Today's Talk

• Macroscopic (hydrodynamic) models for pedestrians.
• Models derived/inspired from vehicular traffic models.
• Singular terms to enforce capacity constraints .
• Structure-preserving numerical analysis.

But There Are Other Models for Pedestrians!

• Reviews: Bellomo, Gibelli, Quaini, Reali (2022) , Bellomo, Liao, Quaini, Russo, Siettos (2023) .
• Force-based models: Reynolds (1987) , Helbing, Molnár (1995) , D'Orsogna, Chuang, Bertozzi, Chayes (2006) .
• Rational behaviour models: Moussaïd, Helbing, Theraulaz (2011) , Degond, Appert-Rolland, Moussaïd, Pettré, Theraulaz (2013) , RB, Carrillo, Degond (2018) , Salam, Tiwari, Klar, Sundar (2023) .
• Other hydrodynamic models: Hughes (2002) , Hughes (2003) , Colombo, Rosini (2005) , Carrillo, Martin, Wolfram (2016) .
• Cellular automata: Burstedde, Klauck, Schadschneider, Zittartz (2001) .
• Mean-field games: Dogbé (2010) , Lachapelle, Wolfram (2011) .
• Multi-scale models: Cristiani, Piccoli, Tosin (2011) .
• Collective learning: Liao, Ren, Yan (2023) .
• Differential games: Barreiro-Gomez, Masmoudi (2023) .

The Fundamental Diagram

$\rho$ is the density, $u$ is the velocity, $J$ is the flux.

Weidmann (1993), Seyfried, Steffen, Klingsch, Boltes (2005), Cao, Seyfried, Zhang, Holl, Song (2017).

Early Traffic Models (1950s)

$${\partial }_t\rho +{\partial }_x(\rho u_{\text{FD}}(\rho ))=0,$$or, defining $J_{\text{FD}}(\rho )=\rho u_{\text{FD}}(\rho )$,$${\partial }_t\rho +J_{\text{FD}}^{\prime }(\rho ){\partial }_x\rho =0.$$Information propagates at speed $J_{\text{FD}}^{\prime }(\rho )-u_{\text{FD}}(\rho )=\rho u_{\text{FD}}^{\prime }(\rho )\le 0.$

Lighthill, Whitham (1955), Richards (1956).

Second-Order Models (1970s)

$$\{\begin{array}{l}{\partial }_t\rho +{\partial }_x(\rho u)=0,\\ {\partial }_{t}u+u{\partial }_{x}u=-\frac{1}{\tau }(u-u_{\text{FD}}(\rho )+\frac{\nu }{\rho }{\partial }_x\rho ),\tau >0,\nu >0.\end{array}$$The ${\partial }_x\rho$ term on the second equation is an attempt to model congestion.

Payne (1971), Whitham (1974).

Requiem for Second-Order Models (1995)

For some increasing function $p(\rho )$, rewrite previous model as$$\{\begin{array}{l}{\partial }_t\rho +{\partial }_x(\rho u)=0,\\ {\partial }_{t}u+u{\partial }_{x}u+\frac{\nu }{\rho }{\partial }_{x}p(\rho )=-\frac{1}{\tau }(u-u_{\text{FD}}(\rho )),\tau >0,\nu >0.\end{array}$$

Characteristic speeds are $u\pm \nu \sqrt{p^{\prime }(\rho )}$: information can reach drivers from behind!

Daganzo (1995).

Resurrection of Second-Order Models (2000)

A new model:$$\{\begin{array}{l}{\partial }_t\rho +{\partial }_x(\rho u)=0,\\ ({\partial }_t+u{\partial }_x)(u+p(\rho ))=0.\end{array}$$$p(\rho )$ is an increasing function, the pseudo-pressure (think of $p(\rho )={\rho }^{\gamma }$ for $\gamma >0$).

Characteristic speeds are $u$ and $u-\rho p^{\prime }(\rho )$.

Aw, Rascle (2000).

