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On the Verification of the Pedestrian Evacuation Model^{ †}

^{1}

^{2}

^{*}

^{†}

^{‡}

## Abstract

**:**

## 1. Introduction

## 2. Governing Equations

#### 2.1. Functional Dependence

#### 2.1.1. Integral Formulation

#### 2.1.2. Differential Formulation

## 3. Numerical Solution

#### 3.1. Splitting Schemes

#### 3.1.1. Splitting Scheme I

#### 3.1.2. Splitting Scheme II

#### 3.2. FV Discretization of the Splitting Scheme I

#### 3.2.1. FV Discretization of the Hyperbolic System (12)

#### 3.2.2. FV Approximation of $\mathit{\mu}$ in (13)

#### 3.2.3. FV Approximation of ${\mathrm{\Psi}}_{D}$

#### 3.2.4. FV Approximation of ${\mathrm{\Psi}}_{BR}$

#### 3.3. Discretization of System (14)

#### 3.4. Splitting Algorithm I

Algorithm 1 Runge-Kutta splitting |

Set the initial condition ${\mathbf{w}}^{0}$ and the method M for the determination of direction $\mathit{\mu}$, $M\in \{D,BR\}$, for$m=1,2,\dots $do(1) $\tilde{\mathbf{w}}:={\mathbf{w}}^{m-1}+{\tau}_{m}\phantom{\rule{4pt}{0ex}}\mathbf{H}\left({\mathbf{w}}^{m-1}\right)$; $\phantom{\rule{1.em}{0ex}}{\mathbf{w}}^{hyp}:=\frac{1}{2}\left({\mathbf{w}}^{m-1}+\tilde{\mathbf{w}}+{\tau}_{m}\phantom{\rule{4pt}{0ex}}\mathbf{H}\left(\tilde{\mathbf{w}}\right)\right)$; (2) ${\mathit{\mu}}_{i}^{hyp}:={\mathrm{\Psi}}_{M,i}\left({\mathbf{w}}^{hyp}\right),i\in I$; (3) $\tilde{\mathbf{w}}:={\mathbf{w}}^{hyp}+{\tau}_{m}\phantom{\rule{4pt}{0ex}}\mathbf{S}\left({\mathbf{w}}^{hyp}\right)$; $\phantom{\rule{1.em}{0ex}}{\tilde{\mathit{\mu}}}_{i}:={\mathrm{\Psi}}_{M,i}\left(\tilde{\mathbf{w}}\right),i\in I$; $\phantom{\rule{1.em}{0ex}}{\mathbf{w}}^{m}:=\frac{1}{2}\left({\mathbf{w}}^{hyp}+\tilde{\mathbf{w}}+{\tau}_{m}\phantom{\rule{4pt}{0ex}}\mathbf{S}\left(\tilde{\mathbf{w}}\right)\right)$; end for |

Algorithm 2 Euler splitting |

for$m=1,2,\dots $do(1) ${\mathbf{w}}^{hyp}:={\mathbf{w}}^{m-1}+{\tau}_{m}\phantom{\rule{4pt}{0ex}}\mathbf{H}\left({\mathbf{w}}^{m-1}\right)$; (2) ${\mathit{\mu}}_{i}^{hyp}:={\mathrm{\Psi}}_{M,i}\left({\mathbf{w}}^{hyp}\right),i\in I$; (3) ${\mathbf{w}}^{m}:={\mathbf{w}}^{hyp}+{\tau}_{m}\phantom{\rule{4pt}{0ex}}\mathbf{S}\left({\mathbf{w}}^{hyp}\right)$; end for |

