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March  2016, 11(1): 29-47. doi: 10.3934/nhm.2016.11.29

Solutions of the Aw-Rascle-Zhang system with point constraints

Boris P. Andreianov 1, , Carlotta Donadello 2, , Ulrich Razafison 2, , Julien Y. Rolland 3, and  Massimiliano D. Rosini 4,

1. 

LMB Laboratoire de Mathématiques CNRS UMR6623, Université de Franche-Comté, 16 route de Gray, 25030 Besançon Cedex, France
2. 

Laboratoire de Mathématiques CNRS UMR 6623, Université de Franche-Comté, 16 route de Gray, 25030 Besançon Cedex
3. 

LMB Laboratoire de Mathématiques CNRS UMR6623, Université de Franche-Comté, 16 route de Gray, 25030 Besancon Cedex, France
4. 

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

Received  April 2015 Revised  September 2015 Published  January 2016

We revisit the entropy formulation and the wave-front tracking construction of physically admissible solutions of the Aw-Rascle and Zhang (ARZ) ``second-order'' model for vehicular traffic. A Kruzhkov-like family of entropies is introduced to select the admissible shocks. This tool allows to define rigorously the appropriate notion of admissible weak solution and to approximate the solutions of the ARZ model with point constraint. Stability of solutions w.r.t. strong convergence is justified. We propose a finite volumes numerical scheme for the constrained ARZ, and we show that it can correctly locate contact discontinuities and take the constraint into account.
Keywords: Aw-Rascle and Zhang model, numerical experiments., admissible solutions, entropies, Road traffic modeling, point constraint, renormalization.
Mathematics Subject Classification: Primary: 35L65; Secondary: 90B2.
Citation: Boris P. Andreianov, Carlotta Donadello, Ulrich Razafison, Julien Y. Rolland, Massimiliano D. Rosini. Solutions of the Aw-Rascle-Zhang system with point constraints. Networks & Heterogeneous Media, 2016, 11 (1) : 29-47. doi: 10.3934/nhm.2016.11.29
References:
[1]

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 (2016), appeared online. doi: 10.1051/m2an/2015078.  Google Scholar

[2]

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, URL http://aimsciences.org/journals/displayArticlesnew.jsp?paperID=10696.  Google Scholar

[3]

B. Andreianov, C. Donadello and M. D. Rosini, Crowd dynamics and conservation laws with nonlocal constraints and capacity drop, Mathematical Models and Methods in Applied Sciences, 24 (2014), 2685-2722. doi: 10.1142/S0218202514500341.  Google Scholar

[4]

B. Andreianov, C. Donadello and M. D. Rosini, A second order model for vehicular traffics with local point constraints on the flow, Mathematical Models and Methods in Applied Sciences (2016), appeared online. doi: 10.1142/S0218202516500172.  Google Scholar

[5]

B. Andreianov, P. Goatin and N. Seguin, Finite volume schemes for locally constrained conservation laws, Numerische Mathematik, 115 (2010), 609-645. doi: 10.1007/s00211-009-0286-7.  Google Scholar

[6]

A. Aw, Existence of a global entropic weak solution for the Aw-Rascle model, Int. J. Evol. Equ., 9 (2014), 53-70.  Google Scholar

[7]

A. Aw and M. Rascle, Resurrection of "second order" models of traffic flow, SIAM Journal on Applied Mathematics, 60 (2000), 916-938. doi: 10.1137/S0036139997332099.  Google Scholar

[8]

F. Bouchut, Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws and Well-Balanced Schemes for Sources, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2004. doi: 10.1007/b93802.  Google Scholar

[9]

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

[10]

C. Chalons and P. Goatin, Transport-equilibrium schemes for computing contact discontinuities in traffic flow modeling, Commun. Math. Sci., 5 (2007), 533-551, URL http://projecteuclid.org/euclid.cms/1188405667. doi: 10.4310/CMS.2007.v5.n3.a2.  Google Scholar

[11]

C. Chalons, P. Goatin and N. Seguin, General constrained conservation laws. application to pedestrian flow modeling, Networks and Heterogeneous Media, 8 (2013), 433-463, URL http://aimsciences.org/journals/displayArticlesnew.jsp?paperID=8648. doi: 10.3934/nhm.2013.8.433.  Google Scholar

[12]

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.  Google Scholar

[13]

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.  Google Scholar

[14]

R. M. Colombo, P. Goatin and M. D. Rosini, A macroscopic model for pedestrian flows in panic situations, Proceedings of the 4th Polish-Japanese Days. GAKUTO International Series. Mathematical Sciences and Applications, 32 (2010), 255-272.  Google Scholar

