June  2013, 8(2): 433-463. doi: 10.3934/nhm.2013.8.433

General constrained conservation laws. Application to pedestrian flow modeling

1. 

Univ. Paris Diderot, Sorbonne Paris Cité, Laboratoire Jacques-Louis Lions, UMR 7598, UPMC, CNRS, F-75205 Paris, France

2. 

INRIA Sophia Antipolis - Méditerranée, EPI OPALE, 2004, route des Lucioles - BP 93, 06902 Sophia Antipolis Cedex

3. 

UPMC Univ Paris 06 and CNRS UMR 7598, Laboratoire J.-L. Lions, 75005 Paris

Received  July 2012 Revised  November 2012 Published  May 2013

We extend the results on conservation laws with local flux constraint obtained in [2, 12] to general (non-concave) flux functions and non-classical solutions arising in pedestrian flow modeling [15]. We first provide a well-posedness result based on wave-front tracking approximations and the Kružhkov doubling of variable technique for a general conservation law with constrained flux. This provides a sound basis for dealing with non-classical solutions accounting for panic states in the pedestrian flow model introduced by Colombo and Rosini [15]. In particular, flux constraints are used here to model the presence of doors and obstacles. We propose a "front-tracking" finite volume scheme allowing to sharply capture classical and non-classical discontinuities. Numerical simulations illustrating the Braess paradox are presented as validation of the method.
Citation: Christophe Chalons, Paola Goatin, Nicolas Seguin. General constrained conservation laws. Application to pedestrian flow modeling. Networks and Heterogeneous Media, 2013, 8 (2) : 433-463. doi: 10.3934/nhm.2013.8.433
References:
[1]

B. Andreianov and N. Seguin, Analysis of a Burgers equation with singular resonant source term and convergence of well-balanced schemes, Discrete Contin. Dyn. Syst., 32 (2012), 1939-1964. doi: 10.3934/dcds.2012.32.1939.

[2]

B. Andreianov, P. Goatin and N. Seguin, Finite volume schemes for locally constrained conservation laws, With supplementary material available online, Numer. Math., 115 (2010), 609-645. doi: 10.1007/s00211-009-0286-7.

[3]

B. Andreianov, K. H. Karlsen and N. H. Risebro, A theory of $L^1$-dissipative solvers for scalar conservation laws with discontinuous flux, Arch. Ration. Mech. Anal., 201 (2011), 27-86. doi: 10.1007/s00205-010-0389-4.

[4]

D. Braess, Über ein Paradoxon aus der Verkehrsplanung, Unternehmensforschung, 12 (1968), 258-268.

[5]

A. Bressan, "Hyperbolic Systems of Conservation Laws. The One-Dimensional Cauchy Problem," Oxford Lecture Series in Mathematics and its Applications, Vol. 20, Oxford University Press, Oxford, 2000.

[6]

R. Bürger, A. García, K. H. Karlsen and J. D. Towers, A family of numerical schemes for kinematic flows with discontinuous flux, J. Engrg. Math., 60 (2008), 387-425. doi: 10.1007/s10665-007-9148-4.

[7]

C. Cancès and N. Seguin, Error estimate for Godunov approximation of locally constrained conservation laws, SIAM J. Numer. Anal., 50 (2012), 3036-3060. doi: 10.1137/110836912.

[8]

C. Chalons, Numerical approximation of a macroscopic model of pedestrian flows, SIAM J. Sci. Comput., 29 (2007), 539-555 (electronic). doi: 10.1137/050641211.

[9]

C. Chalons and P. Goatin, Godunov scheme and sampling technique for computing phase transitions in traffic flow modeling, Interfaces Free Bound., 10 (2008), 197-221. doi: 10.4171/IFB/186.

[10]

G.-Q. Chen and H. Frid, Divergence-measure fields and hyperbolic conservation laws, Arch. Ration. Mech. Anal., 147 (1999), 89-118. doi: 10.1007/s002050050146.

[11]

P. Colella, Glimm's method for gas dynamics, SIAM J. Sci. Statist. Comput., 3 (1982), 76-110. doi: 10.1137/0903007.

[12]

R. M. Colombo and P. Goatin, A well posed conservation law with a variable unilateral constraint, J. Differential Equations, 234 (2007), 654-675. doi: 10.1016/j.jde.2006.10.014.

[13]

R. M. Colombo, P. Goatin and M. D. Rosini, Conservation laws with unilateral constraints in traffic modeling, Applied and Industrial Mathematics in Italy III, Ser. Adv. Math. Appl. Sci., 82, World Sci. Publ., Hackensack, NJ, (2010), 244-255. doi: 10.1142/9789814280303_0022.

[14]

_______, On the modelling and management of traffic, ESAIM Math. Model. Numer. Anal., 45 (2011), 853-872. doi: 10.1051/m2an/2010105.