**Definition**

**1.**

#### 3.5. FV-DG Discretization of the Splitting Scheme II

#### 3.6. FV-DG Approximation of $\mathit{\mu}$ in (15)

#### 3.6.1. FV-DG Approximation of ${\mathrm{\Psi}}_{D}$

#### 3.6.2. FV-DG Approximation of ${\mathrm{\Psi}}_{BR}$

#### 3.6.3. FV-DG Approximation of the Direction $\mathit{\mu}$

#### 3.7. FV-DG Discretization of (16)

#### 3.8. Resulting FV-DG Scheme

**Definition**

**2.**

#### Solution of FV-DG Scheme

## 4. Results

#### 4.1. Domain with One Obstacle

#### 4.2. Domain with Three Obstacles

#### 4.3. H-Shape Domain Evacuation

#### 4.4. T-Shape Domain Evacuation

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Liu, R.; Fu, Z.; Schadschneider, A.; Wen, Q.; Chen, J.; Liu, S. Modeling the effect of visibility on upstairs crowd evacuation by a stochastic FFCA model with finer discretization. Phys. A Stat. Mech. Appl.
**2019**, 531, 121723. [Google Scholar] [CrossRef] - Fu, Z.; Zhan, X.; Luo, L.; Schadschneider, A.; Chen, J. Modeling fatigue of ascending stair evacuation with modified fine discrete floor field cellular automata. Phys. Lett. A
**2019**, 383, 1897–1906. [Google Scholar] [CrossRef] - Twarogowska, M.; Goatin, P.; Duvigneau, R. Macroscopic modeling and simulation of room evacuation. Appl. Math. Model.
**2014**, 38, 5781–5795. [Google Scholar] [CrossRef] - Buchmueller, S.; Weidmann, U. Parameters of pedestrians, Pedestrian Traffic and Walking Facilities. IVT Schriftenreihe
**2007**, 132. [Google Scholar] [CrossRef] - Bellomo, N.; Marasco, A.; Romano, A. From the modelling of driver’s behavior to hydrodynamics models and problems of traffic flow. Nonlinear Anal. RWA
**2002**, 3, 339–363. [Google Scholar] [CrossRef] - Venuti, F.; Bruno, L.; Bellomo, N. Crowd dynamics on a moving platform: Mathematical modelling and application to lively footbridges. Math. Comp. Model.
**2007**, 45, 252–269. [Google Scholar] [CrossRef] - Bellomo, N.; Dogbé, C. On the modelling crowd dynamics from scaling to hyperbolic macroscopic models. Math. Model. Methods Appl. Sci.
**2008**, 18, 1317–1345. [Google Scholar] [CrossRef][Green Version] - Dolejší, V.; Felcman, J.; Kubera, P. FV–DG Method for the Pedestrian Flow Problem. Comput. Fluids
**2019**, 183, 1–15. [Google Scholar] [CrossRef] - Jiang, Y.; Zhang, P.; Wong, S.; Liu, R. A higher-order macroscopic model for pedestrian flows. Phys. A Stat. Mech. Appl.
**2010**, 389, 4623–4635. [Google Scholar] [CrossRef] - Payne, H. Models of Freeway Traffic and Control; Simulation Councils, Incorporated: La Jolla, CA, USA, 1971. [Google Scholar]
- Whitham, G. Linear and Nonlinear Waves; Pure and Applied Mathematics; Wiley: Hoboken, NJ, USA, 1974. [Google Scholar]
- Berres, S.; Ruiz-Baier, R.; Schwandt, H.; Tory, E. An adaptive finite-volume method for a model of two-phase pedestrian flow. Netw. Heterog. Media
