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Some exact solutions to the Lighthill–Whitham–Richards–Payne traffic flow equations : II. Moderate congestion
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Infeld, E., Rowlands, G. (George) and Skorupski, A. A. (2014) Some exact solutions to the Lighthill–Whitham–Richards–Payne traffic flow equations : II. Moderate congestion. Journal of Physics A: Mathematical and Theoretical, Volume 47 (Number 41). 415203. doi:10.1088/1751-8113/47/41/415203 ISSN 1751-8113.
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Official URL: http://dx.doi.org/10.1088/1751-8113/47/41/415203
Abstract
We find a further class of exact solutions to the Lighthill–Whitham– Richards–Payne (LWRP) traffic flow equations. As before, using two consecutive Lagrangian transformations, a linearization is achieved. Next, depending on the initial density, we either obtain exact formulae for the dependence of the car density and velocity on $x,t$, 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 not only that of Rowlands et al (2013 J. Phys. A: Math. Theor. 46 365202 (part I)) but also the two-soliton solution to the Korteweg–de Vries equation. This paper can be read independently of part I. This explains unavoidable repetitions. Possible uses of both papers in checking numerical codes are indicated. Since LWRP, numerous more elaborate models, including multiple lanes, traffic jams, tollgates, etc, abound in the literature. However, we present an exact solution. These are few and far between, other than found by inverse scattering. The literature for various models, including ours, is given. The methods used here and in part I may be useful in solving other problems, such as shallow water flow.
Item Type: | Journal Article | ||||||||||
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Divisions: | Faculty of Science, Engineering and Medicine > Science > Physics | ||||||||||
Journal or Publication Title: | Journal of Physics A: Mathematical and Theoretical | ||||||||||
Publisher: | IOP Publishing Ltd | ||||||||||
ISSN: | 1751-8113 | ||||||||||
Official Date: | 17 October 2014 | ||||||||||
Dates: |
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Volume: | Volume 47 | ||||||||||
Number: | Number 41 | ||||||||||
Article Number: | 415203 | ||||||||||
DOI: | 10.1088/1751-8113/47/41/415203 | ||||||||||
Status: | Peer Reviewed | ||||||||||
Publication Status: | Published | ||||||||||
Access rights to Published version: | Restricted or Subscription Access |
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