Rescaled Aw-Rascle Model (2008)

Make the pseudo-pressure singular, and rescale:$$\{\begin{array}{l}{\partial }_t\rho +{\partial }_x(\rho u)=0,\\ ({\partial }_t+u{\partial }_x)(u+\epsilon p(\rho ))=0,0<\epsilon \ll 1,\\ p(\rho )=({\rho }^{-1}-{\rho }_{\text{max}}^{-1}{)}^{-\gamma },\gamma >0.\end{array}$$The singular term enforces the capacity constraint $\rho \le {\rho }_{\text{max}}$.

Berthelin, Degond, Delitala, Rascle (2008), Berthelin, Degond, Blanc, Moutari, Royer (2008).

The Desired Velocity $\omega$

Rewrite$$\{\begin{array}{l}{\partial }_t\rho +{\partial }_x(\rho u)=0,\\ ({\partial }_t+u{\partial }_x)(u+\epsilon p(\rho ))=0,\end{array}$$as$$\{\begin{array}{l}{\partial }_t\rho +{\partial }_x(\rho u)=0,\\ ({\partial }_t+u{\partial }_x)\omega =0,\\ u=\omega -\epsilon p(\rho ).\end{array}$$

$\omega$, the desired velocity, is advected by the model.

Properties of the Model

Rewrite model in advection-diffusion form,$$\{\begin{array}{l}{\partial }_t\rho +\mathrm{\nabla }\cdot (\rho \omega )=\epsilon \mathrm{\nabla }\cdot (\rho \mathrm{\nabla }\varphi (\rho )),\\ ({\partial }_t+u\cdot \mathrm{\nabla })\omega =0,\\ u=\omega -\epsilon \mathrm{\nabla }\varphi (\rho ).\end{array}$$

The first equation enforces capacity bound.

The second equation introduces steering behaviour.

$N\to \mathrm{\infty }$ Limit

$$\{\begin{array}{l}{\partial }_{t}f(t,x,w)+\mathrm{\nabla }\cdot \left(U_f^{R}f\right)=0,\\ U_f^R(t,x,w)=w-\epsilon \mathrm{\nabla }[\varphi ({\rho }_f^R)](t,x),\\ {\rho }_f^R(t,x)=\frac{1}{R^2}{\int }_{{\mathbb{R}}^2\times {\mathbb{R}}^2}M\left(\frac{|x-y|}{R}\right)f(t,y,w)\mathrm{d}y\mathrm{d}w.\end{array}$$

$R\to 0$ Limit

$$\{\begin{array}{l}{\partial }_{t}f+\mathrm{\nabla }\cdot \left(U_{f}f\right)=0,\\ U_f(t,x,w)=w-\epsilon \mathrm{\nabla }[\varphi ({\rho }_f)](t,x),\\ {\rho }_f(t,x)={\int }_{{\mathbb{R}}^2}f(t,x,w)\mathrm{d}w.\end{array}$$

Oligokinetic Ansatz

The local kinetic model admits a measure-valued solution$$\mu (t,x,w)=\sum _{i=1}^P{\rho }_i(t,x)\delta (w-{\omega }_i(t,x))$$if $({\rho }_i,{\omega }_i)$ satisfy the multi-fluid model$$\{\begin{array}{l}{\partial }_t{\rho }_i+\mathrm{\nabla }\cdot \left({\rho }_i{u}_i\right)=0,\\ {\partial }_t{\omega }_i+(u_i\cdot \mathrm{\nabla }){\omega }_i=0,\\ u_i={\omega }_i-\epsilon \mathrm{\nabla }\varphi (\rho ),\\ \rho =\sum _{i=1}^P{\rho }_i.\end{array}$$

Stiffness Issues (Congestion)

Model in conservative variables (1D):$$\{\begin{array}{l}{\partial }_t\rho +{\partial }_{x}q=\epsilon {\partial }_x(\rho {\partial }_x\varphi (\rho )),\\ {\partial }_{t}q+{\partial }_x({\rho }^{-1}{q}^2)=\epsilon {\partial }_x(q{\partial }_x\varphi (\rho )).\end{array}$$

Recall $q=\rho \omega$ and the equation for $\omega$,$${\partial }_t\omega +\omega {\partial }_x\omega =\epsilon {\partial }_x\varphi (\rho ){\partial }_x\omega .$$Ignoring diffusive effects, $\omega$ satisfies Burgers' equation. It tends to develop shocks, which correspond to delta-shock waves on the density equation.