**2011**, 6, 401–423. [Google Scholar] [CrossRef] - Dridi, M.H. Simulation of high density pedestrian flow: Microscopic model. Open J. Model. Simul.
**2015**, 3, 81–95. [Google Scholar] [CrossRef][Green Version] - Marno, P. Crowded-Macroscopic and Microscopic Models for Pedestrian Dynamics. Ph.D. Thesis, University of Reading, Reading, UK, 2002. [Google Scholar]
- Felcman, J.; Kubera, P. A cellular automaton model for a pedestrian flow problem. Math. Model. Nat. Phenom.
**2021**, 16, 11. [Google Scholar] [CrossRef] - Dogbe, C. On the modelling of crowd dynamics by generalized kinetic models. J. Math. Anal. Appl.
**2012**, 387, 512–532. [Google Scholar] [CrossRef][Green Version] - Toro, E.F. Riemann Solvers and Numerical Methods for Fluid Dynamics; Springer: Berlin, Germany, 1997. [Google Scholar]
- Dijkstra, E.W. A note on two problems in connexion with graphs. Numer. Math.
**1959**, 1, 269–271. [Google Scholar] [CrossRef][Green Version] - Cormen, T.H.; Leiserson, C.E.; Rivest, R.L.; Stein, C. Introduction to Algorithms, 2nd ed.; The MIT Press: Cambridge, MA, USA, 2001. [Google Scholar]
- Bornemann, F.; Rasch, C. Finite-element Discretization of Static Hamilton-Jacobi Equations based on a Local Variational Principle. Comput. Vis. Sci.
**2006**, 9, 57–69. [Google Scholar] [CrossRef][Green Version] - Felcman, J.; Dolejší, V.; Kubera, P. Discontinuous Galerkin Method for the Pedestrian Flow Problem. In ICNAAM 2017 AIP Conference Proceedings 1978:1; Simos, T.E., Tsitouras, C., Eds.; American Institute of Physics: Melville, NY, USA, 2018; pp. 0300191–0300194. [Google Scholar] [CrossRef]
- Kubera, P.; Felcman, J. On a numerical flux for the pedestrian flow equations. J. Appl. Math. Stat. Inform.
**2015**, 11, 79–96. [Google Scholar] [CrossRef][Green Version] - Dolejší, V.; Feistauer, M. Discontinuous Galerkin Method; Springer: New York, NY, USA, 2015. [Google Scholar]
- Petrášová, T. Application of the Dijkstra’s Algorithm in the Pedestrian Flow Problem. Bachelor’s Thesis, Charles University in Prague, Prague, Czech Republic, 2016. [Google Scholar]
- Deuflhard, P. Newton Methods for Nonlinear Problems; Springer Series in Computational Mathematics; Springer: Berlin/Heidelberg, Germany, 2004; Volume 35. [Google Scholar]
- Dolejší, V.; Roskovec, F.; Vlasák, M. Residual based error estimates for the space-time discontinuous Galerkin method applied to the compressible flows. Comput. Fluids
**2015**, 117, 304–324. [Google Scholar] [CrossRef] - Felcman, J.; Kubera, P. On the Eikonal Equation in the Pedestrian Flow Problem. In ICNAAM 2016 AIP Conference Proceedings 1863:1; Simos, T.E., Tsitouras, C., Eds.; American Institute of Physics: Melville, NY, USA, 2017; pp. 5600241–5600244. [Google Scholar]