[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.  Google Scholar

[16]

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

[17]

R. M. Colombo, P. Goatin and M. D. Rosini, Conservation laws with unilateral constraints in traffic modeling, in Applied and Industrial Mathematics in Italy III, 82 (2010), 244-255. doi: 10.1142/9789814280303_0022.  Google Scholar

[18]

Colombo, M. Rinaldo, P. Goatin and M. D. Rosini, On the modelling and management of traffic, ESAIM: M2AN, 45 (2011), 853-872. doi: 10.1051/m2an/2010105.  Google Scholar

[19]

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.  Google Scholar

[20]

C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, vol. 325 of Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, Berlin, 2000. doi: 10.1007/3-540-29089-3_14.  Google Scholar

[21]

R. E. Ferreira and C. I. Kondo, Glimm method and wave-front tracking for the Aw-Rascle traffic flow model, Far East J. Math. Sci. (FJMS), 43 (2010), 203-223.  Google Scholar

[22]

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.  Google Scholar

[23]

M. Godvik and H. Hanche-Olsen, Existence of solutions for the Aw-Rascle traffic flow model with vacuum, Journal of Hyperbolic Differential Equations, 5 (2008), 45-63. doi: 10.1142/S0219891608001428.  Google Scholar

[24]

H. Holden and N. H. Risebro, Front Tracking for Hyperbolic Conservation Laws, vol. 152 of Applied Mathematical Sciences, Springer, New York, 2011. doi: 10.1007/978-3-642-23911-3.  Google Scholar

[25]

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

[26]

M. Lighthill and G. Whitham, On kinematic waves. II. A theory of traffic flow on long crowded roads, in Royal Society of London. Series A, Mathematical and Physical Sciences, 229 (1955), 317-345. doi: 10.1098/rspa.1955.0089.  Google Scholar

[27]

E. Panov, Generalized solutions of the Cauchy problem for a transport equation with discontinuous coefficients, in Instability in models connected with fluid flows. II, vol. 7 of Int. Math. Ser. (N. Y.), Springer, New York, 2008, 23-84. doi: 10.1007/978-0-387-75219-8_2.  Google Scholar

[28]

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

[29]

M. D. Rosini, Nonclassical interactions portrait in a macroscopic pedestrian flow model, Journal of Differential Equations, 246 (2009), 408-427, URL http://www.sciencedirect.com/science/article/pii/S002203960800140X. doi: 10.1016/j.jde.2008.03.018.  Google Scholar

[30]

M. Rosini, The initial-boundary value problem and the constraint, in Macroscopic Models for Vehicular Flows and Crowd Dynamics: Theory and Applications, Understanding Complex Systems, Springer International Publishing, 2013, 63-91. doi: 10.1007/978-3-319-00155-5_6.  Google Scholar

[31]

M. Rosini, Numerical applications, in Macroscopic Models for Vehicular Flows and Crowd Dynamics: Theory and Applications, Understanding Complex Systems, Springer International Publishing, 2013, 167-173. Google Scholar

[32]

H. Zhang, A non-equilibrium traffic model devoid of gas-like behavior, Transportation Research Part B: Methodological, 36 (2002), 275-290. doi: 10.1016/S0191-2615(00)00050-3.  Google Scholar

show all references

References:
[1]

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 (2016), appeared online. doi: 10.1051/m2an/2015078.  Google Scholar

[2]

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, URL http://aimsciences.org/journals/displayArticlesnew.jsp?paperID=10696.  Google Scholar

[3]

B. Andreianov, C. Donadello and M. D. Rosini, Crowd dynamics and conservation laws with nonlocal constraints and capacity drop, Mathematical Models and Methods in Applied Sciences, 24 (2014), 2685-2722. doi: 10.1142/S0218202514500341.  Google Scholar

[4]

B. Andreianov, C. Donadello and M. D. Rosini, A second order model for vehicular traffics with local point constraints on the flow, Mathematical Models and Methods in Applied Sciences (2016), appeared online. doi: 10.1142/S0218202516500172.  Google Scholar

[5]

B. Andreianov, P. Goatin and N. Seguin, Finite volume schemes for locally constrained conservation laws, Numerische Mathematik, 115 (2010), 609-645. doi: 10.1007/s00211-009-0286-7.  Google Scholar

[6]

A. Aw, Existence of a global entropic weak solution for the Aw-Rascle model, Int. J. Evol. Equ., 9 (2014), 53-70.  Google Scholar

[7]

