Name of the Explicit scheme

Discretization of traffic flow model

Upwind

${\rho }_j^{n+1}={\rho }_j^n-\beta \left({\rho }_j^n-{\rho }_{j-1}^n\right)$

Downwind

${\rho }_j^{n+1}={\rho }_j^n-\beta \left({\rho }_{j+1}^n-{\rho }_j^n\right)$

FTCS

${\rho }_j^{n+1}={\rho }_j^n-\alpha \left({\rho }_{j+1}^n-{\rho }_{j-1}^n\right)$

Lax-Friedrichs

${\rho }_j^{n+1}=\left(1/2\right)\left({\rho }_{j+1}^n+{\rho }_{j-1}^n\right)-\alpha \left({\rho }_{j+1}^n-{\rho }_{j-1}^n\right)$

Tolesa

${\rho }_j^{n+1}=\left(1/4\right)\left({\rho }_{j+1}^n+2{\rho }_j^n+{\rho }_{j-1}^n\right)-\alpha \left({\rho }_{j+1}^n-u_{j-1}^n\right)+{\alpha }^2\left({\rho }_{j+1}^n-2{\rho }_j^n+{\rho }_{j-1}^n\right)$