A modified flux-wave formula for the solution of second-order macroscopic traffic flow models

In this paper, a second-order Godunov-type finite volume technique based upon the wave propagation algorithm has been defined which mainly addresses second-order, one-dimensional, macroscopic traffic flow models. Four widely used models that have fundamental differences in terms of the equations form were chosen to evaluate this formula. These models use Riemann’s solution propagating different jump discontinuity from each finite-volume cell interface. The proposed modified flux-wave formula is well-balanced and includes the related source term within the flux-differencing from each cell interface. Therefore, no additional numerical treatment is required with respect to the source terms. The numerical results obtained herein have been compared with high-order relaxations schemes including fifth-order WENO and second-order MUSCL (for three models in a straight homogeneous road) and Roe decomposition technique (for two models in a ring road). The results show that the defined approach is capable to provide stable and realistic response for the macroscopic traffic models without numerical diffusion. Additionally, a very good agreement achieved with the reference solution and also higher order schemes. It is also inferred that the plausibility and positivity conserving conditions have been maintained for the modified flux wave approach for all of the examined models.