Figure 1.
The free phase $F$ and the congested phase $C$ resulting from (1.1) in the coordinates, from left to right, $(\rho,\eta)$ and $(\rho, \rho v)$
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June
2017, 12(2): 259-275.
doi: 10.3934/nhm.2017011
Existence of solutions to a boundary value problem for a phase transition traffic model
Dipartimento di Matematica e Applicazioni, Università di Milano Bicocca, via R. Cozzi 55, 20125 Milano, Italy |
We consider the initial boundary value problem for the phase transition traffic model introduced in [
Keywords:
Hyperbolic systems of conservation laws,
continuum traffic models,
wave-front tracking,
boundary value problem.
Mathematics Subject Classification:
Primary: 35L65; Secondary: 90B20.
Citation:
Francesca Marcellini. Existence of solutions to a boundary value problem for a phase transition traffic model. Networks and Heterogeneous Media,
2017, 12
(2)
: 259-275.
doi: 10.3934/nhm.2017011
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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] |
D. Amadori and R. M. Colombo,
Continuous dependence for 2×2 conservation laws with boundary, J. Differential Equations, 138 (1997), 229-266.
doi: 10.1006/jdeq.1997.3274. |
| [3] |
A. Aw and M. Rascle, Resurrection of "second order" models of traffic flow, SIAM J. Appl. Math., 60 (2000), 916-938 (electronic).
doi: 10.1137/S0036139997332099. |
| [4] |
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. |
| [5] |
A. Bressan,
Hyperbolic Systems of Conservation Laws, vol. 20 of Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, Oxford, 2000, The one-dimensional Cauchy problem. |
| [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, Phase transitions in hyperbolic conservation laws, in Progress in analysis, Vol. I, II (Berlin, 2001), World Sci. Publ., River Edge, NJ, 2003,1279-1287. |
| [8] |
R. M. Colombo and F. Marcellini,
A mixed ODE-PDE model for vehicular traffic, Mathematical Methods in the Applied Sciences, 38 (2015), 1292-1302.
doi: 10.1002/mma.3146. |
| [9] |
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. |
| [10] |
C. M. Dafermos,
Hyperbolic Conservation Laws in Continuum Physics, vol. 325 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 3rd edition, Springer-Verlag, Berlin, 2010.
doi: 10.1007/978-3-642-04048-1. |
| [11] |
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. |
| [12] |
M. Garavello,
Boundary value problem for a phase transition model, Netw. Heterog. Media, 11 (2016), 89-105.
doi: 10.3934/nhm.2016.11.89. |
| [13] |
M. Garavello and F. Marcellini, The godunov method for a 2-phase model, preprint, arXiv: 1703.05135. |
| [14] |
M. Garavello and F. Marcellini, The riemann problem at a junction for a phase-transition traffic model, Discrete Contin. Dyn. Syst. Ser. A, to appear. |
| [15] |
M. Garavello and B. Piccoli,
Traffic Flow on Networks, vol. 1 of AIMS Series on Applied Mathematics, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2006, Conservation laws models. |
| [16] |
M. Garavello and B. Piccoli,
Coupling of Lighthill-Whitham-Richards and phase transition models, J. Hyperbolic Differ. Equ., 10 (2013), 577-636.
doi: 10.1142/S0219891613500215. |
| [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] |
H. Holden and N. H. Risebro,
Front Tracking for Hyperbolic Conservation Laws, vol. 152 of Applied Mathematical Sciences, 2nd edition, Springer, Heidelberg, 2015.
doi: 10.1007/978-3-662-47507-2. |
| [19] |
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. |
| [20] |
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. |
| [21] |
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. |
| [22] |
P. I. Richards,
Shock waves on the highway, Operations Res., 4 (1956), 42-51.
doi: 10.1287/opre.4.1.42. |
| [23] |
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] |
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] |
D. Amadori and R. M. Colombo,
Continuous dependence for 2×2 conservation laws with boundary, J. Differential Equations, 138 (1997), 229-266.
doi: 10.1006/jdeq.1997.3274. |
| [3] |
A. Aw and M. Rascle, Resurrection of "second order" models of traffic flow, SIAM J. Appl. Math., 60 (2000), 916-938 (electronic).
doi: 10.1137/S0036139997332099. |
| [4] |
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. |
| [5] |
A. Bressan,
Hyperbolic Systems of Conservation Laws, vol. 20 of Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, Oxford, 2000, The one-dimensional Cauchy problem. |
| [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, Phase transitions in hyperbolic conservation laws, in Progress in analysis, Vol. I, II (Berlin, 2001), World Sci. Publ., River Edge, NJ, 2003,1279-1287. |
| [8] |
R. M. Colombo and F. Marcellini,
A mixed ODE-PDE model for vehicular traffic, Mathematical Methods in the Applied Sciences, 38 (2015), 1292-1302.
doi: 10.1002/mma.3146. |
| [9] |
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. |
| [10] |
C. M. Dafermos,
Hyperbolic Conservation Laws in Continuum Physics, vol. 325 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 3rd edition, Springer-Verlag, Berlin, 2010.
doi: 10.1007/978-3-642-04048-1. |
| [11] |
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. |
| [12] |
M. Garavello,
Boundary value problem for a phase transition model, Netw. Heterog. Media, 11 (2016), 89-105.
doi: 10.3934/nhm.2016.11.89. |
| [13] |
M. Garavello and F. Marcellini, The godunov method for a 2-phase model, preprint, arXiv: 1703.05135. |
| [14] |
M. Garavello and F. Marcellini, The riemann problem at a junction for a phase-transition traffic model, Discrete Contin. Dyn. Syst. Ser. A, to appear. |
| [15] |
M. Garavello and B. Piccoli,
Traffic Flow on Networks, vol. 1 of AIMS Series on Applied Mathematics, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2006, Conservation laws models. |
| [16] |
M. Garavello and B. Piccoli,
Coupling of Lighthill-Whitham-Richards and phase transition models, J. Hyperbolic Differ. Equ., 10 (2013), 577-636.
doi: 10.1142/S0219891613500215. |
| [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] |
H. Holden and N. H. Risebro,
Front Tracking for Hyperbolic Conservation Laws, vol. 152 of Applied Mathematical Sciences, 2nd edition, Springer, Heidelberg, 2015.
doi: 10.1007/978-3-662-47507-2. |
| [19] |
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. |
| [20] |
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. |
| [21] |
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. |
| [22] |
P. I. Richards,
Shock waves on the highway, Operations Res., 4 (1956), 42-51.
doi: 10.1287/opre.4.1.42. |
| [23] |
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. |
Figure 2.
Wave interactions in a road. Above, from left to right, the cases $2-1/1-2$ and $\mathcal{LW}-\mathcal{PT}/\mathcal{PT}-2$ . Below, from left to right, the cases $1-1/1$ and $\mathcal{PT}-1/\mathcal{PT}$
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