On the performance of HLL, HLLC, and Rusanov solvers for hyperbolic traffic models

This paper investigates the performances of approximate Riemann solvers (ARSs) for hyperbolic traffic models from the family of generic second-order traffic flow modeling. Three approximate Riemann solvers are selected, including the HLL, HLLC, and Rusanov solvers, and evaluated comprehensively against the model by Zhang (2002) and a variant of the phase-transition model by Colombo (2002) with a continuous solution domain. The ARSs are investigated using extensive numerical tests, covering all possible waves arising in different Riemann problems, including shockwaves, rarefaction waves, and contact waves. We first investigate ARSs' performances with the Euler-Upwind spatiotemporal discretization scheme. The results show that Rusanov and HLL solvers capture the solutions to the Riemann problems for both models. However, the HLLC solver fails to remain stable for the model by Colombo (2002) in specific Riemann problems due to its wave-speed mechanism which must be tailored from one model to another. Therefore, Rusanov and HLL solver are identified as more desirable to HLLC for traffic models. This paper further examines the performances of Rusanov and HLL solvers using various highresolution spatial schemes coupled with the third-order TVD-Runge Kutta scheme for temporal discretization. The spatial schemes considered include the second-order MINMOD and MUSCL schemes, the cell-based thirdorder upwind (CBTOU) scheme, and the fifth-order WENO-JS and WENO-Z schemes. It is shown that various combinations of the schemes perform differently regarding numerical diffusions and oscillations. For instance, while the combination of the HLL solver with MUSCL and WENO-type schemes perform better than other combinations regarding numerical diffusions, they can lead to oscillatory behavior near intermediate states even though TVD-RK is used for temporal discretization. The paper discusses the implications of such performances for real-world traffic scenarios consisting of multiple bottlenecks.

Automated summary

This paper investigates the performances of three approximate Riemann solvers in traffic flow models, finding that Rusanov and HLL solvers are more suitable for certain models. Additionally, the paper analyzes the effects of combining different spatial schemes, showing variations in handling numerical diffusions and oscillations.