Dr Rafael Bailo — Pedestrian models with congestion effects

Pedestrian models with congestion effects

Comparison of first and second-order schemes (§4.1.2)

By field:

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

By mesh size:

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

By field:

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

By mesh size:

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

By field:

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

By mesh size:

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

By field:

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

By mesh size:

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

By field:

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

By mesh size:

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

By field:

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

By mesh size:

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

Singular limit behaviour (§4.1.3)

By field:

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

By mesh size:

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

By field:

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

By mesh size:

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

By field:

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

By mesh size:

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

By field:

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

By mesh size:

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

By field:

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

By mesh size:

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

By field:

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

Crowd density, $\rho$

Rescaled congestion, $\varepsilon\phi$

Flux, $J$

Desired momentum, $q$

Desired velocity, $\omega$

Physical velocity, $u$

By mesh size:

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

$\Delta x = 2^{-5}$

$\Delta x = 2^{-6}$

$\Delta x = 2^{-7}$

$\Delta x = 2^{-8}$

$\Delta x = 2^{-9}$

$\Delta x = 2^{-10}$

Congestion and the numerical fundamental diagram (§4.2)

Narrowing experiments

Short narrowing:

Steady states:

Animations:

Steady states:

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Steady states:

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Steady states:

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Steady states:

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Steady states:

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Medium narrowing:

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Steady states:

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Long narrowing:

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Steady states:

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Steady states:

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Steady states:

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Steady states:

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Short narrowing:

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Steady states:

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Steady states:

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Steady states:

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Steady states:

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Medium narrowing:

Steady states:

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Steady states:

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Steady states:

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Steady states:

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Steady states:

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Steady states:

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Long narrowing:

Steady states:

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Steady states:

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Steady states:

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Steady states:

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Steady states:

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Steady states:

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Short narrowing:

Steady states:

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Steady states:

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Steady states:

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Steady states:

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Steady states:

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Steady states:

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Medium narrowing:

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Steady states:

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Steady states:

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Steady states:

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Long narrowing:

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Steady states:

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Steady states:

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Steady states:

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Steady states:

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Short narrowing:

Steady states:

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Steady states:

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Steady states:

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Steady states:

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Steady states:

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Steady states:

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Medium narrowing:

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Steady states:

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Steady states:

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Steady states:

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Steady states:

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Long narrowing:

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Steady states:

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Steady states:

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Steady states:

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Steady states:

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Steady states:

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Short narrowing:

Steady states:

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Steady states:

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Steady states:

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Steady states:

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Medium narrowing:

Steady states:

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Steady states:

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Steady states:

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Steady states:

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Steady states:

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Steady states:

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Long narrowing:

Steady states:

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Steady states:

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Steady states:

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Steady states:

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Steady states:

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Short narrowing:

Steady states:

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Steady states:

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Steady states:

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Steady states:

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Steady states:

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Medium narrowing:

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Steady states:

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Steady states:

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Long narrowing:

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Obstacle experiments

Circular pillar:

Steady states:

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Steady states:

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Steady states:

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Steady states:

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Steady states:

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Steady states:

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Square pillar:

Steady states:

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Steady states:

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Steady states:

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Circular pillar:

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Square pillar:

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Steady states:

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Circular pillar:

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Square pillar:

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Steady states:

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Steady states:

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Steady states:

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Circular pillar:

Steady states:

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Steady states:

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Steady states:

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Steady states:

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Square pillar:

Steady states:

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Steady states:

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Steady states:

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Steady states:

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Steady states:

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Steady states:

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Circular pillar:

Steady states:

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Steady states:

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Steady states:

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Steady states:

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Steady states:

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Steady states:

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Square pillar:

Steady states:

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Steady states:

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Steady states:

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Steady states:

Animations:

Steady states:

Animations:

Steady states:

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Circular pillar:

Steady states:

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Steady states:

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Steady states:

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Steady states:

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Steady states:

Animations:

Steady states:

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Square pillar:

Steady states:

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Steady states:

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Steady states:

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Steady states:

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Steady states:

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Steady states:

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Crowd collision and model instability (§4.3)

Final states

$\varepsilon = 1.0$

$\varepsilon = 0.1$

$\varepsilon = 0.01$

$\varepsilon = 0.001$

$\varepsilon = 0.0001$

Final states

$\varepsilon = 1.0$

$\varepsilon = 0.1$

$\varepsilon = 0.01$

$\varepsilon = 0.001$

$\varepsilon = 0.0001$

Final states

$\varepsilon = 1.0$

$\varepsilon = 0.1$

$\varepsilon = 0.01$

$\varepsilon = 0.001$

$\varepsilon = 0.0001$

Final states

$\varepsilon = 1.0$

$\varepsilon = 0.1$

$\varepsilon = 0.01$

$\varepsilon = 0.001$

$\varepsilon = 0.0001$

Final states

$\varepsilon = 1.0$

$\varepsilon = 0.1$

$\varepsilon = 0.01$

$\varepsilon = 0.001$

$\varepsilon = 0.0001$

Final states

$\varepsilon = 1.0$

$\varepsilon = 0.1$

$\varepsilon = 0.01$

$\varepsilon = 0.001$

$\varepsilon = 0.0001$

Pedestrian models with congestion effects

Pedro Aceves-Sánchez · Rafael Bailo · Pierre Degond · Zoé Mercier
Mathematical Models and Methods in Applied Sciences, 34 (6), 2024.

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