Without specialised schemes, the bound $\rho \le {\rho }_{\text{max}}$ will not hold.

Semi-Discrete & Semi-Implicit Scheme

$$\begin{array}{rl}& \frac{{\rho }^{n+1}-{\rho }^n}{\mathrm{\Delta }t}+{\partial }_x\left({\rho }^n{\omega }^n\right)=\epsilon {\partial }_x\left({\rho }^n{\partial }_x{\varphi }^{n+1}\right),\\ & \frac{q^{n+1}-q^n}{\mathrm{\Delta }t}+{\partial }_x\left(q^n{\omega }^n\right)=\epsilon {\partial }_x\left(q^n{\partial }_x{\varphi }^{n+1}\right),\\ & q^n={\rho }^n{\omega }^n,\\ & {\varphi }^n=\varphi \left({\rho }^n\right),\\ & \varphi \left(\rho \right)={({\rho }^{-1}-{\rho }_{\text{max}}^{-1})}^{-\gamma }.\end{array}$$

Elliptic Problem on ${\varphi }^{n+1}$

The equation for the density becomes$$-\epsilon \mathrm{\Delta }t{\partial }_x\left({\rho }^n{\partial }_x{\varphi }^{n+1}\right)+\rho \left({\varphi }^{n+1}\right)={\rho }^n-\mathrm{\Delta }t{\partial }_x\left({\rho }^n{\omega }^n\right),$$where $\rho (\varphi )$ is the inverse of $\varphi (\rho )$. Existence and well-posedness of ${\varphi }^{n+1}$ holds provided the RHS is positive. This will hold for sufficiently small $\mathrm{\Delta }t$, independently of $\epsilon$.

Once ${\varphi }^{n+1}$ is found, the update of $q^{n+1}$ is explicit.

Hyperbolic Transport

$$\begin{array}{rl}& \frac{{\rho }_i^{n+1}-{\rho }_i^n}{\mathrm{\Delta }t}+\frac{F_{i+1/2}^n-F_{i-1/2}^n}{\mathrm{\Delta }x}=\epsilon \frac{D_{i+1/2}^{n+1}-D_{i-1/2}^{n+1}}{\mathrm{\Delta }x},\\ & \frac{q_i^{n+1}-q_i^n}{\mathrm{\Delta }t}+\frac{G_{i+1/2}^n-G_{i-1/2}^n}{\mathrm{\Delta }x}=\epsilon \frac{C_{i+1/2}^{n+1}-C_{i-1/2}^{n+1}}{\mathrm{\Delta }x}.\end{array}$$The first-order scheme uses an upwind discretisation for the transport terms:$$\begin{array}{rl}& F_{i+1/2}^n={\rho }_i^n{\left({\omega }_{i+1/2}^n\right)}^{+}+{\rho }_{i+1}^n{\left({\omega }_{i+1/2}^n\right)}^{-},\\ & G_{i+1/2}^n=q_i^n{\left({\omega }_{i+1/2}^n\right)}^{+}+q_{i+1}^n{\left({\omega }_{i+1/2}^n\right)}^{-},\\ & {\omega }_{i+1/2}^n=\frac{{\omega }_i^n+{\omega }_{i+1}^n}{2}.\end{array}$$We also construct a second-order flux using a minmod limiter.