[15]

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

[16]

_______, Existence of nonclassical solutions in a pedestrian flow model, Nonl. Analysis: RWA, 10 (2009), 2716-2728. doi: 10.1016/j.nonrwa.2008.08.002.

[17]

M. G. Crandall and L. Tartar, Some relations between nonexpansive and order preserving mappings, Proc. AMS, 78 (1980), 385-390. doi: 10.1090/S0002-9939-1980-0553381-X.

[18]

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.

[19]

M. L. Delle Monache and P. Goatin, Scalar conservation laws with moving density constraints arising in traffic flow modeling, INRIA Research Report, no. 8119, October 2012.

[20]

R. Eymard, T. Gallouët and R. Herbin, Finite volume methods, in "Handbook of Numerical Analysis," Vol. VII, Handb. Numer. Anal., VII, North-Holland, Amsterdam, (2000), 713-1020.

[21]

M. Garavello and P. Goatin, The Aw-Rascle traffic model with locally constrained flow, J. Math. Anal. Appl., 378 (2011), 634-648. doi: 10.1016/j.jmaa.2011.01.033.

[22]

M. Garavello and B. Piccoli, "Traffic Flow on Networks. Conservation Laws Models," AIMS Series on Applied Mathematics, 1, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2006.

[23]

I. M. Gel'fand, Some problems in the theory of quasi-linear equations, Uspehi Mat. Nauk, 14 (1959), 87-158.

[24]

D. Helbing, A. Johansson, and H. Z. Al-Abideen, Dynamics of crowd disasters: An empirical study, Physical Review E, 75 (2007). doi: 10.1103/PhysRevE.75.046109.

[25]

H. Holden and N. H. Risebro, "Front Tracking for Hyperbolic Conservation Laws," Applied Mathematical Sciences, Vol. 152, Springer-Verlag, New York, 2002. doi: 10.1007/978-3-642-56139-9.

[26]

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

[27]

P. G. LeFloch, "Hyperbolic Systems of Conservation Laws. The Theory of Classical and Nonclassical Shock Waves," Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2002. doi: 10.1007/978-3-0348-8150-0.

[28]

T. P. Liu, The Riemann problem for general systems of conservation laws, J. Differential Equations, 18 (1975), 218-234. doi: 10.1016/0022-0396(75)90091-1.

[29]

J. Málek, J. Nečas, M. Rokyta and M. Růžička, "Weak and Measure-Valued Solutions to Evolutionary PDEs," Applied Mathematics and Mathematical Computation, 13, Chapman & Hall, London, 1996.

[30]

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

[31]

M. D. Rosini, Nonclassical interactions portrait in a macroscopic pedestrian flow model, J. Differential Equations, 246 (2009), 408-427. doi: 10.1016/j.jde.2008.03.018.

[32]

B. Temple, Global solution of the Cauchy problem for a class of $2\times 2$ nonstrictly hyperbolic conservation laws, Adv. in Appl. Math., 3 (1982), 335-375. doi: 10.1016/S0196-8858(82)80010-9.

[33]

A. I. Vol'pert, Spaces $BV$ and quasilinear equations, Mat. Sb. (N.S.), 73(115) (1967), 255-302.

show all references

References:
[1]

B. Andreianov and N. Seguin, Analysis of a Burgers equation with singular resonant source term and convergence of well-balanced schemes, Discrete Contin. Dyn. Syst., 32 (2012), 1939-1964. doi: 10.3934/dcds.2012.32.1939.

[2]

B. Andreianov, P. Goatin and N. Seguin, Finite volume schemes for locally constrained conservation laws, With supplementary material available online, Numer. Math., 115 (2010), 609-645. doi: 10.1007/s00211-009-0286-7.

[3]

B. Andreianov, K. H. Karlsen and N. H. Risebro, A theory of $L^1$-dissipative solvers for scalar conservation laws with discontinuous flux, Arch. Ration. Mech. Anal., 201 (2011), 27-86. doi: 10.1007/s00205-010-0389-4.

[4]

D. Braess, Über ein Paradoxon aus der Verkehrsplanung, Unternehmensforschung, 12 (1968), 258-268.

[5]

A. Bressan, "Hyperbolic Systems of Conservation Laws. The One-Dimensional Cauchy Problem," Oxford Lecture Series in Mathematics and its Applications, Vol. 20, Oxford University Press, Oxford, 2000.

[6]

R. Bürger, A. García, K. H. Karlsen and J. D. Towers, A family of numerical schemes for kinematic flows with discontinuous flux, J. Engrg. Math., 60 (2008), 387-425. doi: 10.1007/s10665-007-9148-4.

[7]

C. Cancès and N. Seguin, Error estimate for Godunov approximation of locally constrained conservation laws, SIAM J. Numer. Anal., 50 (2012), 3036-3060. doi: 10.1137/110836912.

