The KG, PW, Zhang and proposed models are discretized using the decomposition technique developed by Roe [27] to evaluate their performance. This technique can be used to approximate the nonlinear system of equations
$$ {G}_t+f{(G)}_x=S(G), $$
(14)
where G is the vector of data variables, f(G) is the vector of functions of these variables, and S(G) is the vector of source terms. The subscripts t and x denote partial derivatives with respect to time and distance, respectively. Equation (14) is then given by
$$ \frac{\partial G}{\partial t}+\frac{\partial f}{\partial G}\frac{\partial G}{\partial x}=S(G). $$
(15)
Let A(G) be the Jacobian matrix of the system. Then (15) can be expressed as
$$ \frac{\partial G}{\partial t}+A(G)\frac{\partial G}{\partial x}=S(G). $$
(16)
Setting the source terms in (16) to zero gives the quasilinear form
$$ \frac{\partial G}{\partial t}+A(G)\frac{\partial G}{\partial x}=0. $$
(17)
The data variables are density ρ and flow ρυ in the KG, PW, Zhang and proposed models. Roe’s technique is used to linearize the Jacobian matrix A(G) by decomposing it into eigenvalues and eigenvectors. This is based on the realistic assumption that the data variables, eigenvalues and eigenvectors remain conserved for small changes in time and distance. This technique is widely employed because it is able to capture the effects of abrupt changes in the data variables.
Consider a road divided into M equidistant segments and N equal duration time steps. The total length is x_{
M
} so a road segment has length δx = x_{
M
}/M, and the total time period is t_{
N
} so a time step is δt = t_{
N
}/N. The Jacobian matrix is approximated for road segments \( \left({x}_i+\frac{\delta x}{2},{x}_i\frac{\delta x}{2}\right) \). This matrix is determined for all M segments in every time interval (t_{n + 1}, t_{
n
}), where t_{n + 1} − t_{
n
} = δt.
Let ΔG be a small change in the data variables G and Δf the corresponding change in the functions of these variables. Further, let G_{
i
} be the average value of the data variables in the ith segment. The change in flux at the boundary between the ith and (i + 1)th segments is
$$ \Delta {f}_{i+\frac{1}{2}}=A\left({G}_{i+\frac{1}{2}}\right)\Delta G, $$
(18)
where \( A\left({G}_{i+\frac{1}{2}}\right) \) is the Jacobian matrix at the segment boundary, and \( {G}_{i+\frac{1}{2}} \) is the vector of data variables at the boundary obtained using Roe’s technique. The flux approximates the change in traffic density and flow at the segment boundary. We have that \( A\left({G}_{i+\frac{1}{2}}\right)=e\Lambda {e}^{1} \), where Λ is a diagonal matrix of the eigenvalues [λ_{1}, λ_{2}, ⋯, λ_{
p
}] of the Jacobian matrix and e is the corresponding eigenvector matrix. From [28], the eigenvalues should be positive so that
$$ \Delta {f}_{i+\frac{1}{2}}=e\left\Lambda \right{e}^{1}\left({G}_{i+1}{G}_i\right), $$
(19)
where the approximation ΔG = (G_{i + 1} − G_{
i
}) is used. The flux at the boundary between segments i and i + 1 at time n is then approximated by
$$ {f}_{i+\frac{1}{2}}^n\left({G}_i^n,{G}_{i+1}^n\right)=\frac{1}{2}\left(f\left({G}_i^n\right)+f\left({G}_{i+1}^n\right)\right)\frac{1}{2}\Delta {f}_{i+\frac{1}{2}}, $$
(20)
where \( f\left({G}_i^n\right) \) and \( f\left({G}_{i+1}^n\right) \) denote the values of the functions of the data variables in road segments i and i + 1 respectively, at time n. Substituting (19) into (20) gives
$$ {f}_{i+\frac{1}{2}}^n\left({G}_i^n,{G}_{i+1}^n\right)=\frac{1}{2}\left(f\left({G}_i^n\right)+f\left({G}_{i+1}^n\right)\right)\frac{1}{2}e\left\Lambda \right{e}^{1}\left({G}_{i+1}^n{G}_i^n\right). $$
(21)
This approximates the change in density and flow without considering the source.
