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Numerical solution of nonlinear equations of traffic flow density using spectral methods by filter

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Abstract

This paper introduces an innovative approach that marries the spectral method with a time-dependent partial differential equation filter to tackle the phenomenon of shock waves in traffic flow modeling. Through the strategic application of Discrete low-pass filters, this method effectively mitigates shock-induced deviations, leading to significantly more accurate results compared to conventional spectral techniques. We conduct a thorough examination of the stability conditions inherent to this approach, providing valuable insights into its robustness. To substantiate its effectiveness, we present a series of numerical examples illustrating the method’s prowess in delivering precise solutions. Comparative analysis against established methods such as Lax and Cu reveals a marked superiority in accuracy. This work not only contributes a novel numerical technique to the field of traffic flow modeling but also addresses a persistent challenge, offering a promising avenue for further research and practical applications.

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Acknowledgements

We thank the anonymous reviewers for helpful comments, which lead to definite improvement in the manuscript.

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The authors declare that this research received no grant from any funding agency in the public, commercial, or not-for-profit sectors.

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Authors and Affiliations

  1. Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran

    Seyed Esmaeil Sadat Najafi, Tofigh Allahviranloo & Mohsen Rostamy Malkhalifeh

  2. Faculty of Engineering and Natural Sciences, Istinye University, Istanbul, Turkey

    Tofigh Allahviranloo

  3. Department of Applied Mathematics, Faculty of Science, Imam Khomeini International University, Qazvin, 34149-16818, Iran

    Saeid Abbasbandy

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Najafi, S.E.S., Allahviranloo, T., Abbasbandy, S. et al. Numerical solution of nonlinear equations of traffic flow density using spectral methods by filter. J. Appl. Math. Comput. 71, 743–763 (2025). https://doi.org/10.1007/s12190-024-02252-8

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