Implications and numerical application of the asymptotical shock wave model

Abstract

According to the Lighthill and Whitham’s shock wave model, a shock wave exists even in a homogeneous speed condition. They referred this wave as unobservable-- analogous to a radio wave that cannot be seen. Recent research has attempted to identify how such a counterintuitive conclusion results from the Lighthill and Whitham’s shock wave model, and derive a new asymptotical shock wave model. The asymptotical model showed that the shock wave in a homogenous speed traffic stream is identical to the ambient vehicle speed. Thus, no radio wave-like shock wave exists. However, performance tests of the asymptotical model using numerical values have not yet been performed. We investigated the new asymptotical model by examining the implications of the new model, and tested it using numerical values based on a test scenario. Our investigation showed that the only difference between both models is in the third term of the equations, and that this difference has a crucial role in the model output. Incorporation of model parameterαis another distinctive feature of the asymptotical model. This parameter makes the asymptotical model more flexible. In addition, due to various choices of α values, model calibration to accommodate various traffic flow situations is achievable. In Lighthill and Whitham’s model, this is not possible. Our numerical test results showed that the new model yields significantly different outputs: the predicted shock wave speeds of the asymptotical model tend to lean toward the downstream direction in most cases compared to the shock wave speeds of Lighthill and Whitham’s model for the same test environment. Statistical tests of significance also indicate that the outputs of the new model are significantly different than the corresponding outputs of Lighthill and Whitham’s model.