LighthillWhithamRichards

Lighthill-Whitham-Richards (LWR) is a macroscopic traffic flow model describing the evolution of vehicle density on a road as a conservation law. Developed independently by M. J. Lighthill and G. B. Whitham in 1955 and by P. I. Richards in 1956, the model treats traffic as a continuous fluid-like medium. The state variable is density ρ(x,t) [vehicles per unit length], and the flow q(ρ) = ρ V(ρ) is given by a fundamental diagram relating density to average speed V(ρ). The governing equation is ∂ρ/∂t + ∂q(ρ)/∂x = 0, a first-order hyperbolic partial differential equation. Solutions describe kinematic waves of constant flow and density; discontinuities in ρ arise as shock waves, while smooth changes form rarefaction waves. Boundary conditions and external inputs such as on-ramps can produce complex traffic patterns including stop-and-go waves.

In practice, LWR provides a simple, widely used framework for traffic planning and simulation. It is attractive for its mathematical tractability and compatibility with conservation principles, but it assumes instantaneous adaptation to local conditions, homogeneous driver behavior, and a prescribed fundamental diagram. Extensions such as second-order models (e.g., Payne-Whitham, Aw-Rascle-Zhang) introduce velocity dynamics and relaxation terms to capture non-equilibrium effects, while numerical methods often rely on Godunov-type schemes.