Some exact solutions to the Lighthill-Whitham-Richards-Payne traffic flow equations
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Publication:2854118
DOI10.1088/1751-8113/46/36/365202zbMATH Open1291.90059arXiv1302.6948OpenAlexW2025023169MaRDI QIDQ2854118
Author name not available (Why is that?)
Publication date: 17 October 2013
Published in: (Search for Journal in Brave)
Abstract: We find a class of exact solutions to the Lighthill Whitham Richards Payne (LWRP) traffic flow equations. Using two consecutive lagrangian transformations, a linearization is achieved. Next, depending on the initial density, we either apply (again two) Lambert functions and obtain exact formulas for the dependence of the car density and velocity on x and t, or else, failing that, the same result in a parametric representation. The calculation always involves two possible factorizations of a consistency condition. Both must be considered. In physical terms, the lineup usually separates into two offshoots at different velocities. Each velocity soon becomes uniform. This outcome in many ways resembles the two soliton solution to the Korteweg-de Vries equation. We check general conservation requirements. Although traffic flow research has developed tremendously since LWRP, this calculation, being exact, may open the door to solving similar problems, such as gas dynamics or water flow in rivers. With this possibility in mind, we outline the procedure in some detail at the end.
Full work available at URL: https://arxiv.org/abs/1302.6948
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