🚦 LWR Traffic Flow: Treating the Motorway as a Fluid

About this simulation

The Lighthill-Whitham-Richards (LWR) model treats a stream of traffic like a compressible fluid: instead of tracking every driver, it tracks a continuous density field ρ(x,t) — vehicles per metre of road — and a flow (flux) that depends on that density. When the road narrows or an incident cuts capacity, density piles up upstream faster than it can escape downstream, and a sharp boundary — a shockwave — forms and travels backward against the direction of traffic, exactly like the jam wave you feel arriving in a real hold-up long after the incident itself has cleared.

How it works

• Fundamental diagram — speed falls linearly with density (the Greenshields relation) v(ρ) = v_max·(1 − ρ/ρ_jam), so flow q = ρ·v rises, peaks, then falls as the road fills up.
• Godunov scheme — the road is split into cells; each timestep, the numerical flux crossing every cell boundary is the correct entropy-satisfying solution of the LWR Riemann problem, which is what produces sharp shocks instead of smeared-out blur.
• Bottleneck — a capacity cap at one location mimics a lane closure or incident; whatever flow it cannot pass gets rejected upstream, raising density there until a visible traffic jam (dark red) queues back from the pinch point.
• Vehicles (the instanced boxes) are advected through the same speed field the density solves for, so their bunching and colour directly visualise ρ(x,t).

Controls

• Inflow density — how much demand enters from the left edge of the modelled stretch.
• Bottleneck severity — how much the capacity is choked at the narrowing; 0% removes it, high values create a hard jam.
• Free-flow speed — the speed limit vehicles approach when density is near zero.
• Simulation speed — how fast the PDE and vehicles advance; slow it down to watch the shockwave crawl backward.
• ↺ Reset road — clears the density field and all vehicles. Drag to rotate, scroll to zoom.

Did you know?

The shockwave speed follows the Rankine-Hugoniot jump condition s = (q₂ − q₁) / (ρ₂ − ρ₁) — the same formula used for shocks in gas dynamics. It is why a jam can appear to move upstream even though every single car in it is still crawling forward.