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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  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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[25]

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

[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.  Google Scholar

[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.  Google Scholar

[28]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[25]

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

[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.  Google Scholar

[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.  Google Scholar

[28]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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