For the source decomposition of the KG model in (3), the PW model [15, 20] in (12) and the Zhang model in (14), we have
$$ {S}_1\left({G}_i^n\right)={\rho}_i^n\left(\frac{\upsilon \left({\rho}_i^n\right){\upsilon}_i^n}{\tau}\right), $$
(22)
and for the proposed model in (11)
$$ {S}_2\left({G}_i^n\right)={\rho}_i^n\left(\frac{\upsilon^2\left({\rho}_i^n\right){\left({\upsilon}_i^n\right)}^2}{{b\upsilon}_s}\right). $$
(23)
The updated data variables for the KG, PW, Zhang and proposed models are
$$ {G}_i^{n+1}={G}_i^n\frac{\delta t}{\delta x}\left({f}_{i+\frac{\delta }{2}}^n{f}_{i+\frac{\delta x}{2}}^n\right)+\delta t{S}_y\left({G}_i^n\right),y=1,2. $$
(24)
Both the KG and proposed models have the same expressions on the lefthand side (LHS) as shown in (3) and (11). Therefore, the Jacobian matrix A(G) is the same for these models and results in the same eigenvalues and eigenvectors, and also average velocity and density. A(G) as well as the corresponding eigenvalues and eigenvectors, average density and velocity for the KG model were derived in [21]. Assuming the right hand side (RHS) of the KG and proposed models are in quasilinear form, traffic flow alignment (harmonization) is not considered so that \( {S}_y(G)=\left(\genfrac{}{}{0pt}{}{0}{0}\right) \). Then (24) takes the form
$$ G=\left(\genfrac{}{}{0pt}{}{\rho }{\rho \upsilon}\right),f(G)=\left(\genfrac{}{}{0pt}{}{f_1}{f_2}\right)=\left(\genfrac{}{}{0pt}{}{\rho \upsilon}{\frac{{\left(\rho \upsilon \right)}^2}{\rho }+\frac{\upsilon^2\left(\rho \right){\upsilon}^2}{2{d}_{tr}}\rho}\right)\ \mathrm{and}\ {S}_y(G)=\left(\genfrac{}{}{0pt}{}{0}{0}\right). $$
(25)
The LHS of (3) and (11) is approximated using the Jacobian matrix \( (G)=\frac{\partial f}{\partial G} \), which is obtained from (25).
The Jacobian matrix \( A(G)=\frac{\partial f}{\partial G} \) from (25) is
$$ A(G)=\left(\genfrac{}{}{0pt}{}{0}{{\upsilon}^2+\left(\frac{\upsilon^2\left(\rho \right){\upsilon}^2}{2{d}_{tr}}\right)}\genfrac{}{}{0pt}{}{1}{2\upsilon}\right). $$
(26)
The eigenvalues obtained from (26) in Appendix A are
$$ {\lambda}_{1,2}=\upsilon \pm \sqrt{\left(\frac{\upsilon^2\left(\rho \right){\upsilon}^2}{2{d}_{tr}}\right)}. $$
(27)
These show that when a transition occurs, the velocity changes according to the equilibrium velocity distribution and the average velocity.
For a traffic flow to be strictly hyperbolic, the eigenvectors must be distinct and real [29]. The eigenvectors corresponding to the eigenvalues in (27) are distinct and real when the equilibrium velocity is greater than the average velocity, i.e.
