Figure 1.
Representation of the phase plane in the fixed and in the bus reference frames
-
Previous Article
On free boundary problem for compressible navier-stokes equations with temperature-dependent heat conductivity
- DCDS-B Home
- This Issue
-
Next Article
Averaging approach to cyclicity of hopf bifurcation in planar linear-quadratic polynomial discontinuous differential systems
December
2017, 22(10): 3921-3952.
doi: 10.3934/dcdsb.2017202
Moving bottlenecks for the Aw-Rascle-Zhang traffic flow model
| 1. | Inria Sophia Antipolis -Méditerranée, Université Côte d'Azur, Inria, CNRS, LJAD, 2004 route des Lucioles -BP 93,06902 Sophia Antipolis Cedex, France |
| 2. | Laboratoire de Mathématiques de Versailles, UMR 8100, Université de Versailles Saint-Quentin-en-Yvelines, 45 avenue des Etats-Unis, 78035 Versailles cedex, France |
We introduce a second order model for traffic flow with moving bottlenecks. The model consists of the × 2$ Aw-Rascle-Zhang system with a point-wise flow constraint whose trajectory is governed by an ordinary differential equation. We define two Riemann solvers, characterize the corresponding invariant domains and propose numerical strategies, which are effective in capturing the non-classical shocks due to the constraint activation.
Keywords:
Traffic flow models,
moving bottlenecks,
unilateral flux constraints,
Riemann solvers,
finite volume schemes.
Mathematics Subject Classification:
Primary:35L65;Secondary:90B20.
Citation:
Stefano Villa, Paola Goatin, Christophe Chalons. Moving bottlenecks for the Aw-Rascle-Zhang traffic flow model. Discrete & Continuous Dynamical Systems - B,
2017, 22
(10)
: 3921-3952.
doi: 10.3934/dcdsb.2017202
![]()
References:
| [1] |
N. Aguillon,
Capturing nonclassical shocks in nonlinear elastodynamic with a conservative finite volume scheme, Interfaces Free Bound., 18 (2016), 137-159.
doi: 10.4171/IFB/360. |
| [2] |
N. Aguillon and C. Chalons,
Nondiffusive conservative schemes based on approximate Riemann solvers for Lagrangian gas dynamics, ESAIM Math. Model. Numer. Anal., 50 (2016), 1887-1916.
doi: 10.1051/m2an/2016010. |
| [3] |
B. Andreianov, C. Donadello and M. D. Rosini,
A second-order model for vehicular traffics with local point constraints on the flow, Math. Models Methods Appl. Sci., 26 (2016), 751-802.
doi: 10.1142/S0218202516500172. |
| [4] |
B. Andreianov, P. Goatin and N. Seguin,
Finite volume schemes for locally constrained conservation laws, Numer. Math., 115 (2010), 609-645.
doi: 10.1007/s00211-009-0286-7. |
| [5] |
B. P. Andreianov, C. Donadello, U. Razafison, J. Y. Rolland and M. D. Rosini,
Solutions of the Aw-Rascle-Zhang system with point constraints, Netw. Heterog. Media, 11 (2016), 29-47.
doi: 10.3934/nhm.2016.11.29. |
| [6] |
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. |
| [7] |
B. Boutin, C. Chalons, F. Lagoutiére and P. G. LeFloch,
A convergent and conservative scheme for nonclassical solutions based on kinetic relations. I, Interfaces Free Bound., 10 (2008), 399-421.
doi: 10.4171/IFB/195. |
| [8] |
G. Bretti and B. Piccoli,
A tracking algorithm for car paths on road networks, SIAM J. Appl. Dyn. Syst., 7 (2008), 510-531.
doi: 10.1137/070697768. |
| [9] |
C. Chalons, M. L. Delle Monache and P. Goatin, A conservative scheme for non-classical solutions to a strongly coupled PDE-ODE problem, Preprint, 2014. |
| [10] |
C. Chalons and P. Goatin,
Transport-equilibrium schemes for computing contact discontinuities in traffic flow modeling, Commun. Math. Sci., 5 (2007), 533-551.
doi: 10.4310/CMS.2007.v5.n3.a2. |
| [11] |
C. Chalons, P. Goatin and N. Seguin,
General constrained conservation laws. Application to pedestrian flow modeling, Netw. Heterog. Media, 8 (2013), 433-463.
doi: 10.3934/nhm.2013.8.433. |
| [12] |
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. |
| [13] |
R. M. Colombo, P. Goatin and M. D. Rosini,
On the modelling and management of traffic, ESAIM Math. Model. Numer. Anal., 45 (2011), 853-872.
doi: 10.1051/m2an/2010105. |
| [14] |
M. L. Delle Monache and P. Goatin,
A front tracking method for a strongly coupled PDE-ODE system with moving density constraints in traffic flow, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 435-447.
