March 2017, 10(1): 215-237. doi: 10.3934/krm.2017009

Deterministic particle approximation of the Hughes model in one space dimension

1. 

DISIM, Università degli Studi dell'Aquila, via Vetoio 1 (Coppito), 67100 L'Aquila (AQ), Italy

2. 

Instytut Matematyki, Uniwersytet Marii Curie-Skłodowskiej, plac Marii Curie-Skłodowskiej 1, 20-031 Lublin, Poland

3. 

Dipartimento di Matematica ed Informatica, Università di Catania, Viale Andrea Doria 6,95125 Catania, Italy

* Corresponding author: Marco Di Francesco

Received  March 2016 Revised  July 2016 Published  November 2016

In this paper we present a new approach to the solution to a generalized version of Hughes' models for pedestrian movements based on a follow-the-leader many particle approximation. In particular, we provide a rigorous global existence result under a smallness assumption on the initial data ensuring that the trace of the solution along the turning curve is zero for all positive times. We also focus briefly on the approximation procedure for symmetric data and Riemann type data. Two different numerical approaches are adopted for the simulation of the model, namely the proposed particle method and a Godunov type scheme. Several numerical tests are presented, which are in agreement with the theoretical prediction.

Citation: Marco Di Francesco, Simone Fagioli, Massimiliano Daniele Rosini, Giovanni Russo. Deterministic particle approximation of the Hughes model in one space dimension. Kinetic & Related Models, 2017, 10 (1) : 215-237. doi: 10.3934/krm.2017009
References:
[1]

D. Amadori and M. Di Francesco, The one-dimensional Hughes model for pedestrian flow: Riemann-type solutions, Acta Math. Sci. Ser. B Engl. Ed., 32 (2012), 259-280, http://dx.doi.org/10.1016/S0252-9602(12)60016-2. doi: 10.1016/S0252-9602(12)60016-2.

[2]

D. Amadori, P. Goatin and M. D. Rosini, Existence results for Hughes’ model for pedestrian flows, J. Math. Anal. Appl., 420 (2014), 387-406, http://dx.doi.org/10.1016/j.jmaa.2014.05.072. doi: 10.1016/j.jmaa.2014.05.072.

[3]

B. Andreianov, C. Donadello, U. Razafison and M. D. Rosini, Riemann problems with non{local point constraints and capacity drop, Mathematical Biosciences and Engineering, 12 (2015), 259-278, http://aimsciences.org/journals/displayArticlesnew.jsp?paperID=10696.

[4]

B. Andreianov, C. Donadello and M. D. Rosini, Crowd dynamics and conservation laws with nonlocal constraints and capacity drop, Math. Models Methods Appl. Sci., 24 (2014), 2685-2722, . doi: 10.1142/S0218202514500341.

[5]

B. Andreianov, C. Donadello, U. Razafison and M. D. Rosini, Qualitative behaviour and numerical approximation of solutions to conservation laws with non-local point constraints on the flux and modeling of crowd dynamics at the bottlenecks, ESAIM: M2AN, http://dx.doi.org/10.1051/m2an/2015078.

[6]

C. Bardos, A. Y. le Roux and J.-C. Nédélec, First order quasilinear equations with boundary conditions, Comm. Partial Differential Equations, 4 (1979), 1017-1034, http://dx.doi.org/10.1080/03605307908820117. doi: 10.1080/03605307908820117.

[7]

N. Bellomo and A. Bellouquid, On the modeling of crowd dynamics: Looking at the beautiful shapes of swarms, Networks and Heterogeneous Media, 6 (2011), 383-399. doi: 10.3934/nhm.2011.6.383.

[8]

D. BraessA. Nagurney and T. Wakolbinger, On a paradox of traffic planning, Transportation Science, 39 (2005), 446-450. doi: 10.1287/trsc.1050.0127.

[9]

A. Bressan, Hyperbolic Systems of Conservation Laws vol. 20 of Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, Oxford, 2000.

[10]

M. Burger, M. Di Francesco, P. A. Markowich and M.-T. Wolfram, Mean field games with nonlinear mobilities in pedestrian dynamics, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1311-1333, http://dx.doi.org/10.3934/dcdsb.2014.19.1311. doi: 10.3934/dcdsb.2014.19.1311.

[11]

J. A. Carrillo, S. Martin and M. -T. Wolfram, A local version of the hughes model for pedestrian flow, to appear in Mathematical Models and Methods in the Applied Sciences, 2015, Url: http://arxiv.org/abs/1501.07054.

