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December  2018, 13(4): 531-547. doi: 10.3934/nhm.2018024

A Godunov type scheme for a class of LWR traffic flow models with non-local flux

Jan Friedrich , Oliver Kolb and  Simone Göttlich ,

University of Mannheim, Department of Mathematics, 68131 Mannheim, Germany

* Corresponding author: Simone Göttlich

Received  February 2018 Revised  April 2018 Published  September 2018

We present a Godunov type numerical scheme for a class of scalar conservation laws with non-local flux arising for example in traffic flow models. The proposed scheme delivers more accurate solutions than the widely used Lax-Friedrichs type scheme. In contrast to other approaches, we consider a non-local mean velocity instead of a mean density and provide $L^∞$ and bounded variation estimates for the sequence of approximate solutions. Together with a discrete entropy inequality, we also show the well-posedness of the considered class of scalar conservation laws. The better accuracy of the Godunov type scheme in comparison to Lax-Friedrichs is proved by a variety of numerical examples.

Keywords: Scalar conservation laws, non-local flux, Godunov scheme, traffic flow models, numerical simulations.
Mathematics Subject Classification: Primary: 35L65, 90B20; Secondary: 65M12.
Citation: 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
References:
[1]

A. Aggarwal, R. M. Colombo and P. Goatin, Nonlocal systems of conservation laws in several space dimensions, SIAM J. Numer. Anal., 53 (2015), 963-983.  doi: 10.1137/140975255.  Google Scholar

[2]

P. Amorim, R. M. Colombo and A. Teixeira, On the numerical integration of scalar nonlocal conservation laws, ESAIM Math. Model. Numer. Anal., 49 (2015), 19-37.  doi: 10.1051/m2an/2014023.  Google Scholar

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D. Armbruster, D. E. Marthaler, C. Ringhofer, K. Kempf and T.-C. Jo, A continuum model for a re-entrant factory, Oper. Res., 54 (2006), 933-950.  doi: 10.1287/opre.1060.0321.  Google Scholar

[4]

F. Betancourt, R. Bürger, K. H. Karlsen and E. M. Tory, On nonlocal conservation laws modelling sedimentation, Nonlinearity, 24 (2011), 855-885.  doi: 10.1088/0951-7715/24/3/008.  Google Scholar

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S. Blandin and P. Goatin, Well-posedness of a conservation law with non-local flux arising in traffic flow modeling, Numer. Math., 132 (2016), 217-241.  doi: 10.1007/s00211-015-0717-6.  Google Scholar

[6]

C. Chalons, P. Goatin and L. M. Villada, High-order numerical schemes for one-dimensional nonlocal conservation laws, SIAM J. Sci. Comput., 40 (2018), A288-A305.  doi: 10.1137/16M110825X.  Google Scholar

[7]

F. A. Chiarello and P. Goatin, Global entropy weak solutions for general non-local traffic flow models with anisotropic kernel, ESAIM: Mathematical Modelling and Numerical Analysis, 52 (2018), 163-180.  doi: 10.1051/m2an/2017066.  Google Scholar

[8]

R. Colombo, M. Garavello and and M. Lécureux-Mercier, A class of nonlocal models for pedestrian traffic, Math. Models Methods Appl. Sci., 22 (2012), 1150023-1150057.  doi: 10.1142/S0218202511500230.  Google Scholar

[9]

R. Colombo, M. Herty and M. Mercier, Control of the continuity equation with a non local flow, ESAIM Control Optim. Calc. Var., 17 (2011), 353-379.  doi: 10.1051/cocv/2010007.  Google Scholar

[10]

P. Goatin and S. Scialanga, The Lighthill-Whitham-Richards traffic flow model with non-local velocity: Analytical study and numerical results, INRIA Research Report, 2015. Google Scholar

[11]

P. Goatin and S. Scialanga, Well-posedness and finite volume approximations of the LWR traffic flow model with non-local velocity, Netw. Heterog. Media, 11 (2016), 107-121.  doi: 10.3934/nhm.2016.11.107.  Google Scholar

[12]

S. Göttlich, S. Hoher, P. Schindler, V. Schleper and A. Verl, Modeling, simulation and validation of material flow on conveyor belts, Appl. Math. Model., 38 (2014), 3295-3313.  doi: 10.1016/j.apm.2013.11.039.  Google Scholar

[13]

S. N. Kružkov, First order quasilinear equations in several independent variables, Mathematics of the USSR-Sbornik, 10 (1970), 217. doi: 10.1070/SM1970v010n02ABEH002156.  Google Scholar

[14]

R. J. LeVeque, Numerical Methods for Conservation Laws, Birkhäuser Verlag, Basel, 1990. doi: 10.1007/978-3-0348-5116-9.  Google Scholar

