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A new model for the emergence of blood capillary networks
A two-dimensional multi-class traffic flow model
1. | Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Sapienza Università di Roma, Via Antonio Scarpa 16, 00161 Rome, Italy |
2. | Department of Mathematics, University of Mannheim, B6 28-29, 68159 Mannheim, Germany |
The aim of this work is to introduce a two-dimensional macroscopic traffic model for multiple populations of vehicles. Starting from the paper [
References:
[1] |
G. Albi, N. Bellomo, L. Fermo, S.-Y. Ha, J. Kim, L. Pareschi, D. Poyato and J. Soler,
Vehicular traffic, crowds, and swarms: From kinetic theory and multiscale methods to applications and research perspectives, Math. Models Methods Appl. Sci., 29 (2019), 1901-2005.
doi: 10.1142/S0218202519500374. |
[2] |
A. Aw, A. Klar, T. Materne and M. Rascle,
Derivation of continuum traffic flow models from microscopic follow-the-leader models, SIAM J. Appl. Math., 63 (2002), 259-278.
doi: 10.1137/S0036139900380955. |
[3] |
A. Aw and M. Rascle,
Resurrection of "second order" models of traffic flow, SIAM J. Appl. Math., 60 (2000), 916-944.
doi: 10.1137/S0036139997332099. |
[4] |
S. Benzoni-Gavage and R. M. Colombo,
An $n$-populations model for traffic flow, European J. Appl. Math., 14 (2003), 587-612.
doi: 10.1017/S0956792503005266. |
[5] |
S. Bertoluzza, S. Falletta, G. Russo and C.-W. Shu, Numerical Solutions of Partial Differential Equations, Advanced Courses in Mathematics, CRM Barcelona, Birkhäuser Verlag, Basel, 2009. |
[6] |
B. N. Chetverushkin, N. G. Churbanova, Y. N. Karamzin and M. A. Trapeznikova, A two-dimensional macroscopic model of traffic flows based on KCFD-schemes, in ECCOMAS CFD 2006: Proceedings of the European Conference on Computational Fluid Dynamics, Egmond aan Zee, The Netherlands, September 5-8, 2006, Citeseer, 2006. Google Scholar |
[7] |
R. M. Colombo,
A $2\times 2$ hyperbolic traffic flow model, Math. Comput. Modelling, 35 (2002), 683-688.
doi: 10.1016/S0895-7177(02)80029-2. |
[8] |
R. M. Colombo and F. Marcellini,
A traffic model aware of real time data, Math. Models Methods Appl. Sci., 26 (2016), 445-467.
doi: 10.1142/S0218202516500081. |
[9] |
E. Cristiani, C. de Fabritiis and B. Piccoli,
A fluid dynamic approach for traffic forecast from mobile sensors data, Commun. Appl. Ind. Math., 1 (2010), 54-71.
doi: 10.1685/2010CAIM487. |
[10] |
C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, 4$^{th}$ edition, Grundlehren der Mathematischen Wissenschaften, 325, Springer-Verlag, Berlin, 2016.
doi: 10.1007/978-3-662-49451-6. |
[11] |
C. F. Daganzo,
A continuum theory of traffic dynamics for freeways with special lanes, Transport. Res. B-Meth., 31 (1997), 83-102.
doi: 10.1016/S0191-2615(96)00017-3. |
[12] |
C. F. Daganzo,
In traffic flow, cellular automata = kinematic waves, Transp. Res. B-Meth., 40 (2006), 396-403.
doi: 10.1016/j.trb.2005.05.004. |
[13] |
S. Fan, M. Herty and B. Seibold,
Comparative model accuracy of a data-fitted generalized Aw-Rascle-Zhang model, Netw. Heterog. Media, 9 (2014), 239-268.
doi: 10.3934/nhm.2014.9.239. |
[14] |
S. Fan and B. Seibold,
Data-fitted first-order traffic models and their second-order generalizations: Comparison by trajectory and sensor data, Transport. Res. Rec., 2391 (2013), 32-43.
doi: 10.3141/2391-04. |
[15] |
M. Garavello, K. Han and B. Piccoli, Models for Vehicular Traffic on Networks, American Institute of Mathematical Sciences, 2016. |
[16] |
D. C. Gazis, R. Herman and R. W. Rothery,
Nonlinear follow-the-leader models of traffic flow, Oper. Res., 9 (1961), 545-567.
