# American Institute of Mathematical Sciences

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 & 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.  doi: 10.3934/dcds.2012.32.1939.  Google Scholar [2] B. Andreianov, P. Goatin and N. Seguin, Finite volume schemes for locally constrained conservation laws,, With supplementary material available online, 115 (2010), 609.  doi: 10.1007/s00211-009-0286-7.  Google Scholar [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.  doi: 10.1007/s00205-010-0389-4.  Google Scholar [4] D. Braess, Über ein Paradoxon aus der Verkehrsplanung,, Unternehmensforschung, 12 (1968), 258.   Google Scholar [5] A. Bressan, "Hyperbolic Systems of Conservation Laws. The One-Dimensional Cauchy Problem,", Oxford Lecture Series in Mathematics and its Applications, (2000).   Google Scholar [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.  doi: 10.1007/s10665-007-9148-4.  Google Scholar [7] C. Cancès and N. Seguin, Error estimate for Godunov approximation of locally constrained conservation laws,, SIAM J. Numer. Anal., 50 (2012), 3036.  doi: 10.1137/110836912.  Google Scholar [8] C. Chalons, Numerical approximation of a macroscopic model of pedestrian flows,, SIAM J. Sci. Comput., 29 (2007), 539.  doi: 10.1137/050641211.  Google Scholar [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.  doi: 10.4171/IFB/186.  Google Scholar [10] G.-Q. Chen and H. Frid, Divergence-measure fields and hyperbolic conservation laws,, Arch. Ration. Mech. Anal., 147 (1999), 89.  doi: 10.1007/s002050050146.  Google Scholar [11] P. Colella, Glimm's method for gas dynamics,, SIAM J. Sci. Statist. Comput., 3 (1982), 76.  doi: 10.1137/0903007.  Google Scholar [12] R. M. Colombo and P. Goatin, A well posed conservation law with a variable unilateral constraint,, J. Differential Equations, 234 (2007), 654.  doi: 10.1016/j.jde.2006.10.014.  Google Scholar [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, 82 (2010), 244.  doi: 10.1142/9789814280303_0022.  Google Scholar [14] _______, On the modelling and management of traffic,, ESAIM Math. Model. Numer. Anal., 45 (2011), 853.  doi: 10.1051/m2an/2010105.  Google Scholar [15] R. M. Colombo and M. D. Rosini, Pedestrian flows and non-classical shocks,, Math. Methods Appl. Sci., 28 (2005), 1553.  doi: 10.1002/mma.624.  Google Scholar [16] _______, Existence of nonclassical solutions in a pedestrian flow model,, Nonl. Analysis: RWA, 10 (2009), 2716.  doi: 10.1016/j.nonrwa.2008.08.002.  Google Scholar [17] M. G. Crandall and L. Tartar, Some relations between nonexpansive and order preserving mappings,, Proc. AMS, 78 (1980), 385.  doi: 10.1090/S0002-9939-1980-0553381-X.  Google Scholar [18] C. M. Dafermos, Polygonal approximations of solutions of the initial value problem for a conservation law,, J. Math. Anal. Appl., 38 (1972), 33.  doi: 10.1016/0022-247X(72)90114-X.  Google Scholar [19] M. L. Delle Monache and P. Goatin, Scalar conservation laws with moving density constraints arising in traffic flow modeling,, INRIA Research Report, (8119).   Google Scholar [20] R. Eymard, T. Gallouët and R. Herbin, Finite volume methods,, in, VII (2000), 713.   Google Scholar [21] M. Garavello and P. Goatin, The Aw-Rascle traffic model with locally constrained flow,, J. Math. Anal. Appl., 378 (2011), 634.  doi: 10.1016/j.jmaa.2011.01.033.  Google Scholar [22] M. Garavello and B. Piccoli, "Traffic Flow on Networks. Conservation Laws Models,", AIMS Series on Applied Mathematics, 1 (2006).   