Analysis of LWR model with fundamental diagram subject to uncertainties

The LWR model is of interest since it is simple and can successfully reproduce some essential features of traffic flow, such as the formation and propagation of traffic disturbances. In this article, we investigate the LWR model from an uncertainty perspective. We attempt to analyse how reliable the LWR model prediction will be if the fundamental diagram (FD) in use is not accurately specified. To fulfil this end, we postulate a flux function (equivalently a FD) driven by a random free flow speed, which accommodates the uncertain feature observed in the speed-density data. We provide essential mathematical properties and solution schemes of the LWR model with the probabilistic FD. In case studies, the approach to evaluate the uncertainty of traffic disturbance propagation with this model is presented. We find that if FD in a LWR model cannot be perfectly specified, the uncertainty associated with the location of a traffic disturbance would increase over time. In contrast, the magnitude of the traffic disturbance can still be accurately predicted.