doi: 10.1287/opre.9.4.545. |
[17] |
P. Goatin,
The Aw-Rascle vehicular traffic flow model with phase transitions, Math. Comput. Modelling, 44 (2006), 287-303.
doi: 10.1016/j.mcm.2006.01.016. |
[18] |
S. K. Godunov,
A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics, Matematicheskii Sbornik, 89 (1959), 271-306.
|
[19] |
D. Helbing, From microscopic to macroscopic traffic models, in A Perspective Look at Nonlinear Media
doi: 10.1007/BFb0104959. |
[20] |
M. Herty, C. Kirchner and S. Moutari,
Multi-class traffic models on road networks, Commun. Math. Sci., 4 (2006), 591-608.
doi: 10.4310/CMS.2006.v4.n3.a6. |
[21] |
M. Herty, A. Fazekas and G. Visconti,
A two-dimensional data-driven model for traffic flow on highways, Netw. Heterog. Media, 13 (2018), 217-240.
doi: 10.3934/nhm.2018010. |
[22] |
M. Herty, S. Moutari and G. Visconti,
Macroscopic modeling of multilane motorways using a two-dimensional second-order model of traffic flow, SIAM J. Appl. Math., 78 (2018), 2252-2278.
doi: 10.1137/17M1151821. |
[23] |
H. Holden and N. H. Risebro,
Models for dense multilane vehicular traffic, SIAM J. Math. Anal., 51 (2019), 3694-3713.
doi: 10.1137/19M124318X. |
[24] |
R. Illner, A. Klar and T. Materne,
Vlasov-Fokker-Planck models for multilane traffic flow, Commun. Math. Sci., 1 (2003), 1-12.
doi: 10.4310/CMS.2003.v1.n1.a1. |
[25] |
E. Kallo, A. Fazekas, S. Lamberty and M. Oeser, Microscopic traffic data obtained from videos recorded on a German motorway, https://data.mendeley.com/datasets/tzckcsrpn6/1, 2019. Google Scholar |
[26] |
A. Klar and R. Wegener,
A hierarchy of models for multilane vehicular traffic. Ⅰ. Modeling, SIAM J. Appl. Math., 59 (1999), 983-1001.
doi: 10.1137/S0036139997326946. |
[27] |
A. Klar and R. Wegener,
A hierarchy of models for multilane vehicular traffic. Ⅱ. Numerical investigations, SIAM J. Appl. Math., 59 (1999), 1002-1011.
doi: 10.1137/S0036139997326958. |
[28] |
A. Klar and R. Wegener,
Kinetic derivation of macroscopic anticipation models for vehicular traffic, SIAM J. Appl. Math., 60 (2000), 1749-1766.
doi: 10.1137/S0036139999356181. |
[29] |
J. A. Laval and L. Leclercq,
A mechanism to describe the formation and propagation of stop-and-go waves in congested freeway traffic, Philos. T. R. Soc. A, 368 (2010), 4519-4541.
doi: 10.1098/rsta.2010.0138. |
[30] |
J.-P. Lebacque,
Two-phase bounded-acceleration traffic flow model: Analytical solutions and applications, Transp. Res. Rec., 1852 (2003), 220-230.
doi: 10.3141/1852-27. |
[31] |
R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2002.
doi: 10.1017/CBO9780511791253.![]() ![]() |
[32] |
M. J. Lighthill and G. B. Whitham,
On kinematic waves. Ⅱ. A theory of traffic flow on long crowded roads, Proc. Roy. Soc. A, 229 (1955), 317-345.
doi: 10.1098/rspa.1955.0089. |
[33] |
P. Nelson, A kinetic model of vehicular traffic and its associated bimodal equilibrium solutions, in Special Issue Devoted to the Proceedings of the 13th International Conference on Transport Theory (Riccione, 1993), 24, 1995,383–409.
doi: 10.1080/00411459508205136. |
[34] |
G. F. Newell,
Nonlinear effects in the dynamics of car following, Oper. Res., 9 (1961), 209-229.
doi: 10.1287/opre.9.2.209. |
[35] |
E. Parzen,
On estimation of a probability density function and mode, Ann. Math. Statist., 33 (1962), 1065-1076.
doi: 10.1214/aoms/1177704472. |
[36] |
W. F. Phillips,
A kinetic model for traffic flow with continuum implications, Transport. Plan. Techn., 5 (1979), 131-138.