Google Scholar [23] I. M. Gel'fand, Some problems in the theory of quasi-linear equations,, Uspehi Mat. Nauk, 14 (1959), 87.   Google Scholar [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.  Google Scholar [25] H. Holden and N. H. Risebro, "Front Tracking for Hyperbolic Conservation Laws,", Applied Mathematical Sciences, (2002).  doi: 10.1007/978-3-642-56139-9.  Google Scholar [26] S. N. Kružhkov, First order quasilinear equations with several independent variables,, Mat. Sb. (N.S.), 81 (1970), 228.   Google Scholar [27] P. G. LeFloch, "Hyperbolic Systems of Conservation Laws. The Theory of Classical and Nonclassical Shock Waves,", Lectures in Mathematics ETH Zürich, (2002).  doi: 10.1007/978-3-0348-8150-0.  Google Scholar [28] T. P. Liu, The Riemann problem for general systems of conservation laws,, J. Differential Equations, 18 (1975), 218.  doi: 10.1016/0022-0396(75)90091-1.  Google Scholar [29] J. Málek, J. Nečas, M. Rokyta and M. Růžička, "Weak and Measure-Valued Solutions to Evolutionary PDEs,", Applied Mathematics and Mathematical Computation, 13 (1996).   Google Scholar [30] E. Yu. Panov, Existence of strong traces for quasi-solutions of multidimensional conservation laws,, J. Hyperbolic Differ. Equ., 4 (2007), 729.  doi: 10.1142/S0219891607001343.  Google Scholar [31] M. D. Rosini, Nonclassical interactions portrait in a macroscopic pedestrian flow model,, J. Differential Equations, 246 (2009), 408.  doi: 10.1016/j.jde.2008.03.018.  Google Scholar [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.  doi: 10.1016/S0196-8858(82)80010-9.  Google Scholar [33] A. I. Vol'pert, Spaces $BV$ and quasilinear equations,, Mat. Sb. (N.S.), 73(115) (1967), 255.   Google Scholar

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.  doi: 10.3934/dcds.2012.32.1939.  Google Scholar [2] B. Andreianov, P. Goatin and N. Seguin, Finite volume schemes for locally constrained conservation laws,, With supplementary material available online, 115 (2010), 609.  doi: 10.1007/s00211-009-0286-7.  Google Scholar [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.  doi: 10.1007/s00205-010-0389-4.  Google Scholar [4] D. Braess, Über ein Paradoxon aus der Verkehrsplanung,, Unternehmensforschung, 12 (1968), 258.   Google Scholar [5] A. Bressan, "Hyperbolic Systems of Conservation Laws. The One-Dimensional Cauchy Problem,", Oxford Lecture Series in Mathematics and its Applications, (2000).   Google Scholar [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.  doi: 10.1007/s10665-007-9148-4.  Google Scholar [7] C. Cancès and N. Seguin, Error estimate for Godunov approximation of locally constrained conservation laws,, SIAM J. Numer. Anal., 50 (2012), 3036.  doi: 10.1137/110836912.  Google Scholar [8] C. Chalons, Numerical approximation of a macroscopic model of pedestrian flows,, SIAM J. Sci. Comput., 29 (2007), 539.  doi: 10.1137/050641211.  Google Scholar [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.  doi: 10.4171/IFB/186.  Google Scholar [10] G.-Q. Chen and H. Frid, Divergence-measure fields and hyperbolic conservation laws,, Arch. Ration. Mech. Anal., 147 (1999), 89.  doi: 10.1007/s002050050146.  Google Scholar [11] P. Colella, Glimm's method for gas dynamics,, SIAM J. Sci. Statist. Comput., 3 (1982), 76.  doi: 10.1137/0903007.  Google Scholar [12] R. M. Colombo and P. Goatin, A well posed conservation law with a variable unilateral constraint,, J. Differential Equations, 234 (2007), 654.  