[12]

R.M. Colombo and M.D. Rosini, Existence of nonclassical solutions in a pedestrian flow model, Nonlinear Anal. Real World Appl., 10 (2009), 2716-2728. doi: 10.1016/j.nonrwa.2008.08.002.

[13]

R. M. Colombo, P. Goatin and M. D. Rosini, A macroscopic model for pedestrian flows in panic situations, in Current advances in nonlinear analysis and related topics, vol. 32 of GAKUTO Internat. Ser. Math. Sci. Appl., Gakkōtosho, Tokyo, 2010,255-272.

[14]

R. M. Colombo and M. D. Rosini, Pedestrian flows and non-classical shocks, Math. Methods Appl. Sci., 28 (2005), 1553-1567, http://dx.doi.org/10.1002/mma.624. doi: 10.1002/mma.624.

[15]

C.M. Dafermos, Polygonal approximations of solutions of the initial value problem for a conservation law, J. Math. Anal. Appl., 38 (1972), 33-41. doi: 10.1016/0022-247X(72)90114-X.

[16]

M. DiFrancesco, S. Fagioli, M. D. Rosini and G. Russo, Follow-the-leader approximation of the Dirichlet boundary value problem for a scalar conservation law, In preparation.

[17]

M. DiFrancesco and M.D. Rosini, Rigorous derivation of nonlinear scalar conservation laws from follow-the-leader type models via many particle limit, Arch. Ration. Mech. Anal., 217 (2015), 831-871. doi: 10.1007/s00205-015-0843-4.

[18]

M. Di Francesco, P. A. Markowich, J.-F. Pietschmann and M.-T. Wolfram, On the Hughes’ model for pedestrian flow: The one-dimensional case, J. Differential Equations, 250 (2011), 1334-1362, http://dx.doi.org/10.1016/j.jde.2010.10.015. doi: 10.1016/j.jde.2010.10.015.

[19]

F. Dubois and P. LeFloch, Boundary conditions for nonlinear hyperbolic systems of conservation laws, J. Differential Equations, 71 (1988), 93-122. doi: 10.1016/0022-0396(88)90040-X.

[20]

N. El-Khatib, P. Goatin and M. D. Rosini, On entropy weak solutions of Hughes’ model for pedestrian motion, Z. Angew. Math. Phys., 64 (2013), 223-251, http://dx.doi.org/10.1007/s00033-012-0232-x. doi: 10.1007/s00033-012-0232-x.

[21]

P. Goatin and M. Mimault, The wave-front tracking algorithm for Hughes’ model of pedestrian motion, SIAM J. Sci. Comput., 35 (2013), B606-B622, http://dx.doi.org/10.1137/120898863. doi: 10.1137/120898863.

[22]

D. HelbingA. Johansson and H.Z. Al-Abideen, An empirical study, Physical Review E (Statistical, Nonlinear, and Soft Matter Physics), 75 (2007), 046109. doi: 10.1103/PhysRevE.75.046109.

[23]

L.F. Henderson, The statistics of crowd fluids, Nature, 229 (1971), 381-383. doi: 10.1038/229381a0.

[24]

R. L. Hughes, A continuum theory for the flow of pedestrians, Transportation Research Part B: Methodological, 36 (2002), 507-535, http://www.sciencedirect.com/science/article/pii/S0191261501000157. doi: 10.1016/S0191-2615(01)00015-7.

[25]

R. L. Hughes, The flow of human crowds, in Annual review of fluid mechanics, vol. 35 of Annu. Rev. Fluid Mech., Annual Reviews, Palo Alto, CA, 2003,169-182, http://dx.doi.org/10.1146/annurev.fluid.35.101101.161136. doi: 10.1146/annurev.fluid.35.101101.161136.

[26]

K. H. Karlsen, K. -A. Lie and N. H. Risebro, A front tracking method for conservation laws with boundary conditions, in Hyperbolic problems: theory, numerics, applications, Vol. Ⅰ (Zürich, 1998), vol. 129 of Internat. Ser. Numer. Math., Birkhäuser, Basel, 1999,493-502.

[27]

S.N. Kruzhkov, First order quasilinear equations with several independent variables., Mat. Sb. (N.S.), 81 (1970), 228-255.

[28]

M.J. Lighthill and G.B. Whitham, On kinematic waves. ii. a theory of traffic flow on long crowded roads, Royal Society of London Proceedings Series A, 229 (1955), 317-345. doi: 10.1098/rspa.1955.0089.