$$ \upsilon \left({\rho}_{i+\frac{1}{2}}\right)>{\upsilon}_{i+\frac{1}{2}}. $$
Conversely, the eigenvectors are imaginary when
$$ \upsilon \left({\rho}_{i+\frac{1}{2}}\right)<{\upsilon}_{i+\frac{1}{2}}, $$
so to maintain the hyperbolic property for the proposed model, the absolute value of the numerator under the radical sign in (27) is employed, which gives
$$ {\lambda}_{1,2}={\upsilon}_{i+\frac{1}{2}}\pm \sqrt{\frac{\left{\upsilon}^2\left({\rho}_{i+\frac{1}{2}}\right){\upsilon}_{i+\frac{1}{2}}^2\right}{2{d}_{tr}}}. $$
(28)
The corresponding eigenvectors of the KG and proposed models are
$$ {e}_{1,2}=\left(\genfrac{}{}{0pt}{}{1}{\upsilon_{i+\frac{1}{2}}\pm \sqrt{\frac{\left{\upsilon}^2\left({\rho}_{i+\frac{1}{2}}\right){\upsilon}_{i+\frac{1}{2}}^2\right}{2{d}_{tr}}}}\right). $$
(29)
The eigenvalues and eigenvectors, average density and velocity for the PW model were derived in [18]. The eigenvalues are
$$ {\lambda}_{1,2}={\upsilon}_{i+\frac{1}{2}}\pm {C}_0, $$
(30)
where C_{0} is the velocity constant. This shows that traffic velocity alignment is at a constant rate C_{0} during transitions. The corresponding eigenvectors are
$$ {e}_{1,2}=\left(\genfrac{}{}{0pt}{}{1}{\upsilon_{i+\frac{1}{2}}\pm {C}_0}\right). $$
(31)
The average velocity at the boundary of segments i and i + 1 for the KG, PW and proposed models is
$$ {\upsilon}_{i+\frac{1}{2}}=\frac{\upsilon_{i+1}\sqrt{\rho_i+1}+{\upsilon}_i\sqrt{\rho_i}}{\sqrt{\rho_i+1}+\sqrt{\rho_i}}. $$
(32)
The average density for these models at the boundary of segments i and i + 1 is given by the geometric mean of the densities in these segments
$$ {\rho}_{i+\frac{1}{2}}=\sqrt{\rho_{i+1}{\rho}_i}. $$
(33)
The eigenvalues for the Zhang model [25] are
$$ {\lambda}_1={\upsilon}_{i+\frac{1}{2}}, $$
(34)
and
$$ {\lambda}_2={\upsilon}_{i+\frac{1}{2}}+\rho v{\left(\rho \right)}_{\rho }, $$
(35)
where the subscript ρ presents the derivative of the equilibrium velocity distribution with respect to density. The corresponding eigenvectors are
$$ {e}_1=\left(\genfrac{}{}{0pt}{}{1}{\upsilon_{i+\frac{1}{2}}v\left({\rho}_{i+\frac{1}{2}}\right){\rho}_{i+\frac{1}{2}}v{\left({\rho}_{i+\frac{1}{2}}\right)}_{\rho }}\right), $$
(36)
$$ {e}_2=\left(\genfrac{}{}{0pt}{}{1}{\upsilon_{i+\frac{1}{2}}v\left({\rho}_{i+\frac{1}{2}}\right)}\right). $$
(37)
The average density \( {\rho}_{i+\frac{1}{2}} \) of the Zhang model at the segment boundary is the same as for the KG, PW and proposed models given in (33). The average velocity \( {v}_{i+\frac{1}{2}} \)at the boundary of segments i and i + 1 for the Zhang model is given in Appendix B.

A.
Entropy Fix
Entropy fix is applied to Roe’s technique to smooth any discontinuities at the segment boundaries. The Jacobian matrix \( A\left({G}_{i+\frac{1}{2}}\right) \) is decomposed into its eigenvalues and eigenvectors to approximate the flux in the road segments (21). The Jacobian matrix for the road segments is then replaced with the entropy fix solution given by
$$ e\left\Lambda \right{e}^{1}, $$
where \( \left\Lambda \right=\left[{\widehat{\lambda}}_1,{\widehat{\lambda}}_2,\cdots, {\widehat{\lambda}}_k,\cdots, {\widehat{\lambda}}_n\right] \) is a diagonal matrix which is a function of the eigenvalues λ_{
k
} of the Jacobian matrix, and e is the corresponding eigenvector matrix. The Harten and Hayman entropy fix scheme [30] is employed here so that
$$ {\widehat{\lambda}}_k=\left\{\begin{array}{c}{\widehat{\delta}}_k\kern0.75em \mathrm{if}\kern1.25em \left{\lambda}_k\right<{\widehat{\delta}}_k\\ {}\ \left{\lambda}_k\right\kern0.75em \mathrm{if}\kern1.25em \left{\lambda}_k\right\ge {\widehat{\delta}}_k\end{array}\right. $$
(38)
with
$$ {\widehat{\delta}}_k=\max \left(0,\kern0.75em {\lambda}_{i+\frac{1}{2}}{\lambda}_i,\kern0.75em {\lambda}_{i+1}{\lambda}_{i+\frac{1}{2}}\right). $$
(39)
This ensures that the \( {\widehat{\lambda}}_k \) are not negative and similar at the segment boundaries. The Jacobian matrix eΛe^{−1} for the proposed, KG and Zhang models are given in Appendix C. The corresponding flux is obtained from (21) using f(G_{
i
}) and f(G_{i + 1}) and substituting eΛe^{−1} for \( A\left({G}_{i+\frac{1}{2}}\right) \), the updated data variables, ρ and ρυ, are then obtained at time n using (24).