[8]

C. Chalons, Numerical approximation of a macroscopic model of pedestrian flows, SIAM J. Sci. Comput., 29 (2007), 539-555 (electronic). doi: 10.1137/050641211.

[9]

C. Chalons and P. Goatin, Godunov scheme and sampling technique for computing phase transitions in traffic flow modeling, Interfaces Free Bound., 10 (2008), 197-221. doi: 10.4171/IFB/186.

[10]

G.-Q. Chen and H. Frid, Divergence-measure fields and hyperbolic conservation laws, Arch. Ration. Mech. Anal., 147 (1999), 89-118. doi: 10.1007/s002050050146.

[11]

P. Colella, Glimm's method for gas dynamics, SIAM J. Sci. Statist. Comput., 3 (1982), 76-110. doi: 10.1137/0903007.

[12]

R. M. Colombo and P. Goatin, A well posed conservation law with a variable unilateral constraint, J. Differential Equations, 234 (2007), 654-675. doi: 10.1016/j.jde.2006.10.014.

[13]

R. M. Colombo, P. Goatin and M. D. Rosini, Conservation laws with unilateral constraints in traffic modeling, Applied and Industrial Mathematics in Italy III, Ser. Adv. Math. Appl. Sci., 82, World Sci. Publ., Hackensack, NJ, (2010), 244-255. doi: 10.1142/9789814280303_0022.

[14]

_______, On the modelling and management of traffic, ESAIM Math. Model. Numer. Anal., 45 (2011), 853-872. doi: 10.1051/m2an/2010105.

[15]

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

[16]

_______, Existence of nonclassical solutions in a pedestrian flow model, Nonl. Analysis: RWA, 10 (2009), 2716-2728. doi: 10.1016/j.nonrwa.2008.08.002.

[17]

M. G. Crandall and L. Tartar, Some relations between nonexpansive and order preserving mappings, Proc. AMS, 78 (1980), 385-390. doi: 10.1090/S0002-9939-1980-0553381-X.

[18]

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.

[19]

M. L. Delle Monache and P. Goatin, Scalar conservation laws with moving density constraints arising in traffic flow modeling, INRIA Research Report, no. 8119, October 2012.

[20]

R. Eymard, T. Gallouët and R. Herbin, Finite volume methods, in "Handbook of Numerical Analysis," Vol. VII, Handb. Numer. Anal., VII, North-Holland, Amsterdam, (2000), 713-1020.

[21]

M. Garavello and P. Goatin, The Aw-Rascle traffic model with locally constrained flow, J. Math. Anal. Appl., 378 (2011), 634-648. doi: 10.1016/j.jmaa.2011.01.033.

[22]

M. Garavello and B. Piccoli, "Traffic Flow on Networks. Conservation Laws Models," AIMS Series on Applied Mathematics, 1, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2006.

[23]

I. M. Gel'fand, Some problems in the theory of quasi-linear equations, Uspehi Mat. Nauk, 14 (1959), 87-158.

[24]

D. Helbing, A. Johansson, and H. Z. Al-Abideen, Dynamics of crowd disasters: An empirical study, Physical Review E, 75 (2007). doi: 10.1103/PhysRevE.75.046109.

[25]

H. Holden and N. H. Risebro, "Front Tracking for Hyperbolic Conservation Laws," Applied Mathematical Sciences, Vol. 152, Springer-Verlag, New York, 2002. doi: 10.1007/978-3-642-56139-9.

[26]

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

[27]

P. G. LeFloch, "Hyperbolic Systems of Conservation Laws. The Theory of Classical and Nonclassical Shock Waves," Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2002. doi: 10.1007/978-3-0348-8150-0.

[28]

T. P. Liu, The Riemann problem for general systems of conservation laws, J. Differential Equations, 18 (1975), 218-234. doi: 10.1016/0022-0396(75)90091-1.

[29]

J. Málek, J. Nečas, M. Rokyta and M. Růžička, "Weak and Measure-Valued Solutions to Evolutionary PDEs," Applied Mathematics and Mathematical Computation, 13, Chapman & Hall, London, 1996.

[30]

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

[31]

M. D. Rosini, Nonclassical interactions portrait in a macroscopic pedestrian flow model, J. Differential Equations, 246 (2009), 408-427. doi: 10.1016/j.jde.2008.03.018.

[32]

B. Temple, Global solution of the Cauchy problem for a class of $2\times 2$ nonstrictly hyperbolic conservation laws, Adv. in Appl. Math., 3 (1982), 335-375. doi: 10.1016/S0196-8858(82)80010-9.

[33]

A. I. Vol'pert, Spaces $BV$ and quasilinear equations, Mat. Sb. (N.S.), 73(115) (1967), 255-302.

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