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Virtual billiards in pseudo–euclidean spaces: Discrete hamiltonian and contact integrability
The Riemann Problem at a Junction for a Phase Transition Traffic Model
Department of Mathematics and its Applications, University of Milano Bicocca, Via R. Cozzi 55, 20125 Milano, Italy |
We extend the Phase Transition model for traffic proposed in [
References:
[1] |
A. Aw and M. Rascle,
Resurrection of "second order" models of traffic flow, SIAM J. Appl. Math., 60 (2000), 916-938.
doi: 10.1137/S0036139997332099. |
[2] |
S. Blandin, D. Work, P. Goatin, B. Piccoli and A. Bayen,
A general phase transition model
for vehicular traffic, SIAM J. Appl. Math., 71 (2011), 107-127.
doi: 10.1137/090754467. |
[3] |
G. M. Coclite, M. Garavello and B. Piccoli,
Traffic flow on a road network, SIAM J. Math. Anal., 36 (2005), 1862-1886.
doi: 10.1137/S0036141004402683. |
[4] |
R. M. Colombo,
Hyperbolic phase transitions in traffic flow, SIAM J. Appl. Math., 63 (2002), 708-721.
doi: 10.1137/S0036139901393184. |
[5] |
R. M. Colombo, Phase transitions in hyperbolic conservation laws, In Progress in analysis, Vol. I, II (Berlin, 2001), pages 1279-1287. World Sci. Publ., River Edge, NJ, 2003. |
[6] |
R. M. Colombo, P. Goatin and B. Piccoli,
Road networks with phase transitions, J. Hyperbolic Differ. Equ., 7 (2010), 85-106.
doi: 10.1142/S0219891610002025. |
[7] |
R. M. Colombo and F. Marcellini,
A mixed ODE-PDE model for vehicular traffic, Math. Methods Appl. Sci., 38 (2015), 1292-1302.
doi: 10.1002/mma.3146. |
[8] |
R. M. Colombo, F. Marcellini and M. Rascle,
A 2-phase traffic model based on a speed bound, SIAM J. Appl. Math., 70 (2010), 2652-2666.
doi: 10.1137/090752468. |
[9] |
M. Garavello, K. Han and B. Piccoli,
Models for Vehicular Traffic on Networks Volume 9 of AIMS Series on Applied Mathematics, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2016. |
[10] |
M. Garavello and B. Piccoli,
Traffic flow on a road network using the Aw-Rascle model, Comm. Partial Differential Equations, 31 (2006), 243-275.
doi: 10.1080/03605300500358053. |
[11] |
M. Garavello and B. Piccoli,
Traffic Flow on Networks volume 1 of AIMS Series on Applied Mathematics, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2006. |
[12] |
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. |
[13] |
M. Herty, S. Moutari and M. Rascle,
Optimization criteria for modelling intersections of
vehicular traffic flow, Netw. Heterog. Media, 1 (2006), 275-294.
doi: 10.3934/nhm.2006.1.275. |
[14] |
M. Herty and M. Rascle,
Coupling conditions for a class of second-order models for traffic
flow, SIAM Journal on Mathematical Analysis, 38 (2006), 595-616.
doi: 10.1137/05062617X. |
[15] |
H. Holden and N. H. Risebro,
A mathematical model of traffic flow on a network of unidirectional roads, SIAM J. Math. Anal., 26 (1995), 999-1017.
doi: 10.1137/S0036141093243289. |
[16] |
J. P. Lebacque, X. Louis, S. Mammar, B. Schnetzlera and H. Haj-Salem,
Modélisation du trafic autoroutier au second ordre, Comptes Rendus Mathematique, 346 (2008), 1203-1206.
doi: 10.1016/j.crma.2008.09.024. |
[17] |
M. J. Lighthill and G. B. Whitham,
On kinematic waves. Ⅱ. A theory of traffic flow on long
crowded roads, Proc. Roy. Soc. London. Ser. A., 229 (1955), 317-345.
doi: 10.1098/rspa.1955.0089. |
[18] |
F. Marcellini,
Free-congested and micro-macro descriptions of traffic flow, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 543-556.
doi: 10.3934/dcdss.2014.7.543. |
[19] |
S. Moutari and M. Rascle,
A hybrid Lagrangian model based on the Aw-Rascle traffic flow
model, SIAM J. Appl. Math., 68 (2007), 413-436.