doi: 10.3934/dcdss.2014.7.435. |
| [15] |
M. L. Delle Monache and P. Goatin,
Scalar conservation laws with moving constraints arising in traffic flow modeling: an existence result, J. Differential Equations, 257 (2014), 4015-4029.
doi: 10.1016/j.jde.2014.07.014. |
| [16] |
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. |
| [17] |
M. Garavello and S. Villa, The Cauchy problem for the Aw-Rascle-Zhang traffic model with locally constrained flow, J. Hyperbolic Differ. Equ. , to appear. |
| [18] |
S. N. Kružkov,
First order quasilinear equations with several independent variables, Mat. Sb. (N.S.), 81 (1970), 228-255.
|
| [19] |
P. D. Lax,
Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, CBMF-NSF Regional Conf. Series in Appl. Math. SIAM, Philadelphia, PA, 1973. |
| [20] |
P. LeFloch,
Hyperbolic Systems of Conservation Laws, The theory of classical and nonclassical shock waves, Lectures in Mathematics. Birkhäuser Verlag, Basel, 2002.
doi: 10.1007/978-3-0348-8150-0. |
| [21] |
B. Temple,
Systems of conservation laws with invariant submanifolds, Trans. Amer. Math. Soc., 280 (1983), 781-795.
doi: 10.1090/S0002-9947-1983-0716850-2. |
| [22] |
S. Villa, The Aw-Rascle-Zhang Model with Constraints, Master thesis, Universitá degli Studi di Milano -Bicocca, 2015. arXiv: 1605.00632. |
| [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] |
N. Aguillon,
Capturing nonclassical shocks in nonlinear elastodynamic with a conservative finite volume scheme, Interfaces Free Bound., 18 (2016), 137-159.
doi: 10.4171/IFB/360. |
| [2] |
N. Aguillon and C. Chalons,
Nondiffusive conservative schemes based on approximate Riemann solvers for Lagrangian gas dynamics, ESAIM Math. Model. Numer. Anal., 50 (2016), 1887-1916.
doi: 10.1051/m2an/2016010. |
| [3] |
B. Andreianov, C. Donadello and M. D. Rosini,
A second-order model for vehicular traffics with local point constraints on the flow, Math. Models Methods Appl. Sci., 26 (2016), 751-802.
doi: 10.1142/S0218202516500172. |
| [4] |
B. Andreianov, P. Goatin and N. Seguin,
Finite volume schemes for locally constrained conservation laws, Numer. Math., 115 (2010), 609-645.
doi: 10.1007/s00211-009-0286-7. |
| [5] |
B. P. Andreianov, C. Donadello, U. Razafison, J. Y. Rolland and M. D. Rosini,
Solutions of the Aw-Rascle-Zhang system with point constraints, Netw. Heterog. Media, 11 (2016), 29-47.
doi: 10.3934/nhm.2016.11.29. |
| [6] |
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. |
| [7] |
B. Boutin, C. Chalons, F. Lagoutiére and P. G. LeFloch,
A convergent and conservative scheme for nonclassical solutions based on kinetic relations. I, Interfaces Free Bound., 10 (2008), 399-421.
doi: 10.4171/IFB/195. |
| [8] |
G. Bretti and B. Piccoli,
A tracking algorithm for car paths on road networks, SIAM J. Appl. Dyn. Syst., 7 (2008), 510-531.
doi: 10.1137/070697768. |
| [9] |
C. Chalons, M. L. Delle Monache and P. Goatin, A conservative scheme for non-classical solutions to a strongly coupled PDE-ODE problem, Preprint, 2014. |
| [10] |
C. Chalons and P. Goatin,
Transport-equilibrium schemes for computing contact discontinuities in traffic flow modeling, Commun. Math. Sci., 5 (2007), 533-551.
doi: 10.4310/CMS.2007.v5.n3.a2. |
| [11] |
C. Chalons, P. Goatin and N. Seguin,
General constrained conservation laws. Application to pedestrian flow modeling, Netw. Heterog. Media, 8 (2013), 433-463.
doi: 10.3934/nhm.2013.8.433. |
| [12] |
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. |
| [13] |
R. M. Colombo, P. Goatin and M. D. Rosini,
On the modelling and management of traffic, ESAIM Math. Model. Numer. Anal., 45 (2011), 853-872.
doi: 10.1051/m2an/2010105. |
| [14] |
M. L. Delle Monache and P. Goatin,
A front tracking method for a strongly coupled PDE-ODE system with moving density constraints in traffic flow, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 435-447.
doi: 10.3934/dcdss.2014.7.435. |
| [15] |
M. L. Delle Monache and P. Goatin,
Scalar conservation laws with moving constraints arising in traffic flow modeling: an existence result, J. Differential Equations, 257 (2014), 4015-4029.
