An improved flux wave-HLLE approach for the solution of traffic flow models based on transition velocities

Abstract

n this study, a new numerical method is presented to solve the nonlinear Partial Differential Equations of second-order one-dimensional non-homogeneous traffic flow models based on transition velocities. The proposed Improved Flux Wave-HLLE (IFW-HLLE) method utilizes a particular type of approximate Riemann speed, that is, a unique combination of characteristic speeds and the Roe speed, to reach a solution with positive velocity and density. This method provides an equilibrium between the source terms and flux variations for steady-state conditions when solving the Riemann problem. The spatial variations in traffic density were also based on the transition velocities. For evaluating its performance, the proposed numerical solution is also compared with the results of the Original Roe Method (ORM) for solving widely-used Payne–Whitham (PW), Zhang, and Khan–Gulliver models. Moreover, both straight and circular paths with periodic boundary conditions were modelled to analyse and investigate the traffic flow of a bottleneck. Results demonstrate that the IFW-HLLE method captures more realistic traffic behaviours compared to ORM. Notably, negative and unrealistic velocity values observed in ORM—for the PW and Zhang models (ranging from -120 to 400 m/s and -600 to 1200 m/s)—were effectively corrected with the proposed method (ranges from 16 to 25 m/s and 8 to 16 m/s). Euclidean error norms calculated for 2D velocity profiles showed maximum errors of 2.6976×10⁻² and 4.0835×10⁻³ for straight and circular paths, respectively, confirming the improved accuracy.