Parabolic Terms

$$\begin{array}{rl}& \frac{{\rho }_i^{n+1}-{\rho }_i^n}{\mathrm{\Delta }t}+\frac{F_{i+1/2}^n-F_{i-1/2}^n}{\mathrm{\Delta }x}=\epsilon \frac{D_{i+1/2}^{n+1}-D_{i-1/2}^{n+1}}{\mathrm{\Delta }x},\\ & \frac{q_i^{n+1}-q_i^n}{\mathrm{\Delta }t}+\frac{G_{i+1/2}^n-G_{i-1/2}^n}{\mathrm{\Delta }x}=\epsilon \frac{C_{i+1/2}^{n+1}-C_{i-1/2}^{n+1}}{\mathrm{\Delta }x}.\end{array}$$The schemes use a (second-order) centred discretisation for the diffusion:$$\begin{array}{rlr}& D_{i+1/2}^{n+1}=\frac{({\rho }_i^n+{\rho }_{i+1}^n)({\varphi }_{i+1}^{n+1}-{\varphi }_i^{n+1})}{2\mathrm{\Delta }x},& C_{i+1/2}^{n+1}=\frac{(q_i^n+q_{i+1}^n)({\varphi }_{i+1}^{n+1}-{\varphi }_i^{n+1})}{2\mathrm{\Delta }x}.\end{array}$$

Elliptic Problem on ${\varphi }_i^{n+1}$

The equation for the density becomes$$-\epsilon \mathrm{\Delta }t\frac{D_{i+1/2}^{n+1}-D_{i-1/2}^{n+1}}{\mathrm{\Delta }x}+\rho \left({\varphi }_i^{n+1}\right)={\rho }_i^n-\mathrm{\Delta }t\frac{F_{i+1/2}^n-F_{i-1/2}^n}{\mathrm{\Delta }x},$$where $\rho (\varphi )$ is the inverse of $\varphi (\rho )$. Existence and well-posedness of ${\varphi }^{n+1}$ holds provided the RHS is positive, which only requires a hyperbolic-like CFL, $\mathrm{\Delta }t\sim \mathrm{\Delta }x$, independent of $\epsilon$.

Once ${\varphi }_i^{n+1}$ is found, the update of $q_i^{n+1}$ is explicit.

Comparison of First and Second-Order Schemes

Same test case. $M=2^{10}$ points.

Singular Limit Behaviour

Same test case. $M=2^{10}$ points.

Corridor Experiments

Corridor Experiments

Corridor Experiments

Crowd Collision and Model Instability

Perhaps a Saffman–Taylor instability, not clear yet.

Outlook

• Congestion function $\varphi$ is insensitive to crowd speed.
• Empirical calibration (e.g. match corridor simulations to experimental data).
• Learning the form of the congestion function from data.
• Resolve the model instability, likely through a multi-fluid or kinetic extension to the model.

References II

• N. Bellomo, L. Gibelli, A. Quaini, and A. Reali. Towards a mathematical theory of behavioral human crowds. Math. Models Methods Appl. Sci., 32(02):321–358, 2022.
• N. Bellomo, J. Liao, A. Quaini, L. Russo, and C. Siettos. Human behavioral crowds review, critical analysis and research perspectives. Math. Models Methods Appl. Sci., 33(08):1611–1659, 2023.
• F. Berthelin, P. Degond, V.L. Blanc, S. Moutari, and M.R.A.J. Royer. A traffic-flow model with constraints for the modeling of traffic jams. Math. Models Method Appl. Sci., 18(supp01):1269–1298, 2008.
• F. Berthelin, P. Degond, M. Delitala, and M. Rascle. A Model for the Formation and Evolution of Traffic Jams. Arch. Ration. Mech. An., 187(2):185–220, 2008.

References III

• C. Burstedde, K. Klauck, A. Schadschneider, and J. Zittartz. Simulation of Pedestrian Dynamics Using a Two-Dimensional Cellular Automaton. Phys. A, 295(3-4):507–525, 2001.
• S. Cao, A. Seyfried, J. Zhang, S. Holl, and W. Song. Fundamental diagrams for multidirectional pedestrian flows. J. Stat. Mech: Theory Exp., 2017(3):033404, 2017.
• J.A. Carrillo, S. Martin, and M. Wolfram. An Improved Version of the Hughes Model for Pedestrian Flow. Math. Models Methods Appl. Sci., 26(04):671–697, 2016.
• R.M. Colombo and M.D. Rosini. Pedestrian Flows and Non-Classical Shocks. Math. Meth. Appl. Sci., 28(13):1553–1567, 2005.

