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June  2021, 16(2): 221-256. doi: 10.3934/nhm.2021005

Influence of a slow moving vehicle on traffic: Well-posedness and approximation for a mildly nonlocal model

Institut Denis Poisson, CNRS UMR 7013, Université de Tours, Université d'Orléans, Parc de Grandmont, 37200 Tours cedex, France

* Corresponding author: Abraham Sylla

Received  June 2020 Revised  November 2020 Published  June 2021 Early access  February 2021

In this paper, we propose a macroscopic model that describes the influence of a slow moving large vehicle on road traffic. The model consists of a scalar conservation law with a nonlocal constraint on the flux. The constraint level depends on the trajectory of the slower vehicle which is given by an ODE depending on the downstream traffic density. After proving well-posedness, we first build a finite volume scheme and prove its convergence, and then investigate numerically this model by performing a series of tests. In particular, the link with the limit local problem of [M. L. Delle Monache and P. Goatin, J. Differ. Equ. 257 (2014), 4015–4029] is explored numerically.

Citation: Abraham Sylla. Influence of a slow moving vehicle on traffic: Well-posedness and approximation for a mildly nonlocal model. Networks and Heterogeneous Media, 2021, 16 (2) : 221-256. doi: 10.3934/nhm.2021005
References:
[1]

Ad imurthiS. S. GhoshalR. Dutta and G. D. Veerappa Gowda, Existence and nonexistence of TV bounds for scalar conservation laws with discontinuous flux, Comm. Pure Appl. Math, 64 (2011), 84-115.  doi: 10.1002/cpa.20346.

[2]

J. Aleksić and D. Mitrović, Strong traces for averaged solutions of heterogeneous ultra-parabolic transport equations, J. Hyperbolic Differ. Equ., 10 (2013), 659-676.  doi: 10.1142/S0219891613500239.

[3]

B. AndreianovC. Donadello and M. D. Rosini, Crowd dynamics and conservation laws with nonlocal constraints and capacity drop, Math. Models Methods in Appl., 24 (2014), 2685-2722.  doi: 10.1142/S0218202514500341.

[4]

B. AndreianovC. DonadelloU. Razafison and M. D. Rosini, Qualitative behaviour and numerical approximation of solutions to conservation laws with non-local point constraints on the flux and modeling of crowd dynamics at the bottlenecks, ESAIM: M2AN, 50 (2016), 1269-1287.  doi: 10.1051/m2an/2015078.

[5]

B. AndreianovC. DonadelloU. Razafison and M. D. Rosini, Analysis and approximation of one-dimensional scalar conservation laws with general point constraints on the flux, J. Math. Pures et Appl., 116 (2018), 309-346.  doi: 10.1016/j.matpur.2018.01.005.

[6]

B. AndreianovP. 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.

[7]

B. AndreianovK. H. Karlsen and H. Risebro, A theory of $\text{L}^{1}$-dissipative solvers for scalar conservation laws with discontinuous flux, Arch. Ration. Mech. Anal., 201 (2011), 27-86.  doi: 10.1007/s00205-010-0389-4.

[8]

F. Bouchut and B. Perthame, Kružkov's estimates for scalar conservation laws revisited, Trans. Amer. Math. Soc., 350 (1998), 2847-2870.  doi: 10.1090/S0002-9947-98-02204-1.

[9]

G. BrettiE. CristianiC. LattanzioA. Maurizi and and B. Piccoli, Two algorithms for a fully coupled and consistently macroscopic PDE-ODE system modeling a moving bottleneck on a road, Math. Eng., 1 (2018), 55-83. 

[10]

R. BürgerA. GarcíaK. H. Karlsen and J. D. Towers, A family of numerical schemes for kinematic flows with discontinuous flux, J. Engrg. Math., 60 (2008), 387-425.  doi: 10.1007/s10665-007-9148-4.

[11]

R. BürgerK. H. Karlsen and J. D. Towers, An Engquist-Osher-type scheme for conservation laws with discontinuous flux adapted to flux connections, SIAM J. Numer. Anal., 47 (2009), 1684-1712.  doi: 10.1137/07069314X.

