An Improved Flux Wave-HLLE approach for the Solution of Traffic Flow Models Based on Transition Velocities

Document Type : Research Article

Authors

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

2 Department of Civil Engineering, Shahid Nikbakht Faculty of Engineering, University of Sistan and Baluhestan, Zahedan, Iran

Abstract

In 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.

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References

[1] M.J. Lighthill, G.B. Whitham, On kinematic waves II. A theory of traffic flow on long crowded roads, Proceedings of the Royal Society of London. series a. mathematical and physical sciences, 229(1178) (1955) 317-345.
[2] P.I. Richards, Shock waves on the highway, Operations Research, 4(1) (1956) 42-51.
[3] Mohammadian S (2017). Numerical Study on Traffic Flow Prediction Using Different Second-Order Continuum Traffic Flow Models. Ferdowsi University of Mashhad [in Persian].
[4] H.J. Payne, Model of freeway traffic and control, Mathematical Model of Public System, (1971) 51-61.
[5] G.B. Whitham, Linear and nonlinear waves, John Wiley & Sons, 2011.
[6] A. Delis, I. Nikolos, M. Papageorgiou, High-resolution numerical relaxation approximations to second-order macroscopic traffic flow models, Transportation Research Part C: Emerging Technologies, 44 (2014) 318-349.
[7] Z.H. Khan, T.A. Gulliver, H. Nasir, A. Rehman, K. Shahzada, A macroscopic traffic model based on driver physiological response, Journal of Engineering Mathematics, 115(1) (2019) 21-41.
[8] J. Del Castillo, P. Pintado, F. Benitez, The reaction time of drivers and the stability of traffic flow, Transportation Research Part B: Methodological, 28(1) (1994) 35-60.
[9] C.F. Daganzo, Requiem for second-order fluid approximations of traffic flow, Transportation Research Part B: Methodological, 29(4) (1995) 277-286.
[10] A. Aw, M. Rascle, Resurrection of" second order" models of traffic flow, SIAM Journal on Applied Mathematics, 60(3) (2000) 916-938.
[11] P. Berg, A. Mason, A. Woods, Continuum approach to car-following models, Physical Review E, 61(2) (2000) 1056.
[12] H.M. Zhang, A non-equilibrium traffic model devoid of gas-like behavior, Transportation Research Part B: Methodological, 36(3) (2002) 275-290.
[13] Z.H. Khan, T.A. Gulliver, K.S. Khattak, A. Qazi, A macroscopic traffic model based on reaction velocity, Iranian Journal of Science and Technology, Transactions of Civil Engineering, 44(1) (2020) 139-150.
[14] Z.H. Khan, T.A. Gulliver, A macroscopic traffic model based on transition velocities, Journal of Computational Science, 43 (2020) 101131.
[15] K. Mohamed, M.A. Abdelrahman, The NHRS scheme for the two models of traffic flow, Computational and Applied Mathematics, 42(1) (2023) 53.
[16] S. Moodi, H. Mahdizadeh, J. Shucksmith, M. Rubinato, M. Azhdary Moghaddam, Experimental and numerical modelling of water waves in sewer networks during sewer/surface flow interaction using a coupled ODE‐SWE solver, Journal of Flood Risk Management, 17(1) (2024) e12953.
[17] R.J. LeVeque, Finite volume methods for hyperbolic problems, Cambridge University Press, 2002.
[18] H. Mahdizadeh, P.K. Stansby, B.D. Rogers, On the approximation of local efflux/influx bed discharge in the shallow water equations based on a wave propagation algorithm, International Journal for Numerical Methods in Fluids, 66(10) (2011) 1295-1314.
[19] H. Mahdizadeh, P.K. Stansby, B.D. Rogers, Flood wave modeling based on a two-dimensional modified wave propagation algorithm coupled to a full-pipe network solver, Journal of Hydraulic Engineering, 138(3) (2012) 247-259.
[20] H. Mahdizadeh, S. Sharifi, P. Omidvar, On the approximation of two-dimensional transient pipe flow using a modified wave propagation algorithm, Journal of Fluids Engineering, 140(7) (2018) 071402.
[21] S. Mohammadian, A.M. Moghaddam, A. Sahaf, On the performance of HLL, HLLC, and Rusanov solvers for hyperbolic traffic models, Computers & Fluids, 231 (2021) 105161.
[22] M. Araghi, H. Mahdizadeh, S. Moodi, Numerical modelling of macroscopic traffic flow based on driver physiological response using a modified wave propagation algorithm, Journal of Decisions and Operations Research, 6(3) (2021) 350-364.
[23] M. Araghi, S. Mahdizadeh, H. Mahdizadeh, S. Moodi, Correction to: A modified flux-wave formula for the solution of second-order macroscopic traffic flow models, Nonlinear Dynamics, 107(3) (2022) 3179-3179.
[24] M. Araghi, H. Mahdizadeh, S. Moodi, Applying High-Resolution Wave Propagation Method in Numerical Modeling of Macroscopic Traffic Flow based on Driver Physiological-Psychological Behavior, Journal of Civil and Environmental Engineering, 53(110) (2023) 1-12.
[25] F. van Wageningen-Kessels, H. Van Lint, K. Vuik, S. Hoogendoorn, Genealogy of traffic flow models, EURO Journal on Transportation and Logistics, 4(4) (2015) 445-473.
[26] S. Moodi, H. Mahdizadeh, Numerical Modeling of Water Influx Falling into an Empty Tank using a Modified Wave Propagation Algorithm, Modares Mechanical Engineering, 18(6) (2018) 182-190.
[27] B. Van Leer, Towards the ultimate conservative difference scheme I. The quest of monotonicity, in:  Proceedings of the Third International Conference on Numerical Methods in Fluid Mechanics: Vol. I General Lectures. Fundamental Numerical Techniques July 3–7, 1972 Universities of Paris VI and XI, Springer, 2008, pp. 163-168.
[28] D.S. Bale, R.J. Leveque, S. Mitran, J.A. Rossmanith, A wave propagation method for conservation laws and balance laws with spatially varying flux functions, SIAM Journal on Scientific Computing, 24(3) (2003) 955-978.
[29] R. Courant, K. Friedrichs, H. Lewy, On the partial difference equations of mathematical physics, IBM Journal of Research and Development, 11(2) (1967) 215-234.
[30] P.L. Roe, Approximate Riemann solvers, parameter vectors, and difference schemes, Journal of Computational Physics, 43(2) (1981) 357-372.
AUT Journal of Civil Engineering
Volume 9, Issue 2
2025
Pages 159-170

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