Linear speed-density relationship Discretization $\frac{\partial \rho }{\partial t}+\frac{\partial }{\partial x}\left({v}_{\mathrm{max}}\left(\rho -\frac{{\rho }^{2}}{{\rho }_{\mathrm{max}}}\right)\right)=0$ ${\rho }_{j}^{n+1}=a\left({\rho }_{j-1}^{n}\right)+b\left({\rho }_{j}^{n}\right)+c\left({\rho }_{j+1}^{n}\right)+d{\left({\rho }_{j-1}^{n}\right)}^{2}+e{\left({\rho }_{j}^{n}\right)}^{2}+f{\left({\rho }_{j+1}^{n}\right)}^{2}$ where $a=\left[1/4+{v}_{\mathrm{max}}\left(\Delta t/2\Delta x\right)+{v}_{\mathrm{max}}{\left(\Delta t/2\Delta x\right)}^{2}\right]$ $b=\left[1/2-2{v}_{\mathrm{max}}{\left(\Delta t/2\Delta x\right)}^{2}\right]$ $c=\left[1/4-{v}_{\mathrm{max}}\left(\Delta t/2\Delta x\right)+{v}_{\mathrm{max}}{\left(\Delta t/2\Delta x\right)}^{2}\right]$ $d=\left[-\left({v}_{\mathrm{max}}/{\rho }_{\mathrm{max}}\right)\left(\Delta t/2\Delta x\right)-\left({v}_{\mathrm{max}}/{\rho }_{\mathrm{max}}\right){\left(\Delta t/2\Delta x\right)}^{2}\right]$ $e=\left[2\left({v}_{\mathrm{max}}/{\rho }_{\mathrm{max}}\right){\left(\Delta t/2\Delta x\right)}^{2}\right]$ $f=\left[\left({v}_{\mathrm{max}}/{\rho }_{\mathrm{max}}\right)\left(\Delta t/2\Delta x\right)-\left({v}_{\mathrm{max}}/{\rho }_{\mathrm{max}}\right){\left(\Delta t/\Delta x\right)}^{2}\right]$