[12]

C. Cancès and T. Gallouët, On the time continuity of entropy solutions, J. Evol. Equ., 11 (2011), 43-55.  doi: 10.1007/s00028-010-0080-0.

[13]

C. Cancès and N. Seguin, Error estimate for Godunov approximation of locally constrained conservation laws, SIAM J. Numer. Anal., 50 (2012), 3036-3060.  doi: 10.1137/110836912.

[14]

C. ChalonsM. L. Delle Monache and P. Goatin, A conservative scheme for non-classical solutions to a strongly coupled PDE-ODE problem, Interfaces Free Bound., 19 (2017), 553-570.  doi: 10.4171/IFB/392.

[15]

C. ChalonsP. Goatin and N. Seguin, General constrained conservation laws. Application to pedestrian flow modeling, Networks Heterogen. Media, 8 (2013), 433-463.  doi: 10.3934/nhm.2013.8.433.

[16]

F. A. ChiarelloJ. FriedrichP. GoatinS. Göttlich and O. Kolb, A non-local traffic flow model for 1-to-1 junctions, European Journal of Applied Mathematics, 31 (2020), 1029-1049.  doi: 10.1017/S095679251900038X.

[17]

R. M. Colombo and P. Goatin, A well posed conservation law with a variable unilateral constraint, J. Differ. Equ., 234 (2007), 654-675.  doi: 10.1016/j.jde.2006.10.014.

[18]

R. M. ColomboM. Mercier and M. D. Rosini, Stability and total variation estimates on general scalar balance laws, Commun. Math. Sci., 7 (2009), 37-65.  doi: 10.4310/CMS.2009.v7.n1.a2.

[19]

R. M. Colombo and M. D. Rosini, Pedestrian flows and non-classical shocks, Math. Methods Appl. Sci., 28 (2005), 1553-1567.  doi: 10.1002/mma.624.

[20]

M. L. Delle Monache and P. Goatin, Scalar conservation laws with moving constraints arising in traffic flow modeling: An existence result, J. Differ. Equ., 257 (2014), 4015-4029.  doi: 10.1016/j.jde.2014.07.014.

[21]

M. L. Delle Monache and P. Goatin, Stability estimates for scalar conservation laws with moving flux constraints, Networks Heterogen. Media, 12 (2017), 245-258.  doi: 10.3934/nhm.2017010.

[22]

J. Droniou and R. Eymard, Uniform-in-time convergence result of numerical methods for non-linear parabolic equations, Numer. Math., 132 (2016), 721-766.  doi: 10.1007/s00211-015-0733-6.

[23]

R. Eymard, T. Gallouët and R. Herbin, Finite Volume Methods, North-Holland, Amsterdam, 2000.

[24]

H. Helge and H. Risebro, Front Tracking for Hyperbolic Conservation Laws, Springer-Verlag, New York, 2002. doi: 10.1007/978-3-642-56139-9.

[25]

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

[26]

C. LattanzioA. Maurizi and B. Piccoli, Moving bottlenecks in car traffic flow: A PDE-ODE coupled model, SIAM J. Math. Analysis, 43 (2011), 50-67.  doi: 10.1137/090767224.

[27]

T. Liard and B. Piccoli, Well-Posedness for scalar conservation laws with moving flux constraints, SIAM J. Appl. Math., 79 (2018), 641-667.  doi: 10.1137/18M1172211.

[28]

T. Liard and B. Piccoli, On entropic solutions to conservation laws coupled with moving bottlenecks, preprint, hal-02149946.

[29]

W. NevesE. Y. Panov and J. Silva, Strong traces for conservation laws with general non-autonomous flux, SIAM J. Math. Analysis, 50 (2018), 6049-6081.  doi: 10.1137/17M1159828.

[30]

E. Y. Panov, On the strong pre-compactness property for entropy solutions of a degenerate elliptic equation with discontinuous flux, J. Differ. Equ., 247 (2009), 2821-2870.  doi: 10.1016/j.jde.2009.08.022.

[31]

E. Y. Panov, Existence and strong pre-compactness properties for entropy solutions of a first-order quasilinear equation with discontinuous flux, Arch. Ration. Mech. Anal., 195 (2010), 643-673.  doi: 10.1007/s00205-009-0217-x.

