Control and Optimization in Applied Mathematics

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Numerical Solution of Homogeneous Aw-Rascle Type Traffic Flow Models Using an Improved Wave Propagation-HLLE Approach

Articles in Press

Document Type : Research Article

Authors

1 Department of Civil Engineering, Faculty of Engineering, University of Birjand, Iran

2 Department of Civil Engineering, Faculty of Mining, Civil and Chemical Engineering, Birjand University of Technology, Birjand, Iran

10.30473/coam.2026.75815.1338

Abstract

Homogeneous second-order Aw-Rascle-type models have demonstrated greater effectiveness than their non-homogeneous counterparts in traffic flow modeling. This study addresses the numerical solution of hyperbolic conservation laws governing these models by coupling the second-order HLLE Riemann solver, a Godunov-type finite volume approach, with the wave propagation algorithm. A novel wave-speed selection strategy is proposed by comparing characteristic velocities with Roe speeds, yielding solutions with guaranteed positive density and speed. The proposed IWP-HLLE method is applied to simulate shock, rarefaction, and contact discontinuity waves under homogeneous long-road conditions, eliminating the influence of external source terms and ensuring the homogeneity of the governing hyperbolic equations. Its performance is benchmarked against the MacCormack scheme supplemented by two standard stabilization techniques, namely artificial viscosity (AV) and central differencing (CD). Spatiotemporal distributions and density profiles are examined across four representative traffic scenarios: free flow, congested traffic flow, queue dissolution, and congested flow with non-equilibrium velocity and uniform density. The results demonstrate that the IWP-HLLE approach substantially suppresses numerical oscillations compared to both AV and CD methods while maintaining stability across all test cases.

Highlights

  • A novel IWP-HLLE scheme is proposed by coupling the HLLE Riemann solver with an improved wave propagation algorithm for homogeneous second-order traffic flow models.
  • A new wave-speed selection strategy blending characteristic velocities and Roe speeds guarantees positive density and velocity throughout the simulation domain.
  • The IWP-HLLE method effectively captures shock waves, rarefaction waves, and contact discontinuities in both AR and ARZ models without spurious oscillations.
  • The Monotonized Central (MC) limiter, integrated into the scheme, achieves second-order spatial accuracy and significantly suppresses numerical dispersion near wave fronts.
  • RMSE analysis over four traffic scenarios confirms IWP-HLLE outperforms both artificial viscosity (AV) and central differencing (CD) smoothing techniques in the most test cases.

Keywords

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References
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Control and Optimization in Applied Mathematics

Articles in Press, Accepted Manuscript
Available Online from 09 March 2026
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