[29]

B. MauryA. Roudneff-Chupin and F. Santambrogio, A macroscopic crowd motion model of gradient flow type, Math. Models Methods Appl. Sci., 20 (2010), 1787-1821. doi: 10.1142/S0218202510004799.

[30]

M. Muramatsu and T. Nagatani, Jamming transition in two-dimensional pedestrian traffic, Physica A, 275 (2000), 281-291. doi: 10.1016/S0378-4371(99)00447-1.

[31]

E. Y. Panov, Existence of strong traces for quasi-solutions of multidimensional conservation laws, J. Hyperbolic Differ. Equ., 4 (2007), 729-770, http://dx.doi.org/10.1142/S0219891607001343. doi: 10.1142/S0219891607001343.

[32]

B. Piccoli and A. Tosin, Pedestrian flows in bounded domains with obstacles, Contin. Mech. Thermodyn., 21 (2009), 85-107. doi: 10.1007/s00161-009-0100-x.

[33]

P.I. Richards, Shock waves on the highway, Operations Res., 4 (1956), 42-51. doi: 10.1287/opre.4.1.42.

[34]

M. D. Rosini, Macroscopic Models for Vehicular Flows and Crowd Dynamics: Theory and Applications, Understanding Complex Systems, Springer, Heidelberg, 2013, Classical and non-classical advanced mathematics for real life applications. doi: 10.1007/978-3-319-00155-5.

[35]

A. TreuilleS. Cooper and Z. Popovic, Continuum crowds, ACM Transaction on Graphics, 25 (2006), 1160-1168, Proceedings of SCM SIGGRAPH 2006. doi: 10.1145/1179352.1142008.

[36]

M. Twarogowska, P. Goatin and R. Duvigneau, Macroscopic modeling and simulations of room evacuation, Appl. Math. Model., 38 (2014), 5781-5795, http://dx.doi.org/10.1016/j.apm.2014.03.027. doi: 10.1016/j.apm.2014.03.027.

[37]

A. Vasseur, Strong traces for solutions of multidimensional scalar conservation laws, Arch. Ration. Mech. Anal., 160 (2001), 181-193, http://dx.doi.org/10.1007/s002050100157. doi: 10.1007/s002050100157.

show all references

References:
[1]

D. Amadori and M. Di Francesco, The one-dimensional Hughes model for pedestrian flow: Riemann-type solutions, Acta Math. Sci. Ser. B Engl. Ed., 32 (2012), 259-280, http://dx.doi.org/10.1016/S0252-9602(12)60016-2. doi: 10.1016/S0252-9602(12)60016-2.

[2]

D. Amadori, P. Goatin and M. D. Rosini, Existence results for Hughes’ model for pedestrian flows, J. Math. Anal. Appl., 420 (2014), 387-406, http://dx.doi.org/10.1016/j.jmaa.2014.05.072. doi: 10.1016/j.jmaa.2014.05.072.

[3]

B. Andreianov, C. Donadello, U. Razafison and M. D. Rosini, Riemann problems with non{local point constraints and capacity drop, Mathematical Biosciences and Engineering, 12 (2015), 259-278, http://aimsciences.org/journals/displayArticlesnew.jsp?paperID=10696.

[4]

B. Andreianov, C. Donadello and M. D. Rosini, Crowd dynamics and conservation laws with nonlocal constraints and capacity drop, Math. Models Methods Appl. Sci., 24 (2014), 2685-2722, . doi: 10.1142/S0218202514500341.

[5]

B. Andreianov, C. Donadello, U. Razafison and M. D. Rosini, Qualitative behaviour and numerical approximation of solutions to conservation laws with non-local point constraints on the flux and modeling of crowd dynamics at the bottlenecks, ESAIM: M2AN, http://dx.doi.org/10.1051/m2an/2015078.

[6]

C. Bardos, A. Y. le Roux and J.-C. Nédélec, First order quasilinear equations with boundary conditions, Comm. Partial Differential Equations, 4 (1979), 1017-1034, http://dx.doi.org/10.1080/03605307908820117. doi: 10.1080/03605307908820117.

[7]

N. Bellomo and A. Bellouquid, On the modeling of crowd dynamics: Looking at the beautiful shapes of swarms, Networks and Heterogeneous Media, 6 (2011), 383-399. doi: 10.3934/nhm.2011.6.383.