doi: 10.1080/03081067908717157. |
[37] |
B. Piccoli, K. Han, T. L. Friesz, T. Yao and J. Tang,
Second-order models and traffic data from mobile sensors, Transp. Res. C-Emer., 52 (2015), 32-56.
doi: 10.1016/j.trc.2014.12.013. |
[38] |
L. A. Pipes,
An operational analysis of traffic dynamics, J. Appl. Phys., 24 (1953), 274-281.
doi: 10.1063/1.1721265. |
[39] |
I. Prigogine and F. C. Andrews,
A Boltzmann-like approach for traffic flow, Oper. Res., 8 (1960), 789-797.
doi: 10.1287/opre.8.6.789. |
[40] |
G. Puppo, M. Semplice, A. Tosin and G. Visconti,
Kinetic models for traffic flow resulting in a reduced space of microscopic velocities, Kinet. Relat. Models, 10 (2017), 823-854.
doi: 10.3934/krm.2017033. |
[41] |
P. I. Richards,
Shock waves on the highway, Oper. Res., 4 (1956), 42-51.
doi: 10.1287/opre.4.1.42. |
[42] |
M. Rosenblatt,
Remarks on some nonparametric estimates of a density function, Ann. Math. Statist., 27 (1956), 832-837.
doi: 10.1214/aoms/1177728190. |
[43] |
M. Schönhof and D. Helbing, Empirical features of congested traffic states and their implications for traffic modeling, Transport. Sci., 41 (2007), 135-166. Google Scholar |
[44] |
US Department of Transportation and Federal Highway Administration, Next Generation Simulation (NGSIM), http://ops.fhwa.dot.gov/trafficanalysistools/ngsim.htm. Google Scholar |
[45] |
G. Wong and S. Wong,
A multi-class traffic flow model–an extension of LWR model with heterogeneous drivers, Transport. Res. A-Pol, 36 (2002), 827-841.
doi: 10.1016/S0965-8564(01)00042-8. |
[46] |
H. M. Zhang,
A non-equilibrium traffic model devoid of gas-like behavior, Transp. Res. B, 36 (2002), 275-290.
doi: 10.1016/S0191-2615(00)00050-3. |
[47] |
T. Zhang and Y. X. Zheng,
Two-dimensional Riemann problem for a single conservation law, Trans. Amer. Math. Soc., 312 (1989), 589-619.
doi: 10.1090/S0002-9947-1989-0930070-3. |
show all references
References:
[1] |
G. Albi, N. Bellomo, L. Fermo, S.-Y. Ha, J. Kim, L. Pareschi, D. Poyato and J. Soler,
Vehicular traffic, crowds, and swarms: From kinetic theory and multiscale methods to applications and research perspectives, Math. Models Methods Appl. Sci., 29 (2019), 1901-2005.
doi: 10.1142/S0218202519500374. |
[2] |
A. Aw, A. Klar, T. Materne and M. Rascle,
Derivation of continuum traffic flow models from microscopic follow-the-leader models, SIAM J. Appl. Math., 63 (2002), 259-278.
doi: 10.1137/S0036139900380955. |
[3] |
A. Aw and M. Rascle,
Resurrection of "second order" models of traffic flow, SIAM J. Appl. Math., 60 (2000), 916-944.
doi: 10.1137/S0036139997332099. |
[4] |
S. Benzoni-Gavage and R. M. Colombo,
An $n$-populations model for traffic flow, European J. Appl. Math., 14 (2003), 587-612.
doi: 10.1017/S0956792503005266. |
[5] |
S. Bertoluzza, S. Falletta, G. Russo and C.-W. Shu, Numerical Solutions of Partial Differential Equations, Advanced Courses in Mathematics, CRM Barcelona, Birkhäuser Verlag, Basel, 2009. |
[6] |
B. N. Chetverushkin, N. G. Churbanova, Y. N. Karamzin and M. A. Trapeznikova, A two-dimensional macroscopic model of traffic flows based on KCFD-schemes, in ECCOMAS CFD 2006: Proceedings of the European Conference on Computational Fluid Dynamics, Egmond aan Zee, The Netherlands, September 5-8, 2006, Citeseer, 2006. Google Scholar |
[7] |
R. M. Colombo,
A $2\times 2$ hyperbolic traffic flow model, Math. Comput. Modelling, 35 (2002), 683-688.