**Figure 1.**Examples of finite volume meshes with one circular obstacle at point $[32,5]$ with radius $R=2$ m (

**top**) and with three circular obstacles at points $[32,5]$, $[37,2.5]$ and $[37,7.5]$ with radius $R=1$ m (

**bottom**).

**Figure 2.**Domain with one obstacle, FV(BR), FV(DA) (

**top**) and FV-DG(BR), FV-DG(DA) (

**bottom**). Density distribution at time $t=10,15,20$ s obtained on the finest meshes.

**Figure 3.**Domain with one circular obstacle, the time evolution of the total number of pedestrians $M\left({t}_{m}\right)$ for the finest grid $\#{\mathcal{T}}_{h}=9056$ (FVM) and $\#{\mathcal{T}}_{h}=9082$ (FV-DG), total view (

**left**) and the detailed view close to the end of the evacuation (

**right**).

**Figure 4.**Domain with three obstacles, FV(BR), FV(DA) (

**top**) and FV-DG(BR), FV-DG(DA) (

**bottom**). Density distribution at time $t=10,15,20$ s obtained on the finest meshes.

**Figure 5.**Domain with three circular obstacles, the time evolution of the total number of pedestrians $M\left({t}_{m}\right)$ for the finest grid $\#{\mathcal{T}}_{h}$ = 11,660 (FV) and $\#{\mathcal{T}}_{h}$ = 11,636 (FV-DG), total view (

**left**) and the detailed view close to the end of the evacuation (

**right**).

**Figure 6.**H-shape domain, examples of the computational grids (FV-DG) with $\#{\mathcal{T}}_{h}=2941$ (

**left**) and $\#{\mathcal{T}}_{h}$ = 12,742 (

**right**) elements, the exits at the right-hand side bottom corner are marked by the blue color.

**Figure 7.**H-shape domain, FV(BR), FV(DA) (

**top**) and FV-DG(BR), FV-DG(DA) (

**bottom**). Density distribution obtained on the finest meshes at time $t=5,15,30,45$ s.

**Figure 8.**H-shape domain, the time evolution of the total number of pedestrians $M\left({t}_{m}\right)$ for the finest grid (

**left**) and for the coarsened grid (

**right**).

**Figure 9.**T-shape domain, computational grids (FV-DG) with $\#{\mathcal{T}}_{h}=2578$ (

**left**) and $\#{\mathcal{T}}_{h}$ = 10,848 (

**right**) elements, the exits are marked by the blue color.

**Figure 10.**T-shape domain, the time evolution of the total number of pedestrians $M\left({t}_{m}\right)$ for the finest grid $\#{\mathcal{T}}_{h}$ = 10,310 (FVM) and $\#{\mathcal{T}}_{h}$ = 10,848 (FV-DG), total view (

**left**) and the detailed view close to the end of the evacuation (

**right**).

**Figure 11.**T-shape domain, FV(BR), FV(DA) (

**top**) and FV-DG(BR), FV-DG(DA) (

**bottom**). Density distribution obtained on the finest meshes at time $t=9,15$ s.

**Figure 12.**T-shape domain pedestrian trajectories. First row FV in order Bornemann-Rasch algorithm and Dijkstra’s algorithm, second row FV-DG method in the same order. Pedestrian trajectories for starting position ${x}^{0}=\left[\pm 0.4,-12\right]$ (blue color) and ${x}^{0}=\left[\pm 1,-16\right]$, respectively (green color). The pedestrians starting at ${x}^{0}=\left[\pm 0.4,-12\right]$ cross the line $\left[.,0\right]$ at time $t=9$ s and reach the more distant exit at time $t=18$ s. The pedestrians starting at ${x}^{0}=\left[\pm 1,-16\right]$ cross the line $\left[.,0\right]$ at time $t=15$ s and reach the nearer exit at time $t=23$ s.

**Table 1.**Domain with one obstacle, FV (Splitting Scheme I-left) and FV-DG (Splitting Scheme II-right)-Comparison of computations on series of meshes, $NELEM=\#{\mathcal{T}}_{h}$, ${T}_{evac}^{BR}=$ evacuation time with Bornemann-Rasch algorithm for the determination of $\varphi $ in (8), ${T}_{evac}^{DA}=$ evacuation time with Dijkstra’s algorithm for the determination of $\varphi $ in (6).

$\mathrm{NELEM}$ | ${T}_{\mathrm{evac}}^{\mathrm{BR}}$ [s] | ${T}_{\mathrm{evac}}^{\mathrm{DA}}$ [s] | $\mathrm{NELEM}$ | ${T}_{\mathrm{evac}}^{\mathrm{BR}}$ [s] | ${T}_{\mathrm{evac}}^{\mathrm{DA}}$ [s] |
---|---|---|---|---|---|

1114 | 35.0 | 35.1 | 1126 | 35.2 | 36.2 |

2148 | 34.7 | 34.7 | 2132 | 35.2 | 36.1 |

4610 | 34.9 | 34.7 | 4627 | 35.0 | 35.5 |

9056 | 34.8 | 34.7 | 9082 | 34.8 | 35.4 |

**Table 2.**Domain with three obstacles, FV (left) and FV-DG (right)-Comparison of computations on the series of meshes, $NELEM=\#{\mathcal{T}}_{h}$, ${T}_{evac}^{BR}=$ evacuation time with Bornemann-Rasch algorithm, ${T}_{evac}^{DA}=$ evacuation time with Dijkstra’s algorithm.

