Numerical Approximation of Nonlinear Dispersive-Diffusive Traffic Flow Model with Source Term

Title

Numerical Approximation of Nonlinear Dispersive-Diffusive Traffic Flow Model with Source Term

Author

1. Prof Dr. Amitha M Rao, Professor, N.S.S. College of Commerce & Economics, India
2. Mrs. Veena Shinde-Deore, Professor, Bhavan’s Hazarimal Somani College, Chowpatty, India

Abstract

The study of traffic flow forecasting provides critical information for the planning and operational management of modern transportation systems. This paper uses a nonlinear Partial Differential Equation (PDE) with source term describing traffic phenomena to present a traffic modelling framework for traffic density evolution. The PDE is based on the nonhomogeneous KdV–Burgers equation including convection, diffusion, dispersion and source term. The governing PDE is solved by employing Finite Difference Method (FDM) under two initial conditions: with and without source term. Stability error analysis is carried out to ensure the convergence of the method. The numerical results computed by using MATLAB together with solution graphs of traffic density waves show the influence of source term on the solution of PDE. 

Keywords

Conclusion

In this paper, a nonlinear dispersive-diffusive traffic flow model with source term, has been numerically solved by using FDM. The PDE model is solved by considering  S(x,t)=0 and S(x,t)=sin(πx) e^(-t). Errors have been calculated to ensure the convergence of the method. A comparative analysis between PDE with source term and without source term has been made to understand the influence of source term on the solution of PDE. So, it is concluded that the FDM provides a stable, consistent and convergent computational technique for analyzing nonlinear traffic flow dynamics. And also the present model can be applied to study traffic flow evolution and behaviour which help us in  developing efficient transportation systems.

Author Contrubution

The author confirms sole responsibility for the following: study conception and design, data collection, analysis and interpretation of results, and manuscript preparation.

Funding

The authors did not receive any specific grants from funding agencies in the public, commercial, or non-profit sectors for the research, authorship, and/or publication of this article.

Conflict of Interest

All authors declare that they have no conflicts of interest.

Data Sharing Statement

Not applicable.

Software And Tools Use

Not applicable

Acknowledgements

I thank the following individuals for their expertise and assistance in all aspects of our study and for their help in writing the manuscript. I am also grateful for the insightful comments given by anonymous peer reviewers. Everyone's generosity and expertise have improved this study in myriad ways and saved me from many errors.