doi: 10.1137/060678415. |
[20] |
P. I. Richards,
Shock waves on the highway, Operations Res., 4 (1956), 42-51.
doi: 10.1287/opre.4.1.42. |
[21] |
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. |
show all references
References:
[1] |
A. Aw and M. Rascle,
Resurrection of "second order" models of traffic flow, SIAM J. Appl. Math., 60 (2000), 916-938.
doi: 10.1137/S0036139997332099. |
[2] |
S. Blandin, D. Work, P. Goatin, B. Piccoli and A. Bayen,
A general phase transition model
for vehicular traffic, SIAM J. Appl. Math., 71 (2011), 107-127.
doi: 10.1137/090754467. |
[3] |
G. M. Coclite, M. Garavello and B. Piccoli,
Traffic flow on a road network, SIAM J. Math. Anal., 36 (2005), 1862-1886.
doi: 10.1137/S0036141004402683. |
[4] |
R. M. Colombo,
Hyperbolic phase transitions in traffic flow, SIAM J. Appl. Math., 63 (2002), 708-721.
doi: 10.1137/S0036139901393184. |
[5] |
R. M. Colombo, Phase transitions in hyperbolic conservation laws, In Progress in analysis, Vol. I, II (Berlin, 2001), pages 1279-1287. World Sci. Publ., River Edge, NJ, 2003. |
[6] |
R. M. Colombo, P. Goatin and B. Piccoli,
Road networks with phase transitions, J. Hyperbolic Differ. Equ., 7 (2010), 85-106.
doi: 10.1142/S0219891610002025. |
[7] |
R. M. Colombo and F. Marcellini,
A mixed ODE-PDE model for vehicular traffic, Math. Methods Appl. Sci., 38 (2015), 1292-1302.
doi: 10.1002/mma.3146. |
[8] |
R. M. Colombo, F. Marcellini and M. Rascle,
A 2-phase traffic model based on a speed bound, SIAM J. Appl. Math., 70 (2010), 2652-2666.
doi: 10.1137/090752468. |
[9] |
M. Garavello, K. Han and B. Piccoli,
Models for Vehicular Traffic on Networks Volume 9 of AIMS Series on Applied Mathematics, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2016. |
[10] |
M. Garavello and B. Piccoli,
Traffic flow on a road network using the Aw-Rascle model, Comm. Partial Differential Equations, 31 (2006), 243-275.
doi: 10.1080/03605300500358053. |
[11] |
M. Garavello and B. Piccoli,
Traffic Flow on Networks volume 1 of AIMS Series on Applied Mathematics, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2006. |
[12] |
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. |
[13] |
M. Herty, S. Moutari and M. Rascle,
Optimization criteria for modelling intersections of
vehicular traffic flow, Netw. Heterog. Media, 1 (2006), 275-294.
doi: 10.3934/nhm.2006.1.275. |
[14] |
M. Herty and M. Rascle,
Coupling conditions for a class of second-order models for traffic
flow, SIAM Journal on Mathematical Analysis, 38 (2006), 595-616.
doi: 10.1137/05062617X. |
[15] |
H. Holden and N. H. Risebro,
A mathematical model of traffic flow on a network of unidirectional roads, SIAM J. Math. Anal., 26 (1995), 999-1017.
doi: 10.1137/S0036141093243289. |
[16] |
J. P. Lebacque, X. Louis, S. Mammar, B. Schnetzlera and H. Haj-Salem,
Modélisation du trafic autoroutier au second ordre, Comptes Rendus Mathematique, 346 (2008), 1203-1206.
doi: 10.1016/j.crma.2008.09.024. |
[17] |
M. J. Lighthill and G. B. Whitham,
On kinematic waves. Ⅱ. A theory of traffic flow on long
crowded roads, Proc. Roy. Soc. London. Ser. A., 229 (1955), 317-345.
doi: 10.1098/rspa.1955.0089. |
[18] |
F. Marcellini,
Free-congested and micro-macro descriptions of traffic flow, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 543-556.
doi: 10.3934/dcdss.2014.7.543. |
[19] |
S. Moutari and M. Rascle,
A hybrid Lagrangian model based on the Aw-Rascle traffic flow
model, SIAM J. Appl. Math., 68 (2007), 413-436.
doi: 10.1137/060678415. |
[20] |
P. I. Richards,
Shock waves on the highway, Operations Res., 4 (1956), 42-51.
doi: 10.1287/opre.4.1.42. |
[21] |
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. |










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