doi: 10.1016/S0895-7177(02)80029-2. |
[8] |
R. M. Colombo and F. Marcellini,
A traffic model aware of real time data, Math. Models Methods Appl. Sci., 26 (2016), 445-467.
doi: 10.1142/S0218202516500081. |
[9] |
E. Cristiani, C. de Fabritiis and B. Piccoli,
A fluid dynamic approach for traffic forecast from mobile sensors data, Commun. Appl. Ind. Math., 1 (2010), 54-71.
doi: 10.1685/2010CAIM487. |
[10] |
C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, 4$^{th}$ edition, Grundlehren der Mathematischen Wissenschaften, 325, Springer-Verlag, Berlin, 2016.
doi: 10.1007/978-3-662-49451-6. |
[11] |
C. F. Daganzo,
A continuum theory of traffic dynamics for freeways with special lanes, Transport. Res. B-Meth., 31 (1997), 83-102.
doi: 10.1016/S0191-2615(96)00017-3. |
[12] |
C. F. Daganzo,
In traffic flow, cellular automata = kinematic waves, Transp. Res. B-Meth., 40 (2006), 396-403.
doi: 10.1016/j.trb.2005.05.004. |
[13] |
S. Fan, M. Herty and B. Seibold,
Comparative model accuracy of a data-fitted generalized Aw-Rascle-Zhang model, Netw. Heterog. Media, 9 (2014), 239-268.
doi: 10.3934/nhm.2014.9.239. |
[14] |
S. Fan and B. Seibold,
Data-fitted first-order traffic models and their second-order generalizations: Comparison by trajectory and sensor data, Transport. Res. Rec., 2391 (2013), 32-43.
doi: 10.3141/2391-04. |
[15] |
M. Garavello, K. Han and B. Piccoli, Models for Vehicular Traffic on Networks, American Institute of Mathematical Sciences, 2016. |
[16] |
D. C. Gazis, R. Herman and R. W. Rothery,
Nonlinear follow-the-leader models of traffic flow, Oper. Res., 9 (1961), 545-567.
doi: 10.1287/opre.9.4.545. |
[17] |
P. Goatin,
The Aw-Rascle vehicular traffic flow model with phase transitions, Math. Comput. Modelling, 44 (2006), 287-303.
doi: 10.1016/j.mcm.2006.01.016. |
[18] |
S. K. Godunov,
A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics, Matematicheskii Sbornik, 89 (1959), 271-306.
|
[19] |
D. Helbing, From microscopic to macroscopic traffic models, in A Perspective Look at Nonlinear Media
doi: 10.1007/BFb0104959. |
[20] |
M. Herty, C. Kirchner and S. Moutari,
Multi-class traffic models on road networks, Commun. Math. Sci., 4 (2006), 591-608.
doi: 10.4310/CMS.2006.v4.n3.a6. |
[21] |
M. Herty, A. Fazekas and G. Visconti,
A two-dimensional data-driven model for traffic flow on highways, Netw. Heterog. Media, 13 (2018), 217-240.
doi: 10.3934/nhm.2018010. |
[22] |
M. Herty, S. Moutari and G. Visconti,
Macroscopic modeling of multilane motorways using a two-dimensional second-order model of traffic flow, SIAM J. Appl. Math., 78 (2018), 2252-2278.
doi: 10.1137/17M1151821. |
[23] |
H. Holden and N. H. Risebro,
Models for dense multilane vehicular traffic, SIAM J. Math. Anal., 51 (2019), 3694-3713.
doi: 10.1137/19M124318X. |
[24] |
R. Illner, A. Klar and T. Materne,
Vlasov-Fokker-Planck models for multilane traffic flow, Commun. Math. Sci., 1 (2003), 1-12.
doi: 10.4310/CMS.2003.v1.n1.a1. |
[25] |
E. Kallo, A. Fazekas, S. Lamberty and M. Oeser, Microscopic traffic data obtained from videos recorded on a German motorway, https://data.mendeley.com/datasets/tzckcsrpn6/1, 2019. Google Scholar |
[26] |
A. Klar and R. Wegener,
A hierarchy of models for multilane vehicular traffic. Ⅰ. Modeling, SIAM J. Appl. Math., 59 (1999), 983-1001.
doi: 10.1137/S0036139997326946. |
[27] |
A. Klar and R. Wegener,
A hierarchy of models for multilane vehicular traffic. Ⅱ. Numerical investigations, SIAM J. Appl. Math., 59 (1999), 1002-1011.
