LWR Traffic Flow — Fundamental Diagram

Lighthill-Whitham-Richards model: ∂ρ/∂t + ∂q/∂x = 0, where q=ρ·v(ρ). Shockwaves propagate at the Rankine-Hugoniot speed; rarefaction fans smooth out density gradients.

Fundamental Diagram q(ρ)

Space-Time Diagram ρ(x,t)

Current Density Profile

Left density ρL

Right density ρR

Max speed vmax

Shock position x0