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December  2017, 22(10): 3921-3952. doi: 10.3934/dcdsb.2017202

Moving bottlenecks for the Aw-Rascle-Zhang traffic flow model

Stefano Villa 1, , Paola Goatin 1,, and  Christophe Chalons 2,

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

* Corresponding author: P. Goatin

Received  July 2016 Revised  July 2017 Published  August 2017

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[18]

S. N. Kružkov, First order quasilinear equations with several independent variables, Mat. Sb. (N.S.), 81 (1970), 228-255.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

S. Villa, The Aw-Rascle-Zhang Model with Constraints, Master thesis, Universitá degli Studi di Milano -Bicocca, 2015. arXiv: 1605.00632. Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[18]

S. N. Kružkov, First order quasilinear equations with several independent variables, Mat. Sb. (N.S.), 81 (1970), 228-255.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

S. Villa, The Aw-Rascle-Zhang Model with Constraints, Master thesis, Universitá degli Studi di Milano -Bicocca, 2015. arXiv: 1605.00632. Google Scholar

[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.  Google Scholar

Figure 1.  Representation of the phase plane in the fixed and in the bus reference frames
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Figure 2.  Notations used in the definition of the Riemann solvers
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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
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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)).
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Figure 5.  Representation of the points used in the proof of Lemmas 4.8 and 4.9
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Figure 6.  An example of a discontinuity reconstruction for the Riemann solver $\mathcal{RS}^\alpha_1$.
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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 $y_0=0$.
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Figure 8.  Representation of the reconstruction method (v-component)
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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 $y_0=0$.
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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 $y_0=-0.1$.
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