
u\Delta(\rho v)+\Delta\left(\rho v^{2}\right)=0 \tag{44}\]  
Thus, the solution of Equation (44) is  
\[\
u=\frac{2\Delta(\rho v)+2\sqrt{\left(\Delta\rho v\right)^{2}-\left(\Delta \rho\right)\left(\Delta\rho v^{2}\right)}}{2\Delta\rho}. \tag{45}\]\(\Delta\rho v=\rho_{i+1}v_{i+1}-\rho_{i}v_{i}\), where the subscript \(i\) denotes the road segments. Furthermore, \(\Delta\rho v^{2}=\rho_{i+1}{v_{i+1}}^{2}-\rho_{i}{v_{i}}^{2}\), and \(\Delta\rho=\rho_{i+1}-\rho_{i}\); thus, Equation (45) takes the form  
\[v_{i+\frac{1}{2}}=\frac{v_{i+1}+\sqrt{\rho_{i+1}}+v_{i}\sqrt{\rho_{i}}}{\sqrt{ \rho_{i+1}}+\sqrt{\rho_{i}}}. \tag{46}\]  
Equation (46) denotes the average velocity of the proposed model at \(i\) and \(i+1\). The average density of the proposed model is the geometric mean of the density [27]. \[\rho_{i+\frac{1}{2}}=\sqrt{\rho_{i+1}\rho_{i}} \tag{47}\]  
[PERSON]'s model has the same average density and velocity as the proposed model. The computing array for the performance analysis of the traffic model was created by segmenting the solution domain spatially and temporally.