On a numerical flux for the pedestrian flow equations*

Kubera, P.; Felcman, J.

doi:10.1515/jamsi-2015-0014

Abstract

The pedestrian flow equations are formulated as the hyperbolic problem with a source term, completed by the eikonal equation yielding the desired direction of the pedestrian velocity. The operator splitting consisting of successive discretization of the eikonal equation, ordinary differential equation with the right hand side being the source term and the homogeneous hyperbolic system is proposed. The numerical flux of the Vijayasundaram type is proposed for the finite volume solution of the hyperbolic problem. The Vijayasundaram numerical flux, originally proposed for the hyperbolic problems possessing the homogeneity property is extended for pedestrian flow, where the homogeneity property is lost. The application of the proposed numerical flux is demonstrated on the physically relevant problem.

References

Bellomo, N. and Dogbé, C. 2008. On the modelling crowd dynamics from scaling to hyperbolic macroscopic models. Mathematical Models and Methods in Applied Sciences 18, 1317 – 1345.
Search in Google Scholar
Bornemann, F. and Rasch, C. 2006. Finite-element discretization of static hamilton-jacobi equations based on a local variational principle. Computing and Visualization in Science 9, 2, 57–69.
Search in Google Scholar
Dridi, M. H. 2015. Simulation of high density pedestrian flow: Microscopic model. Open Journal of Modelling and Simulation 3, 1, 81 – 95.
Search in Google Scholar
Eymard, R., Gallouët, T., and Herbin, R. 2000. Finite volume methods. In Handbook of Numerical Analysis, P. Ciarlet and J. Lions, Eds. Vol. VII. North-Holland, 713–1020.
Search in Google Scholar
Feistauer, M., Felcman, J., and Straškraba, I. 2003. Mathematical and computational methods for compressible flow. Clarendon Press.
Search in Google Scholar
Felcman, J. and Havle, O. 2011. On a numerical flux for the shallow water equations. Applied Mathematics and Computation 217, 11, 5160 – 5170. Special issue on Fluid Flow and Heat Transfer.10.1016/j.amc.2010.07.047
Search in Google Scholar
Felcman, J. and Kubera, P. 2015. Modeling pedestrian flow via shallow water equations. (in preparation).
Search in Google Scholar
Furmanek, P., Furst, J., and Kozel, K. 2009. High order finite volume schemes for numerical solution of 2D and 3D transonic flows. KYBERNETIKA 45, 4, 567–579.
Search in Google Scholar
Gallouët, T., Hérard, J.-M., and Seguin, N. 2003. Some approximate godunov schemes to compute shallow-water equations with topography. Comput. Fluids 32, 4, 479–513.
Search in Google Scholar
Helbing, D., Farkas, I. J., Molnár, P., and Vicsek, T. 2002. Simulation of pedestrian crowd in normal and evacuation situations. Pedestrian and Evacuation Dynamics Journal, 21 – 58.
Search in Google Scholar
Marno, P. 2002. Crowded - macroscopic and microscopic models for pedestrian dynamics. Ph.D. thesis, University of Reading.
Search in Google Scholar
Toro, E. F. 1997. Riemann Solvers and Numerical Methods for Fluid Dynamics. Springer, Berlin.10.1007/978-3-662-03490-3
Search in Google Scholar
Twarogowska, M., Goatin, P., and Duvigneau, R. 2013. Numerical study of macroscopic pedestrian flow models. Tech. rep.10.1016/j.trpro.2014.09.063
Search in Google Scholar
Vijayasundaram, G. 1982. Resolution numérique des équations d’euler pour des écoulements transsoniques avec un schéma de godunov en éléments finis. Ph.D. thesis, Paris IV.
Search in Google Scholar

On a numerical flux for the pedestrian flow equations*

Abstract

Paradigm

My account