[32]

J. D. Towers, Convergence of the Godunov scheme for a scalar conservation law with time and space discontinuities, J. Hyperbolic Differ. Equ., 15 (2018), 175-190.  doi: 10.1142/S0219891618500078.

[33]

J. D. Towers, Convergence via OSLC of the Godunov scheme for a scalar conservation law with time and space flux discontinuities, Numer. Math., 139 (2018), 939-969.  doi: 10.1007/s00211-018-0957-3.

show all references

References:
[1]

Ad imurthiS. S. GhoshalR. Dutta and G. D. Veerappa Gowda, Existence and nonexistence of TV bounds for scalar conservation laws with discontinuous flux, Comm. Pure Appl. Math, 64 (2011), 84-115.  doi: 10.1002/cpa.20346.

[2]

J. Aleksić and D. Mitrović, Strong traces for averaged solutions of heterogeneous ultra-parabolic transport equations, J. Hyperbolic Differ. Equ., 10 (2013), 659-676.  doi: 10.1142/S0219891613500239.

[3]

B. AndreianovC. Donadello and M. D. Rosini, Crowd dynamics and conservation laws with nonlocal constraints and capacity drop, Math. Models Methods in Appl., 24 (2014), 2685-2722.  doi: 10.1142/S0218202514500341.

[4]

B. AndreianovC. DonadelloU. Razafison and M. D. Rosini, Qualitative behaviour and numerical approximation of solutions to conservation laws with non-local point constraints on the flux and modeling of crowd dynamics at the bottlenecks, ESAIM: M2AN, 50 (2016), 1269-1287.  doi: 10.1051/m2an/2015078.

[5]

B. AndreianovC. DonadelloU. Razafison and M. D. Rosini, Analysis and approximation of one-dimensional scalar conservation laws with general point constraints on the flux, J. Math. Pures et Appl., 116 (2018), 309-346.  doi: 10.1016/j.matpur.2018.01.005.

[6]

B. AndreianovP. 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.

[7]

B. AndreianovK. H. Karlsen and H. Risebro, A theory of $\text{L}^{1}$-dissipative solvers for scalar conservation laws with discontinuous flux, Arch. Ration. Mech. Anal., 201 (2011), 27-86.  doi: 10.1007/s00205-010-0389-4.

[8]

F. Bouchut and B. Perthame, Kružkov's estimates for scalar conservation laws revisited, Trans. Amer. Math. Soc., 350 (1998), 2847-2870.  doi: 10.1090/S0002-9947-98-02204-1.

[9]

G. BrettiE. CristianiC. LattanzioA. Maurizi and and B. Piccoli, Two algorithms for a fully coupled and consistently macroscopic PDE-ODE system modeling a moving bottleneck on a road, Math. Eng., 1 (2018), 55-83. 

[10]

R. BürgerA. GarcíaK. H. Karlsen and J. D. Towers, A family of numerical schemes for kinematic flows with discontinuous flux, J. Engrg. Math., 60 (2008), 387-425.  doi: 10.1007/s10665-007-9148-4.

[11]

R. BürgerK. H. Karlsen and J. D. Towers, An Engquist-Osher-type scheme for conservation laws with discontinuous flux adapted to flux connections, SIAM J. Numer. Anal., 47 (2009), 1684-1712.  doi: 10.1137/07069314X.

[12]

C. Cancès and T. Gallouët, On the time continuity of entropy solutions, J. Evol. Equ., 11 (2011), 43-55.  doi: 10.1007/s00028-010-0080-0.

[13]

C. Cancès and N. Seguin, Error estimate for Godunov approximation of locally constrained conservation laws, SIAM J. Numer. Anal., 50 (2012), 3036-3060.  doi: 10.1137/110836912.

[14]

C. ChalonsM. L. Delle Monache and P. Goatin, A conservative scheme for non-classical solutions to a strongly coupled PDE-ODE problem, Interfaces Free Bound., 19 (2017), 553-570.  doi: 10.4171/IFB/392.

[15]

C. ChalonsP. Goatin and N. Seguin, General constrained conservation laws. Application to pedestrian flow modeling, Networks Heterogen. Media, 8 (2013), 433-463.  doi: 10.3934/nhm.2013.8.433.

