A dynamically bi-orthogonal solution method for a stochastic Lighthill-Whitham-Richards traffic flow model

Macroscopic traffic flow modeling is essential for describing and forecasting the characteristics of traffic flow. However, the classic Lighthill–Whitham–Richards (LWR) model only provides equilibrium values for steady-state conditions and fails to capture common stochastic variabilities, which are a necessary component of accurate modeling of real-time traffic management and control. In this paper, a stochastic LWR (SLWR) model that randomizes free-flow speed is developed to account for the stochasticity incurred by the heterogeneity of drivers, while holding individual drivers’ behavior constant. The SLWR model follows a conservation law of stochastic traffic density and flow and is formulated as a time-dependent stochastic partial differential equation. The model is solved using a dynamically bi-orthogonal (DyBO) method based on a spatial basis and stochastic basis. Various scenarios are simulated and compared with the Monte Carlo (MC) method, and the results show that the SLWR model can effectively describe dynamic traffic evolutions and reproduce some commonly observed traffic phenomena. Furthermore, the DyBO method shows significant computational advantages over the MC method.