References IV

• E. Cristiani, B. Piccoli, and A. Tosin. Multiscale Modeling of Granular Flows with Application to Crowd Dynamics. Multiscale Model. Simul., 9(1):155–182, 2011.
• M.R. D'Orsogna, Y.L. Chuang, A.L. Bertozzi, and L.S. Chayes. Self-Propelled Particles with Soft-Core Interactions: Patterns, Stability, and Collapse. Phys. Rev. Lett., 96(10):104302, 2006.
• C.F. Daganzo. Requiem for Second-Order Fluid Approximations of Traffic Flow. Transportation Research Part B: Methodological, 29(4):277–286, 1995.
• P. Degond, C. Appert-Rolland, M. Moussaïd, J. Pettré, and G. Theraulaz. A Hierarchy of Heuristic-Based Models of Crowd Dynamics. J. Stat. Phys., 152(6):1033–1068, 2013.

References V

• C. Dogbé. Modeling Crowd Dynamics by the Mean-Field Limit Approach. Math. Comput. Model., 52(9-10):1506–1520, 2010.
• D. Helbing and P. Molnár. Social Force Model for Pedestrian Dynamics. Phys. Rev. E, 51(5):4282–4286, 1995.
• R.L. Hughes. A Continuum Theory for the Flow of Pedestrians. Transportation Research Part B: Methodological, 36(6):507–535, 2002.
• R.L. Hughes. The Flow of Human Crowds. Annu. Rev. Fluid Mech., 35(1):169–182, 2003.

References VI

• A. Lachapelle and M. Wolfram. On a Mean Field Game Approach Modeling Congestion and Aversion in Pedestrian Crowds. Transportation Research Part B: Methodological, 45(10):1572–1589, 2011.
• J. Liao, Y. Ren, and W. Yan. Kinetic modeling of a leader-follower system in crowd evacuation with collective learning. Math. Models Methods Appl. Sci., 33(05):1099–1117, 2023.
• M.J. Lighthill and G.B. Whitham. On Kinematic Waves II - A Theory of Traffic Flow on Long Crowded Roads. Proc. R. Soc. A Math. Phys. Eng. Sci., 229(1178):317–345, 1955.
• M. Moussaïd, D. Helbing, and G. Theraulaz. How simple rules determine pedestrian behavior and crowd disasters. Proc. Natl. Acad. Sci., 108(17):6884–6888, 2011.

References VII

• H.J. Payne. Model of Freeway Traffic and Control. Mathematical Model of Public System, 28:51–61, 1971.
• C.W. Reynolds. Flocks, Herds and Schools: A Distributed Behavioral Model. ACM SIGGRAPH Comput. Graph., 21(4):25–34, 1987.
• P.I. Richards. Shock Waves on the Highway. Oper. Res., 4(1):42–51, 1956.
• P.S.A. Salam, S. Tiwari, A. Klar, and S. Sundar. A numerical investigation of pedestrian dynamics based on rational behaviour in different density scenarios. Phys. A, 624:128933, 2023.

References VIII

• A. Seyfried, B. Steffen, W. Klingsch, and M. Boltes. The fundamental diagram of pedestrian movement revisited. J. Stat. Mech: Theory Exp., 2005(10):P10002–P10002, 2005.
• U. Weidmann. Transporttechnik Der Fussgänger, Transporttechnische Eigenschaften Des Fussgängerverkehrs (Literturauswertung). Institut für Verkehrsplanung, Transporttechnik, Strassen und Eisenbahnbau (IVT), ETH Zürich, 1993.
• G.B. Whitham. Linear and Nonlinear Waves. Wiley, 1974.