[16]

F. A. ChiarelloJ. FriedrichP. GoatinS. Göttlich and O. Kolb, A non-local traffic flow model for 1-to-1 junctions, European Journal of Applied Mathematics, 31 (2020), 1029-1049.  doi: 10.1017/S095679251900038X.

[17]

R. M. Colombo and P. Goatin, A well posed conservation law with a variable unilateral constraint, J. Differ. Equ., 234 (2007), 654-675.  doi: 10.1016/j.jde.2006.10.014.

[18]

R. M. ColomboM. Mercier and M. D. Rosini, Stability and total variation estimates on general scalar balance laws, Commun. Math. Sci., 7 (2009), 37-65.  doi: 10.4310/CMS.2009.v7.n1.a2.

[19]

R. M. Colombo and M. D. Rosini, Pedestrian flows and non-classical shocks, Math. Methods Appl. Sci., 28 (2005), 1553-1567.  doi: 10.1002/mma.624.

[20]

M. L. Delle Monache and P. Goatin, Scalar conservation laws with moving constraints arising in traffic flow modeling: An existence result, J. Differ. Equ., 257 (2014), 4015-4029.  doi: 10.1016/j.jde.2014.07.014.

[21]

M. L. Delle Monache and P. Goatin, Stability estimates for scalar conservation laws with moving flux constraints, Networks Heterogen. Media, 12 (2017), 245-258.  doi: 10.3934/nhm.2017010.

[22]

J. Droniou and R. Eymard, Uniform-in-time convergence result of numerical methods for non-linear parabolic equations, Numer. Math., 132 (2016), 721-766.  doi: 10.1007/s00211-015-0733-6.

[23]

R. Eymard, T. Gallouët and R. Herbin, Finite Volume Methods, North-Holland, Amsterdam, 2000.

[24]

H. Helge and H. Risebro, Front Tracking for Hyperbolic Conservation Laws, Springer-Verlag, New York, 2002. doi: 10.1007/978-3-642-56139-9.

[25]

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

[26]

C. LattanzioA. Maurizi and B. Piccoli, Moving bottlenecks in car traffic flow: A PDE-ODE coupled model, SIAM J. Math. Analysis, 43 (2011), 50-67.  doi: 10.1137/090767224.

[27]

T. Liard and B. Piccoli, Well-Posedness for scalar conservation laws with moving flux constraints, SIAM J. Appl. Math., 79 (2018), 641-667.  doi: 10.1137/18M1172211.

[28]

T. Liard and B. Piccoli, On entropic solutions to conservation laws coupled with moving bottlenecks, preprint, hal-02149946.

[29]

W. NevesE. Y. Panov and J. Silva, Strong traces for conservation laws with general non-autonomous flux, SIAM J. Math. Analysis, 50 (2018), 6049-6081.  doi: 10.1137/17M1159828.

[30]

E. Y. Panov, On the strong pre-compactness property for entropy solutions of a degenerate elliptic equation with discontinuous flux, J. Differ. Equ., 247 (2009), 2821-2870.  doi: 10.1016/j.jde.2009.08.022.

[31]

E. Y. Panov, Existence and strong pre-compactness properties for entropy solutions of a first-order quasilinear equation with discontinuous flux, Arch. Ration. Mech. Anal., 195 (2010), 643-673.  doi: 10.1007/s00205-009-0217-x.

[32]

J. D. Towers, Convergence of the Godunov scheme for a scalar conservation law with time and space discontinuities, J. Hyperbolic Differ. Equ., 15 (2018), 175-190.  doi: 10.1142/S0219891618500078.

[33]

J. D. Towers, Convergence via OSLC of the Godunov scheme for a scalar conservation law with time and space flux discontinuities, Numer. Math., 139 (2018), 939-969.  doi: 10.1007/s00211-018-0957-3.