doi: 10.1137/S0036139997326958. |
[28] |
A. Klar and R. Wegener,
Kinetic derivation of macroscopic anticipation models for vehicular traffic, SIAM J. Appl. Math., 60 (2000), 1749-1766.
doi: 10.1137/S0036139999356181. |
[29] |
J. A. Laval and L. Leclercq,
A mechanism to describe the formation and propagation of stop-and-go waves in congested freeway traffic, Philos. T. R. Soc. A, 368 (2010), 4519-4541.
doi: 10.1098/rsta.2010.0138. |
[30] |
J.-P. Lebacque,
Two-phase bounded-acceleration traffic flow model: Analytical solutions and applications, Transp. Res. Rec., 1852 (2003), 220-230.
doi: 10.3141/1852-27. |
[31] |
R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2002.
doi: 10.1017/CBO9780511791253.![]() ![]() |
[32] |
M. J. Lighthill and G. B. Whitham,
On kinematic waves. Ⅱ. A theory of traffic flow on long crowded roads, Proc. Roy. Soc. A, 229 (1955), 317-345.
doi: 10.1098/rspa.1955.0089. |
[33] |
P. Nelson, A kinetic model of vehicular traffic and its associated bimodal equilibrium solutions, in Special Issue Devoted to the Proceedings of the 13th International Conference on Transport Theory (Riccione, 1993), 24, 1995,383–409.
doi: 10.1080/00411459508205136. |
[34] |
G. F. Newell,
Nonlinear effects in the dynamics of car following, Oper. Res., 9 (1961), 209-229.
doi: 10.1287/opre.9.2.209. |
[35] |
E. Parzen,
On estimation of a probability density function and mode, Ann. Math. Statist., 33 (1962), 1065-1076.
doi: 10.1214/aoms/1177704472. |
[36] |
W. F. Phillips,
A kinetic model for traffic flow with continuum implications, Transport. Plan. Techn., 5 (1979), 131-138.
doi: 10.1080/03081067908717157. |
[37] |
B. Piccoli, K. Han, T. L. Friesz, T. Yao and J. Tang,
Second-order models and traffic data from mobile sensors, Transp. Res. C-Emer., 52 (2015), 32-56.
doi: 10.1016/j.trc.2014.12.013. |
[38] |
L. A. Pipes,
An operational analysis of traffic dynamics, J. Appl. Phys., 24 (1953), 274-281.
doi: 10.1063/1.1721265. |
[39] |
I. Prigogine and F. C. Andrews,
A Boltzmann-like approach for traffic flow, Oper. Res., 8 (1960), 789-797.
doi: 10.1287/opre.8.6.789. |
[40] |
G. Puppo, M. Semplice, A. Tosin and G. Visconti,
Kinetic models for traffic flow resulting in a reduced space of microscopic velocities, Kinet. Relat. Models, 10 (2017), 823-854.
doi: 10.3934/krm.2017033. |
[41] |
P. I. Richards,
Shock waves on the highway, Oper. Res., 4 (1956), 42-51.
doi: 10.1287/opre.4.1.42. |
[42] |
M. Rosenblatt,
Remarks on some nonparametric estimates of a density function, Ann. Math. Statist., 27 (1956), 832-837.
doi: 10.1214/aoms/1177728190. |
[43] |
M. Schönhof and D. Helbing, Empirical features of congested traffic states and their implications for traffic modeling, Transport. Sci., 41 (2007), 135-166. Google Scholar |
[44] |
US Department of Transportation and Federal Highway Administration, Next Generation Simulation (NGSIM), http://ops.fhwa.dot.gov/trafficanalysistools/ngsim.htm. Google Scholar |
[45] |
G. Wong and S. Wong,
A multi-class traffic flow model–an extension of LWR model with heterogeneous drivers, Transport. Res. A-Pol, 36 (2002), 827-841.
doi: 10.1016/S0965-8564(01)00042-8. |
[46] |
H. M. Zhang,
A non-equilibrium traffic model devoid of gas-like behavior, Transp. Res. B, 36 (2002), 275-290.
doi: 10.1016/S0191-2615(00)00050-3. |
[47] |
T. Zhang and Y. X. Zheng,
Two-dimensional Riemann problem for a single conservation law, Trans. Amer. Math. Soc., 312 (1989), 589-619.
doi: 10.1090/S0002-9947-1989-0930070-3. |













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