Linear speed-density relationship

Discretization

$\frac{\partial ρ}{\partial t} + \frac{\partial}{\partial x} (v_{\max} (ρ - \frac{ρ^{2}}{ρ_{\max}})) = 0$

$ρ_{j}^{n + 1} = a (ρ_{j - 1}^{n}) + b (ρ_{j}^{n}) + c (ρ_{j + 1}^{n}) + d {(ρ_{j - 1}^{n})}^{2} + e {(ρ_{j}^{n})}^{2} + f {(ρ_{j + 1}^{n})}^{2}$

where

$a = [1 / 4 + v_{\max} (Δ t / 2 Δ x) + v_{\max} {(Δ t / 2 Δ x)}^{2}]$

$b = [1 / 2 - 2 v_{\max} {(Δ t / 2 Δ x)}^{2}]$

$c = [1 / 4 - v_{\max} (Δ t / 2 Δ x) + v_{\max} {(Δ t / 2 Δ x)}^{2}]$

$d = [- (v_{\max} / ρ_{\max}) (Δ t / 2 Δ x) - (v_{\max} / ρ_{\max}) {(Δ t / 2 Δ x)}^{2}]$

$e = [2 (v_{\max} / ρ_{\max}) {(Δ t / 2 Δ x)}^{2}]$

$f = [(v_{\max} / ρ_{\max}) (Δ t / 2 Δ x) - (v_{\max} / ρ_{\max}) {(Δ t / Δ x)}^{2}]$