Figure 1.  Evolution in time of the subjective density $ \xi $ and the bus velocity $ \dot y $ (1280 cells)
Figure 2.  The numerical solution at different fixed times
Figure 3.  Rates of convergence at time $ T = 13 $
Figure 4.  Evolution in time of the numerical density corresponding to initial data (47) (left) and (48) (right), with 1280 cells
Figure 5.  Evolution in time of the numerical density corresponding to initial data (49) (1280 cells)
Table 1.  Measured errors at time $ T = 13 $
Number of cells $ E_{\rho}^\Delta \ (\times 10^{-2}) $ $ E_{y}^\Delta \ (\times 10^{-3}) $
160 $ 24.053 $ $ 48.0643 $
320 $ 15.731 $ $ 15.939 $
640 $ 9.647 $ $ 7.698 $
1280 $ 6.197 $ $ 3.715 $
2560 $ 3.226 $ $ 1.777 $
5120 $ 1.936 $ $ 0.889 $
10240 $ 1.055 $ $ 0.443 $
Number of cells $ E_{\rho}^\Delta \ (\times 10^{-2}) $ $ E_{y}^\Delta \ (\times 10^{-3}) $
160 $ 24.053 $ $ 48.0643 $
320 $ 15.731 $ $ 15.939 $
640 $ 9.647 $ $ 7.698 $
1280 $ 6.197 $ $ 3.715 $
2560 $ 3.226 $ $ 1.777 $
5120 $ 1.936 $ $ 0.889 $
10240 $ 1.055 $ $ 0.443 $
Table 2.  Measured errors at time $ T = 0.7245 $
Number of cells $ E_{\mathbf{L}^{{1}}}^\Delta \ (\times 10^{-4}) $ $ E_{\mathbf{L}^{{\infty}}}^\Delta \ (\times 10^{-3}) $
160 $ 32.672 $ $ 18.519 $
320 $ 14.236 $ $ 7.341 $
640 $ 5.837 $ $ 3.701 $
1280 $ 3.833 $ $ 4.879 $
2560 $ 3.207 $ $ 6.405 $
5120 $ 2.922 $ $ 7.144 $
10240 $ 2.776 $ $ 7.501 $
20480 $ 2.698 $ $ 7.674 $
40960 $ 2.658 $ $ 7.759 $
Number of cells $ E_{\mathbf{L}^{{1}}}^\Delta \ (\times 10^{-4}) $ $ E_{\mathbf{L}^{{\infty}}}^\Delta \ (\times 10^{-3}) $
160 $ 32.672 $ $ 18.519 $
320 $ 14.236 $ $ 7.341 $
640 $ 5.837 $ $ 3.701 $
1280 $ 3.833 $ $ 4.879 $
2560 $ 3.207 $ $ 6.405 $
5120 $ 2.922 $ $ 7.144 $
10240 $ 2.776 $ $ 7.501 $
20480 $ 2.698 $ $ 7.674 $
40960 $ 2.658 $ $ 7.759 $
Table 3.  Measured errors at time $ T = 0.7245 $
weight function $ E_{\mathbf{L}^{{1}}}^\Delta $ $ E_{\mathbf{L}^{{\infty}}}^\Delta $
$ \mu_1 $ $ 6.810 \times 10^{-3} $ $ 5.489 \times 10^{-2} $
$ \mu_2 $ $ 1.105 \times 10^{-3} $ $ 1.972 \times 10^{-2} $
$ \mu_3 $ $ 2.658 \times 10^{-4} $ $ 7.759 \times 10^{-3} $
$ \mu_4 $ $ 9.232 \times 10^{-5} $ $ 2.913 \times 10^{-3} $
$ \mu_5 $ $ 6.190 \times 10^{-5} $ $ 9.110 \times 10^{-4} $
weight function $ E_{\mathbf{L}^{{1}}}^\Delta $ $ E_{\mathbf{L}^{{\infty}}}^\Delta $
$ \mu_1 $ $ 6.810 \times 10^{-3} $ $ 5.489 \times 10^{-2} $
$ \mu_2 $ $ 1.105 \times 10^{-3} $ $ 1.972 \times 10^{-2} $
$ \mu_3 $ $ 2.658 \times 10^{-4} $ $ 7.759 \times 10^{-3} $
$ \mu_4 $ $ 9.232 \times 10^{-5} $ $ 2.913 \times 10^{-3} $
$ \mu_5 $ $ 6.190 \times 10^{-5} $ $ 9.110 \times 10^{-4} $
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