[8]

D. BraessA. Nagurney and T. Wakolbinger, On a paradox of traffic planning, Transportation Science, 39 (2005), 446-450. doi: 10.1287/trsc.1050.0127.

[9]

A. Bressan, Hyperbolic Systems of Conservation Laws vol. 20 of Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, Oxford, 2000.

[10]

M. Burger, M. Di Francesco, P. A. Markowich and M.-T. Wolfram, Mean field games with nonlinear mobilities in pedestrian dynamics, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 1311-1333, http://dx.doi.org/10.3934/dcdsb.2014.19.1311. doi: 10.3934/dcdsb.2014.19.1311.

[11]

J. A. Carrillo, S. Martin and M. -T. Wolfram, A local version of the hughes model for pedestrian flow, to appear in Mathematical Models and Methods in the Applied Sciences, 2015, Url: http://arxiv.org/abs/1501.07054.

[12]

R.M. Colombo and M.D. Rosini, Existence of nonclassical solutions in a pedestrian flow model, Nonlinear Anal. Real World Appl., 10 (2009), 2716-2728. doi: 10.1016/j.nonrwa.2008.08.002.

[13]

R. M. Colombo, P. Goatin and M. D. Rosini, A macroscopic model for pedestrian flows in panic situations, in Current advances in nonlinear analysis and related topics, vol. 32 of GAKUTO Internat. Ser. Math. Sci. Appl., Gakkōtosho, Tokyo, 2010,255-272.

[14]

R. M. Colombo and M. D. Rosini, Pedestrian flows and non-classical shocks, Math. Methods Appl. Sci., 28 (2005), 1553-1567, http://dx.doi.org/10.1002/mma.624. doi: 10.1002/mma.624.

[15]

C.M. Dafermos, Polygonal approximations of solutions of the initial value problem for a conservation law, J. Math. Anal. Appl., 38 (1972), 33-41. doi: 10.1016/0022-247X(72)90114-X.

[16]

M. DiFrancesco, S. Fagioli, M. D. Rosini and G. Russo, Follow-the-leader approximation of the Dirichlet boundary value problem for a scalar conservation law, In preparation.

[17]

M. DiFrancesco and M.D. Rosini, Rigorous derivation of nonlinear scalar conservation laws from follow-the-leader type models via many particle limit, Arch. Ration. Mech. Anal., 217 (2015), 831-871. doi: 10.1007/s00205-015-0843-4.

[18]

M. Di Francesco, P. A. Markowich, J.-F. Pietschmann and M.-T. Wolfram, On the Hughes’ model for pedestrian flow: The one-dimensional case, J. Differential Equations, 250 (2011), 1334-1362, http://dx.doi.org/10.1016/j.jde.2010.10.015. doi: 10.1016/j.jde.2010.10.015.

[19]

F. Dubois and P. LeFloch, Boundary conditions for nonlinear hyperbolic systems of conservation laws, J. Differential Equations, 71 (1988), 93-122. doi: 10.1016/0022-0396(88)90040-X.

[20]

N. El-Khatib, P. Goatin and M. D. Rosini, On entropy weak solutions of Hughes’ model for pedestrian motion, Z. Angew. Math. Phys., 64 (2013), 223-251, http://dx.doi.org/10.1007/s00033-012-0232-x. doi: 10.1007/s00033-012-0232-x.

[21]

P. Goatin and M. Mimault, The wave-front tracking algorithm for Hughes’ model of pedestrian motion, SIAM J. Sci. Comput., 35 (2013), B606-B622, http://dx.doi.org/10.1137/120898863. doi: 10.1137/120898863.

[22]

D. HelbingA. Johansson and H.Z. Al-Abideen, An empirical study, Physical Review E (Statistical, Nonlinear, and Soft Matter Physics), 75 (2007), 046109. doi: 10.1103/PhysRevE.75.046109.

[23]

L.F. Henderson, The statistics of crowd fluids, Nature, 229 (1971), 381-383. doi: 10.1038/229381a0.

[24]

R. L. Hughes, A continuum theory for the flow of pedestrians, Transportation Research Part B: Methodological, 36 (2002), 507-535, http://www.sciencedirect.com/science/article/pii/S0191261501000157. doi: 10.1016/S0191-2615(01)00015-7.

[25]

R. L. Hughes, The flow of human crowds, in Annual review of fluid mechanics, vol. 35 of Annu. Rev. Fluid Mech., Annual Reviews, Palo Alto, CA, 2003,169-182, http://dx.doi.org/10.1146/annurev.fluid.35.101101.161136. doi: 10.1146/annurev.fluid.35.101101.161136.

