Comparison of numerical schemes for macroscopic traffic flow models

Accurate and reliable numerical algorithms for the solution of macroscopic traffic flow models are very important to ensure that the models' properties are accurately represented. An inappropriate numerical scheme may result in deviate or erratic model behavior, such as large numerical dissipations and numerical instabilities. A new computation scheme for the numerical approximation of solutions of Payne-type models is presented. This scheme is proved to satisfy the positive constraints of traffic variables and thus prevents traffic from moving upstream. In addition, an automated model calibration approach is proposed; it is applied to determine the parameters of the considered macroscopic model for different numerical schemes (Steger-Warming, MacCormack, and the proposed numerical scheme). Predictions from the various numerical approximation models are compared with observations from the validation data set. On the basis of this comparison, it is concluded that the proposed numerical scheme is both more accurate and more robust than the other schemes considered. It yields a small mean squared error and fast computation due to the ability to use a larger time-step; it also provides accurate results in regions in which the gradients of the speeds and densities are high.