A. Aw and M. Rascle, Resurrection of "second order" models of traffic flow, SIAM Journal on Applied Mathematics, 60 (2000), 916-938. doi: 10.1137/S0036139997332099.  Google Scholar

[8]

F. Bouchut, Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws and Well-Balanced Schemes for Sources, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2004. doi: 10.1007/b93802.  Google Scholar

[9]

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

[10]

C. Chalons and P. Goatin, Transport-equilibrium schemes for computing contact discontinuities in traffic flow modeling, Commun. Math. Sci., 5 (2007), 533-551, URL http://projecteuclid.org/euclid.cms/1188405667. doi: 10.4310/CMS.2007.v5.n3.a2.  Google Scholar

[11]

C. Chalons, P. Goatin and N. Seguin, General constrained conservation laws. application to pedestrian flow modeling, Networks and Heterogeneous Media, 8 (2013), 433-463, URL http://aimsciences.org/journals/displayArticlesnew.jsp?paperID=8648. doi: 10.3934/nhm.2013.8.433.  Google Scholar

[12]

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.  Google Scholar

[13]

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.  Google Scholar

[14]

R. M. Colombo, P. Goatin and M. D. Rosini, A macroscopic model for pedestrian flows in panic situations, Proceedings of the 4th Polish-Japanese Days. GAKUTO International Series. Mathematical Sciences and Applications, 32 (2010), 255-272.  Google Scholar

[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.  Google Scholar

[16]

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

[17]

R. M. Colombo, P. Goatin and M. D. Rosini, Conservation laws with unilateral constraints in traffic modeling, in Applied and Industrial Mathematics in Italy III, 82 (2010), 244-255. doi: 10.1142/9789814280303_0022.  Google Scholar

[18]

Colombo, M. Rinaldo, P. Goatin and M. D. Rosini, On the modelling and management of traffic, ESAIM: M2AN, 45 (2011), 853-872. doi: 10.1051/m2an/2010105.  Google Scholar

[19]

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.  Google Scholar

[20]

C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, vol. 325 of Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, Berlin, 2000. doi: 10.1007/3-540-29089-3_14.  Google Scholar

[21]

R. E. Ferreira and C. I. Kondo, Glimm method and wave-front tracking for the Aw-Rascle traffic flow model, Far East J. Math. Sci. (FJMS), 43 (2010), 203-223.  Google Scholar

[22]

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.  Google Scholar

[23]

M. Godvik and H. Hanche-Olsen, Existence of solutions for the Aw-Rascle traffic flow model with vacuum, Journal of Hyperbolic Differential Equations, 5 (2008), 45-63. doi: 10.1142/S0219891608001428.  Google Scholar

[24]

H. Holden and N. H. Risebro, Front Tracking for Hyperbolic Conservation Laws, vol. 152 of Applied Mathematical Sciences, Springer, New York, 2011. doi: 10.1007/978-3-642-23911-3.  Google Scholar

[25]

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

[26]

M. Lighthill and G. Whitham, On kinematic waves. II. A theory of traffic flow on long crowded roads, in Royal Society of London. Series A, Mathematical and Physical Sciences, 229 (1955), 317-345. doi: 10.1098/rspa.1955.0089.  Google Scholar

[27]

E. Panov, Generalized solutions of the Cauchy problem for a transport equation with discontinuous coefficients, in Instability in models connected with fluid flows. II, vol. 7 of Int. Math. Ser. (N. Y.), Springer, New York, 2008, 23-84. doi: 10.1007/978-0-387-75219-8_2.  Google Scholar

[28]

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

[29]

M. D. Rosini, Nonclassical interactions portrait in a macroscopic pedestrian flow model, Journal of Differential Equations, 246 (2009), 408-427, URL http://www.sciencedirect.com/science/article/pii/S002203960800140X. doi: 10.1016/j.jde.2008.03.018.  Google Scholar

[30]

M. Rosini, The initial-boundary value problem and the constraint, in Macroscopic Models for Vehicular Flows and Crowd Dynamics: Theory and Applications, Understanding Complex Systems, Springer International Publishing, 2013, 63-91. doi: 10.1007/978-3-319-00155-5_6.  Google Scholar

[31]

M. Rosini, Numerical applications, in Macroscopic Models for Vehicular Flows and Crowd Dynamics: Theory and Applications, Understanding Complex Systems, Springer International Publishing, 2013, 167-173. Google Scholar

[32]

H. Zhang, A non-equilibrium traffic model devoid of gas-like behavior, Transportation Research Part B: Methodological, 36 (2002), 275-290. doi: 10.1016/S0191-2615(00)00050-3.  Google Scholar

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