doi: 10.1016/j.jde.2006.10.014.  Google Scholar [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, 82 (2010), 244.  doi: 10.1142/9789814280303_0022.  Google Scholar [14] _______, On the modelling and management of traffic,, ESAIM Math. Model. Numer. Anal., 45 (2011), 853.  doi: 10.1051/m2an/2010105.  Google Scholar [15] R. M. Colombo and M. D. Rosini, Pedestrian flows and non-classical shocks,, Math. Methods Appl. Sci., 28 (2005), 1553.  doi: 10.1002/mma.624.  Google Scholar [16] _______, Existence of nonclassical solutions in a pedestrian flow model,, Nonl. Analysis: RWA, 10 (2009), 2716.  doi: 10.1016/j.nonrwa.2008.08.002.  Google Scholar [17] M. G. Crandall and L. Tartar, Some relations between nonexpansive and order preserving mappings,, Proc. AMS, 78 (1980), 385.  doi: 10.1090/S0002-9939-1980-0553381-X.  Google Scholar [18] C. M. Dafermos, Polygonal approximations of solutions of the initial value problem for a conservation law,, J. Math. Anal. Appl., 38 (1972), 33.  doi: 10.1016/0022-247X(72)90114-X.  Google Scholar [19] M. L. Delle Monache and P. Goatin, Scalar conservation laws with moving density constraints arising in traffic flow modeling,, INRIA Research Report, (8119).   Google Scholar [20] R. Eymard, T. Gallouët and R. Herbin, Finite volume methods,, in, VII (2000), 713.   Google Scholar [21] M. Garavello and P. Goatin, The Aw-Rascle traffic model with locally constrained flow,, J. Math. Anal. Appl., 378 (2011), 634.  doi: 10.1016/j.jmaa.2011.01.033.  Google Scholar [22] M. Garavello and B. Piccoli, "Traffic Flow on Networks. Conservation Laws Models,", AIMS Series on Applied Mathematics, 1 (2006).   Google Scholar [23] I. M. Gel'fand, Some problems in the theory of quasi-linear equations,, Uspehi Mat. Nauk, 14 (1959), 87.   Google Scholar [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.  Google Scholar [25] H. Holden and N. H. Risebro, "Front Tracking for Hyperbolic Conservation Laws,", Applied Mathematical Sciences, (2002).  doi: 10.1007/978-3-642-56139-9.  Google Scholar [26] S. N. Kružhkov, First order quasilinear equations with several independent variables,, Mat. Sb. (N.S.), 81 (1970), 228.   Google Scholar [27] P. G. LeFloch, "Hyperbolic Systems of Conservation Laws. The Theory of Classical and Nonclassical Shock Waves,", Lectures in Mathematics ETH Zürich, (2002).  doi: 10.1007/978-3-0348-8150-0.  Google Scholar [28] T. P. Liu, The Riemann problem for general systems of conservation laws,, J. Differential Equations, 18 (1975), 218.  doi: 10.1016/0022-0396(75)90091-1.  Google Scholar [29] J. Málek, J. Nečas, M. Rokyta and M. Růžička, "Weak and Measure-Valued Solutions to Evolutionary PDEs,", Applied Mathematics and Mathematical Computation, 13 (1996).   Google Scholar [30] E. Yu. Panov, Existence of strong traces for quasi-solutions of multidimensional conservation laws,, J. Hyperbolic Differ. Equ., 4 (2007), 729.  doi: 10.1142/S0219891607001343.  Google Scholar [31] M. D. Rosini, Nonclassical interactions portrait in a macroscopic pedestrian flow model,, J. Differential Equations, 246 (2009), 408.  doi: 10.1016/j.jde.2008.03.018.  Google Scholar [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.  doi: 10.1016/S0196-8858(82)80010-9.  Google Scholar [33] A. I. Vol'pert, Spaces $BV$ and quasilinear equations,, Mat. Sb. (N.S.), 73(115) (1967), 255.   Google Scholar
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