We consider a generalized version of Hughes’ macroscopic model for crowd motion in the one-dimensional case. It consists in a scalar conservation law accounting for the conservation of the number of pedestrians, coupled with an eikonal equation giving the direction of the flux depending on pedestrian density. As a result of this non-trivial coupling, we have to deal with a conservation law with space–time discontinuous flux, whose discontinuity depends non-locally on the density itself. We propose a definition of entropy weak solution, which allows us to recover a maximum principle. Moreover, we study the structure of the solutions to Riemann-type problems, and we construct them explicitly for small times, depending on the choice of the running cost in the eikonal equation. In particular, aiming at the optimization of the evacuation time, we propose a strategy that is optimal in the case of high densities. All results are illustrated by numerical simulations.
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This research was supported by the ERC Starting Grant 2010 under the project “TRAffic Management by Macroscopic Models” and the Polonium 2011 (French–Polish cooperation program) under the project “CROwd Motion Modeling and Management”. The third author was partially supported by Narodowe Centrum Nauki, grant 4140.
An erratum to this article is available at http://dx.doi.org/10.1007/s00033-014-0448-z.
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El-Khatib, N., Goatin, P. & Rosini, M.D. On entropy weak solutions of Hughes’ model for pedestrian motion. Z. Angew. Math. Phys. 64, 223–251 (2013). https://doi.org/10.1007/s00033-012-0232-x
Mathematics Subject Classification (2000)
- Primary 35L65
- Secondary 35F21
- Pedestrian flow
- Conservation laws
- Eikonal equation
- Weak entropy solutions
- Numerical approximations