[26]

K. H. Karlsen, K. -A. Lie and N. H. Risebro, A front tracking method for conservation laws with boundary conditions, in Hyperbolic problems: theory, numerics, applications, Vol. Ⅰ (Zürich, 1998), vol. 129 of Internat. Ser. Numer. Math., Birkhäuser, Basel, 1999,493-502.

[27]

S.N. Kruzhkov, First order quasilinear equations with several independent variables., Mat. Sb. (N.S.), 81 (1970), 228-255.

[28]

M.J. Lighthill and G.B. Whitham, On kinematic waves. ii. a theory of traffic flow on long crowded roads, Royal Society of London Proceedings Series A, 229 (1955), 317-345. doi: 10.1098/rspa.1955.0089.

[29]

B. MauryA. Roudneff-Chupin and F. Santambrogio, A macroscopic crowd motion model of gradient flow type, Math. Models Methods Appl. Sci., 20 (2010), 1787-1821. doi: 10.1142/S0218202510004799.

[30]

M. Muramatsu and T. Nagatani, Jamming transition in two-dimensional pedestrian traffic, Physica A, 275 (2000), 281-291. doi: 10.1016/S0378-4371(99)00447-1.

[31]

E. Y. Panov, Existence of strong traces for quasi-solutions of multidimensional conservation laws, J. Hyperbolic Differ. Equ., 4 (2007), 729-770, http://dx.doi.org/10.1142/S0219891607001343. doi: 10.1142/S0219891607001343.

[32]

B. Piccoli and A. Tosin, Pedestrian flows in bounded domains with obstacles, Contin. Mech. Thermodyn., 21 (2009), 85-107. doi: 10.1007/s00161-009-0100-x.

[33]

P.I. Richards, Shock waves on the highway, Operations Res., 4 (1956), 42-51. doi: 10.1287/opre.4.1.42.

[34]

M. D. Rosini, Macroscopic Models for Vehicular Flows and Crowd Dynamics: Theory and Applications, Understanding Complex Systems, Springer, Heidelberg, 2013, Classical and non-classical advanced mathematics for real life applications. doi: 10.1007/978-3-319-00155-5.

[35]

A. TreuilleS. Cooper and Z. Popovic, Continuum crowds, ACM Transaction on Graphics, 25 (2006), 1160-1168, Proceedings of SCM SIGGRAPH 2006. doi: 10.1145/1179352.1142008.

[36]

M. Twarogowska, P. Goatin and R. Duvigneau, Macroscopic modeling and simulations of room evacuation, Appl. Math. Model., 38 (2014), 5781-5795, http://dx.doi.org/10.1016/j.apm.2014.03.027. doi: 10.1016/j.apm.2014.03.027.

[37]

A. Vasseur, Strong traces for solutions of multidimensional scalar conservation laws, Arch. Ration. Mech. Anal., 160 (2001), 181-193, http://dx.doi.org/10.1007/s002050100157. doi: 10.1007/s002050100157.

Figure 1.  Evolution of $R(t,x)$ with initial data $\bar{\rho}(x)=0.25$ at times $t=0$, $t=0.5$ and $t=1$. In the particle simulations the blu dots represent particles positions, whereas the red line is the discretized density. The magenta vertical line describes the turning point evolution
Figure 2.  Evolution of $R(t,x)$ with initial data $\bar{\rho}(x)=0.6$ at times $t=0$, $t=0.5$ and $t=1$
Figure 3.  Evolution of $R(t,x)$ with initial data $\bar{\rho}(x)$ given in (24)
Figure 4.  Evolution of $R(t,x)$ with initial data $\bar{\rho}(x)$ given in (25)
Figure 5.  Evolution of $R(t,x)$ with initial data $\bar{\rho}(x)$ given in (26)
Figure 6.  Mass transfer across the turning point and non-classical shock with initial data $\bar{\rho}$ given in (26)
Figure 7.  Comparison between the Follow-the-Leader scheme (in red) and the Godunov scheme (in blu)
Figure 8.  Increasing the number of particles, and then the cells integration for the Godunov method, the agreement between the two methods greatly improves. Here we consider the initial datum $\bar{\rho}=0.3\times\,\mathbf{1}_{[-1,0]}+0.7\times \,\mathbf{1}_{(0,1]}$ and we set $N=1000$ with $1500$ time iterations
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