$\mathrm{NELEM}$ | ${T}_{\mathrm{evac}}^{\mathrm{BR}}$ [s] | ${T}_{\mathrm{evac}}^{\mathrm{DA}}$ [s] | $\mathrm{NELEM}$ | ${T}_{\mathrm{evac}}^{\mathrm{BR}}$ [s] | ${T}_{\mathrm{evac}}^{\mathrm{DA}}$ [s] |
---|---|---|---|---|---|

1622 | 34.5 | 34.9 | 1653 | 35.0 | 35.6 |

2653 | 34.6 | 34.5 | 2616 | 34.9 | 35.5 |

7115 | 34.7 | 34.5 | 7084 | 34.9 | 35.1 |

11,660 | 35.0 | 34.6 | 11,636 | 34.7 | 35.0 |

**Table 3.**H-shape domain, FV (left) and FV-DG (right), $NELEM=\#{\mathcal{T}}_{h}$, ${T}_{evac}^{BR}=$ evacuation time with Bornemann-Rasch algorithm, ${T}_{evac}^{DA}=$ evacuation time with Dijkstra’s algorithm.

$\mathrm{NELEM}$ | ${T}_{\mathrm{evac}}^{\mathrm{BR}}$ [s] | ${T}_{\mathrm{evac}}^{\mathrm{DA}}$ [s] | $\mathrm{NELEM}$ | ${T}_{\mathrm{evac}}^{\mathrm{BR}}$ [s] | ${T}_{\mathrm{evac}}^{\mathrm{DA}}$ [s] |
---|---|---|---|---|---|

2994 | 90.8 | 81.0 | 2941 | 87.5 | 79.1 |

12,835 | 90.5 | 81.0 | 12,742 | 90.5 | 80.0 |

**Table 4.**T-shape domain, FV (left) and FV-DG (right), $NELEM=\#{\mathcal{T}}_{h}$, ${T}_{evac}^{BR}=$ evacuation time with Bornemann-Rasch algorithm, ${T}_{evac}^{DA}=$ evacuation time with Dijkstra’s algorithm.

$\mathrm{NELEM}$ | ${T}_{\mathrm{evac}}^{\mathrm{BR}}$ [s] | ${T}_{\mathrm{evac}}^{\mathrm{DA}}$ [s] | $\mathrm{NELEM}$ | ${T}_{\mathrm{evac}}^{\mathrm{BR}}$ [s] | ${T}_{\mathrm{evac}}^{\mathrm{DA}}$ [s] |
---|---|---|---|---|---|

2678 | 26.7 | 25.4 | 2578 | 28.1 | 27.9 |

10,310 | 26.7 | 25.2 | 10,848 | 28.4 | 26.6 |

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**MDPI and ACS Style**

Kubera, P.; Felcman, J. On the Verification of the Pedestrian Evacuation Model. *Mathematics* **2021**, *9*, 1525.
https://doi.org/10.3390/math9131525

**AMA Style**

Kubera P, Felcman J. On the Verification of the Pedestrian Evacuation Model. *Mathematics*. 2021; 9(13):1525.
https://doi.org/10.3390/math9131525

**Chicago/Turabian Style**

Kubera, Petr, and Jiří Felcman. 2021. "On the Verification of the Pedestrian Evacuation Model" *Mathematics* 9, no. 13: 1525.
https://doi.org/10.3390/math9131525