Advertisement

A Numerical Scheme for Evacuation Dynamics

  • Maria GokieliEmail author
  • Andrzej Szczepańczyk
Conference paper
  • 98 Downloads
Part of the Lecture Notes in Computer Science book series (LNCS, volume 12044)

Abstract

We give a stability condition for a semi–implicit numerical scheme and prove unconditional stability for an implicit scheme for a nonlinear advection – diffusion equation, meant as a model of crowd dynamics. Numerical stability is given for a wider class of equations and schemes.

Keywords

Finite elements method CFL condition Stability 

References

  1. 1.
    Borsche, R., Colombo, R.M., Garavello, M., Meurer, A.: Differential equations modeling crowd interactions. J. Nonlinear Sci. 25, 827–859 (2015)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Colombo, R.M., Gokieli, M., Rosini, M.D.: Modeling crowd dynamics through hyperbolic - elliptic equations. In: Non-Linear Partial Differential Equations, Mathematical Physics, and Stochastic Analysis – The Helge Holden Anniversary Volume, pp. 111–128. EMS Series of Congress Reports, May 2018Google Scholar
  3. 3.
    Gokieli, M.: An advection-diffusion equation as model for crowd evacuation (to appear)Google Scholar
  4. 4.
    Hughes, R.L.: A continuum theory for the flow of pedestrians. Transp. Res. Part B: Methodol. 36(6), 507–535 (2002) CrossRefGoogle Scholar
  5. 5.
    Hughes, R.L.: The flow of human crowds. Annu. Rev. Fluid Mech. 35(1), 169–182 (2003) MathSciNetCrossRefGoogle Scholar
  6. 6.
    Jiang, Y., Zhou, S., Tian, F.-B.: Macroscopic pedestrian flow model with degrading spatial information. J. Comput. Sci. 10, 36–44 (2015)CrossRefGoogle Scholar
  7. 7.
    Kachroo, P.: Pedestrian Dynamics: Mathematical Theory and Evacuation Control. CRC Press, Boca Raton (2009)CrossRefGoogle Scholar
  8. 8.
    Kamga, J.-B.A., Després, B.: CFL condition and boundary conditions for DGM approximation of convection-diffusion. SIAM J. Numer. Anal. 44(6), 2245–2269 (2006)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Szczepańczyk, A.: Master’s thesis. University of Warsaw, Interdisciplinary Centre for Mathematical and Computational Modelling (ICM) (to appear)Google Scholar
  10. 10.
    Twarogowska, M., Goatin, P., Duvigneau, R.: Macroscopic modeling and simulations of room evacuation. Appl. Math. Model. 38(24), 5781–5795 (2014)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Faculty of Mathematics and Natural Sciences - School of Exact SciencesCardinal Stefan Wyszyński UniversityWarsawPoland
  2. 2.ICMUniversity of WarsawWarsawPoland

Personalised recommendations