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Existence of optima and equilibria for traffic flow on networks
Coupling of microscopic and phase transition models at boundary
1. | Dipartimento di Matematica e Applicazioni, Università di Milano Bicocca, via R. Cozzi 53 - Edificio U5, 20125 - Milano |
2. | Joseph and Loretta Lopez Chair Professor of Mathematics, Department of Mathematical Sciences and Program Director, Center for Computational and Integrative Biology, Rutgers University - Camden, 311 N 5th Street, Camden, NJ 08102 |
References:
[1] |
D. Amadori, Initial-boundary value problems for nonlinear systems of conservation laws, NoDEA Nonlinear Differential Equations Appl., 4 (1997), 1-42.
doi: 10.1007/PL00001406. |
[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), 916-938.
doi: 10.1137/S0036139997332099. |
[4] |
S. Blandin, J. Argote, A. M. Bayen and D. B. Work, Phase transition model of non-stationary traffic flow: Definition, properties and solution method, Transportation Research Part B: Methodological, 52 (2013), 31-55. |
[5] |
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. |
[6] |
R. M. Colombo, Hyperbolic phase transitions in traffic flow, SIAM J. Appl. Math., 63 (2002), 708-721.
doi: 10.1137/S0036139901393184. |
[7] |
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. |
[8] |
C. F. Daganzo, Requiem for second-order fluid approximations of traffic flow, Transportation Research Part B, 29 (1995), 277-286.
doi: 10.1016/0191-2615(95)00007-Z. |
[9] |
F. Dubois and P. LeFloch, Boundary conditions for nonlinear hyperbolic systems of conservation laws, J. Differential Equations, 71 (1988), 93-122.
doi: 10.1016/0022-0396(88)90040-X. |
[10] |
M. Garavello and B. Piccoli, Coupling of lwr and phase transition models at boundary, Journal of Hyperbolic Differential Equations, 10 (2013), 577-636. |
[11] |
D. C. Gazis, R. Herman and R. W. Rothery, Nonlinear follow-the-leader models of traffic flow, Operations Res., 9 (1961), 545-567.
doi: 10.1287/opre.9.4.545. |
[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] |
D. Helbing, From microscopic to macroscopic traffic models, in "A perspective look at nonlinear media,'' Lecture Notes in Phys., Springer, (1998), 122-139.
doi: 10.1007/BFb0104959. |
[14] |
D. Helbing, A. Hennecke, V. Shvetsov and M. Treiber, Micro- and macro-simulation of freeway traffic, Traffic flow-modelling and simulation, Math. Comput. Modelling, 35 (2002), 517-547.
doi: 10.1016/S0895-7177(02)80019-X. |
[15] |
C. Lattanzio and B. Piccoli, Coupling of microscopic and macroscopic traffic models at boundaries, Math. Models Methods Appl. Sci., 20 (2010), 2349-2370.
doi: 10.1142/S0218202510004945. |
[16] |
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. |
[17] |
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. |
[18] |
H. J. Payne, Models of freeway traffic and control, in mathematical models of public systems, Simul. Counc. Proc., 1 (1971). |
[19] |
I. Prigogine and R. Herman, Kinetic theory of vehicular traffic, American Elsevier Pub. Co., 1971. |
[20] |
P. I. Richards, Shock waves on the highway, Operations Res., 4 (1956), 42-51.
doi: 10.1287/opre.4.1.42. |
[21] |
G. B. Whitham, "Linear and Nonlinear Waves,'' Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. |
[22] |
D. B. Work, S. Blandin, O.-P. Tossavainen, B. Piccoli and A. M. Bayen, A traffic model for velocity data assimilation, Appl. Math. Res. Express. AMRX, 1 (2010), 1-35. |
[23] |
H. M. 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] |
D. Amadori, Initial-boundary value problems for nonlinear systems of conservation laws, NoDEA Nonlinear Differential Equations Appl., 4 (1997), 1-42.
doi: 10.1007/PL00001406. |
[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), 916-938.
doi: 10.1137/S0036139997332099. |
[4] |
S. Blandin, J. Argote, A. M. Bayen and D. B. Work, Phase transition model of non-stationary traffic flow: Definition, properties and solution method, Transportation Research Part B: Methodological, 52 (2013), 31-55. |
[5] |
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. |
[6] |
R. M. Colombo, Hyperbolic phase transitions in traffic flow, SIAM J. Appl. Math., 63 (2002), 708-721.
doi: 10.1137/S0036139901393184. |
[7] |
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. |
[8] |
C. F. Daganzo, Requiem for second-order fluid approximations of traffic flow, Transportation Research Part B, 29 (1995), 277-286.
doi: 10.1016/0191-2615(95)00007-Z. |
[9] |
F. Dubois and P. LeFloch, Boundary conditions for nonlinear hyperbolic systems of conservation laws, J. Differential Equations, 71 (1988), 93-122.
doi: 10.1016/0022-0396(88)90040-X. |
[10] |
M. Garavello and B. Piccoli, Coupling of lwr and phase transition models at boundary, Journal of Hyperbolic Differential Equations, 10 (2013), 577-636. |
[11] |
D. C. Gazis, R. Herman and R. W. Rothery, Nonlinear follow-the-leader models of traffic flow, Operations Res., 9 (1961), 545-567.
doi: 10.1287/opre.9.4.545. |
[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] |
D. Helbing, From microscopic to macroscopic traffic models, in "A perspective look at nonlinear media,'' Lecture Notes in Phys., Springer, (1998), 122-139.
doi: 10.1007/BFb0104959. |
[14] |
D. Helbing, A. Hennecke, V. Shvetsov and M. Treiber, Micro- and macro-simulation of freeway traffic, Traffic flow-modelling and simulation, Math. Comput. Modelling, 35 (2002), 517-547.
doi: 10.1016/S0895-7177(02)80019-X. |
[15] |
C. Lattanzio and B. Piccoli, Coupling of microscopic and macroscopic traffic models at boundaries, Math. Models Methods Appl. Sci., 20 (2010), 2349-2370.
doi: 10.1142/S0218202510004945. |
[16] |
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. |
[17] |
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. |
[18] |
H. J. Payne, Models of freeway traffic and control, in mathematical models of public systems, Simul. Counc. Proc., 1 (1971). |
[19] |
I. Prigogine and R. Herman, Kinetic theory of vehicular traffic, American Elsevier Pub. Co., 1971. |
[20] |
P. I. Richards, Shock waves on the highway, Operations Res., 4 (1956), 42-51.
doi: 10.1287/opre.4.1.42. |
[21] |
G. B. Whitham, "Linear and Nonlinear Waves,'' Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. |
[22] |
D. B. Work, S. Blandin, O.-P. Tossavainen, B. Piccoli and A. M. Bayen, A traffic model for velocity data assimilation, Appl. Math. Res. Express. AMRX, 1 (2010), 1-35. |
[23] |
H. M. 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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