doi: 10.1016/j.jde.2014.07.014. |
| [16] |
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. |
| [17] |
M. Garavello and S. Villa, The Cauchy problem for the Aw-Rascle-Zhang traffic model with locally constrained flow, J. Hyperbolic Differ. Equ. , to appear. |
| [18] |
S. N. Kružkov,
First order quasilinear equations with several independent variables, Mat. Sb. (N.S.), 81 (1970), 228-255.
|
| [19] |
P. D. Lax,
Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, CBMF-NSF Regional Conf. Series in Appl. Math. SIAM, Philadelphia, PA, 1973. |
| [20] |
P. LeFloch,
Hyperbolic Systems of Conservation Laws, The theory of classical and nonclassical shock waves, Lectures in Mathematics. Birkhäuser Verlag, Basel, 2002.
doi: 10.1007/978-3-0348-8150-0. |
| [21] |
B. Temple,
Systems of conservation laws with invariant submanifolds, Trans. Amer. Math. Soc., 280 (1983), 781-795.
doi: 10.1090/S0002-9947-1983-0716850-2. |
| [22] |
S. Villa, The Aw-Rascle-Zhang Model with Constraints, Master thesis, Universitá degli Studi di Milano -Bicocca, 2015. arXiv: 1605.00632. |
| [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.
Notations used in the definition of the Riemann solvers
Figure 3.
Example of an invariant domain (the shaded area) for $\mathcal{RS}^\alpha_1$ (right) and $\mathcal{RS}^\alpha_2$ (left), for $v_1>V_b$ (case (ⅱ) in Theorems 4.2 and 4.3 respectively). Geometrically, the condition (10) in Theorem 4.2 states that the first Lax curves passing through the points of intersection between the constraint line and the second Lax curves $v=v_2$ and $v=v_1$ respectively, are both above the first Lax curve $w=w_2$ . A similar interpretation applies to condition (12) in Theorem 4.3
Figure 4.
Representation of the points used in Section 4.2.1. The invariant domain is the colored area. If by contradiction $h_\alpha(v_1)<w_2$ (case (a)), then the two points $(\rho^*, v^*)$ and $(\hat{\rho}, \hat{v})$ cannot both belong to $\mathcal{D}_{v_1, v_2, w_1, w_2}$ . The same is for $(\rho^*, v^*)$ and $(\check{\rho}_1, \check{v}_1)$ if $h_\alpha(v_2)<w_2$ (case (b)).
Figure 5.
Representation of the points used in the proof of Lemmas 4.8 and 4.9
Figure 6.
An example of a discontinuity reconstruction for the Riemann solver $\mathcal{RS}^\alpha_1$ .
Figure 7.
Comparison between the solution obtained with both conditions (16) and (17) (the dot-dashed line) and the one obtained when only (17) is enforced (the continuous line): the first has undesirable oscillation, while the latter is correct. The initial data are: $(\rho^l, v^l)=(7, 3)$ , $(\rho^r, v^r)=(6, 4)$ , $\alpha = 0.4$ , $V_b=1.5$ , $R = 15$ and $y_0=0$ .
Figure 8.
Representation of the reconstruction method (v-component)
Figure 9.
Solutions obtained with the discontinuity reconstruction method for the $(\rho, v)$ coordinates. The dot-dashed line is obtained without the correction for the contact discontinuity, while the continuous line is obtained with the correction: in the first case, the velocity downstream the shock is overestimated, in the latter it is correct. The initial data are $(\rho^l, v^l)=(\rho^r, v^r)=(7, 3)$ , $V_b=1$ , $\alpha = 0.25$ and $y_0=0$ .
Figure 10.
Spatio-temporal evolution of traffic density and bus trajectory given by $\mathcal{RS}^\alpha_1$ (top) and $\mathcal{RS}^\alpha_2$ (bottom) corresponding to the data $(\rho^l, v^l)=(9, 1)$ for $x<0$ , $(\rho^r, v^r)=(2, 8)$ for $x>0$ , $V_b=4$ , $\alpha =0.5$ , $R=15$ and $y_0=-0.1$ .
| [1] |
Alberto Bressan, Fang Yu. Continuous Riemann solvers for traffic flow at a junction. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4149-4171. doi: 10.3934/dcds.2015.35.4149 |
| [2] |
Florent Berthelin, Thierry Goudon, Bastien Polizzi, Magali Ribot. Asymptotic problems and numerical schemes for traffic flows with unilateral constraints describing the formation of jams. Networks & Heterogeneous Media, 2017, 12 (4) : 591-617. doi: 10.3934/nhm.2017024 |
| [3] |
Maria Laura Delle Monache, Paola Goatin. A front tracking method for a strongly coupled PDE-ODE system with moving density constraints in traffic flow. Discrete & Continuous Dynamical Systems - S, 2014, 7 (3) : 435-447. doi: 10.3934/dcdss.2014.7.435 |
| [4] |
Raimund Bürger, Kenneth H. Karlsen, John D. Towers. On some difference schemes and entropy conditions for a class of multi-species kinematic flow models with discontinuous flux. Networks & Heterogeneous Media, 2010, 5 (3) : 461-485. doi: 10.3934/nhm.2010.5.461 |
| [5] |
Paola Goatin, Sheila Scialanga. Well-posedness and finite volume approximations of the LWR traffic flow model with non-local velocity. Networks & Heterogeneous Media, 2016, 11 (1) : 107-121. doi: 10.3934/nhm.2016.11.107 |
| [6] |
Jan Friedrich, Oliver Kolb, Simone Göttlich. A Godunov type scheme for a class of LWR traffic flow models with non-local flux. Networks & Heterogeneous Media, 2018, 13 (4) : 531-547. doi: 10.3934/nhm.2018024 |
| [7] |
Alexander Kurganov, Anthony Polizzi. Non-oscillatory central schemes for traffic flow models with Arrhenius look-ahead dynamics. Networks & Heterogeneous Media, 2009, 4 (3) : 431-451. doi: 10.3934/nhm.2009.4.431 |
| [8] |
Maria Laura Delle Monache, Paola Goatin. Stability estimates for scalar conservation laws with moving flux constraints. Networks & Heterogeneous Media, 2017, 12 (2) : 245-258. doi: 10.3934/nhm.2017010 |
| [9] |
Boris Andreianov, Mostafa Bendahmane, Kenneth H. Karlsen, Charles Pierre. Convergence of discrete duality finite volume schemes for the cardiac bidomain model. Networks & Heterogeneous Media, 2011, 6 (2) : 195-240. doi: 10.3934/nhm.2011.6.195 |
| [10] |
Alberto Bressan, Anders Nordli. The Riemann solver for traffic flow at an intersection with buffer of vanishing size. Networks & Heterogeneous Media, 2017, 12 (2) : 173-189. doi: 10.3934/nhm.2017007 |
| [11] |
Alberto Bressan, Khai T. Nguyen. Conservation law models for traffic flow on a network of roads. Networks & Heterogeneous Media, 2015, 10 (2) : 255-293. doi: 10.3934/nhm.2015.10.255 |
| [12] |
Tong Li. Qualitative analysis of some PDE models of traffic flow. Networks & Heterogeneous Media, 2013, 8 (3) : 773-781. doi: 10.3934/nhm.2013.8.773 |
| [13] |
Paola Goatin. Traffic flow models with phase transitions on road networks. Networks & Heterogeneous Media, 2009, 4 (2) : 287-301. doi: 10.3934/nhm.2009.4.287 |
| [14] |
Michael Herty, Lorenzo Pareschi. Fokker-Planck asymptotics for traffic flow models. Kinetic & Related Models, 2010, 3 (1) : 165-179. doi: 10.3934/krm.2010.3.165 |
| [15] |
Johanna Ridder, Wen Shen. Traveling waves for nonlocal models of traffic flow. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 4001-4040. doi: 10.3934/dcds.2019161 |
| [16] |
Mohamed Benyahia, Massimiliano D. Rosini. A macroscopic traffic model with phase transitions and local point constraints on the flow. Networks & Heterogeneous Media, 2017, 12 (2) : 297-317. doi: 10.3934/nhm.2017013 |
| [17] |
Zhangxin Chen. On the control volume finite element methods and their applications to multiphase flow. Networks & Heterogeneous Media, 2006, 1 (4) : 689-706. doi: 10.3934/nhm.2006.1.689 |
| [18] |
Raimund Bürger, Antonio García, Kenneth H. Karlsen, John D. Towers. Difference schemes, entropy solutions, and speedup impulse for an inhomogeneous kinematic traffic flow model. Networks & Heterogeneous Media, 2008, 3 (1) : 1-41. doi: 10.3934/nhm.2008.3.1 |
| [19] |
Michael Herty, Reinhard Illner. Analytical and numerical investigations of refined macroscopic traffic flow models. Kinetic & Related Models, 2010, 3 (2) : 311-333. doi: 10.3934/krm.2010.3.311 |
| [20] |
Gabriella Puppo, Matteo Semplice, Andrea Tosin, Giuseppe Visconti. Kinetic models for traffic flow resulting in a reduced space of microscopic velocities. Kinetic & Related Models, 2017, 10 (3) : 823-854. doi: 10.3934/krm.2017033 |
2017 Impact Factor: 0.972
Tools
Metrics
Other articles
by authors
[Back to Top]