[15]

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.  Google Scholar

[16]

P. I. Richards, Shock waves on the highway, Oper. Res., 4 (1956), 42-51.  doi: 10.1287/opre.4.1.42.  Google Scholar

show all references

References:
[1]

A. Aggarwal, R. M. Colombo and P. Goatin, Nonlocal systems of conservation laws in several space dimensions, SIAM J. Numer. Anal., 53 (2015), 963-983.  doi: 10.1137/140975255.  Google Scholar

[2]

P. Amorim, R. M. Colombo and A. Teixeira, On the numerical integration of scalar nonlocal conservation laws, ESAIM Math. Model. Numer. Anal., 49 (2015), 19-37.  doi: 10.1051/m2an/2014023.  Google Scholar

[3]

D. Armbruster, D. E. Marthaler, C. Ringhofer, K. Kempf and T.-C. Jo, A continuum model for a re-entrant factory, Oper. Res., 54 (2006), 933-950.  doi: 10.1287/opre.1060.0321.  Google Scholar

[4]

F. Betancourt, R. Bürger, K. H. Karlsen and E. M. Tory, On nonlocal conservation laws modelling sedimentation, Nonlinearity, 24 (2011), 855-885.  doi: 10.1088/0951-7715/24/3/008.  Google Scholar

[5]

S. Blandin and P. Goatin, Well-posedness of a conservation law with non-local flux arising in traffic flow modeling, Numer. Math., 132 (2016), 217-241.  doi: 10.1007/s00211-015-0717-6.  Google Scholar

[6]

C. Chalons, P. Goatin and L. M. Villada, High-order numerical schemes for one-dimensional nonlocal conservation laws, SIAM J. Sci. Comput., 40 (2018), A288-A305.  doi: 10.1137/16M110825X.  Google Scholar

[7]

F. A. Chiarello and P. Goatin, Global entropy weak solutions for general non-local traffic flow models with anisotropic kernel, ESAIM: Mathematical Modelling and Numerical Analysis, 52 (2018), 163-180.  doi: 10.1051/m2an/2017066.  Google Scholar

[8]

R. Colombo, M. Garavello and and M. Lécureux-Mercier, A class of nonlocal models for pedestrian traffic, Math. Models Methods Appl. Sci., 22 (2012), 1150023-1150057.  doi: 10.1142/S0218202511500230.  Google Scholar

[9]

R. Colombo, M. Herty and M. Mercier, Control of the continuity equation with a non local flow, ESAIM Control Optim. Calc. Var., 17 (2011), 353-379.  doi: 10.1051/cocv/2010007.  Google Scholar

[10]

P. Goatin and S. Scialanga, The Lighthill-Whitham-Richards traffic flow model with non-local velocity: Analytical study and numerical results, INRIA Research Report, 2015. Google Scholar

[11]

P. Goatin and S. Scialanga, Well-posedness and finite volume approximations of the LWR traffic flow model with non-local velocity, Netw. Heterog. Media, 11 (2016), 107-121.  doi: 10.3934/nhm.2016.11.107.  Google Scholar

[12]

S. Göttlich, S. Hoher, P. Schindler, V. Schleper and A. Verl, Modeling, simulation and validation of material flow on conveyor belts, Appl. Math. Model., 38 (2014), 3295-3313.  doi: 10.1016/j.apm.2013.11.039.  Google Scholar

[13]

S. N. Kružkov, First order quasilinear equations in several independent variables, Mathematics of the USSR-Sbornik, 10 (1970), 217. doi: 10.1070/SM1970v010n02ABEH002156.  Google Scholar

[14]

R. J. LeVeque, Numerical Methods for Conservation Laws, Birkhäuser Verlag, Basel, 1990. doi: 10.1007/978-3-0348-5116-9.  Google Scholar

[15]

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.  Google Scholar

[16]

P. I. Richards, Shock waves on the highway, Oper. Res., 4 (1956), 42-51.  doi: 10.1287/opre.4.1.42.  Google Scholar

Figure 1.  Illustration of a non-local traffic flow model either given by (1)-(3) or (4)-(6)
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Figure 2.  Space discretization and downstream kernel $\eta = Nh$ for $N = 2$ in gray
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Figure 3.  Comparison of the Godunov and LxF scheme for $v(\rho) = 1-\rho$, $h = 0.01$ at $T = 0.1$
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Figure 4.  Comparison of the Godunov and LxF scheme for $v(\rho) = 1-\rho$, $h = 0.01$ at $T = 1$
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Figure 5.  Comparison of the Godunov and LxF scheme for $v(\rho) = 1-\rho^5$, $h = 0.01$ at $T = 0.05$
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Figure 6.  Approximate solutions at $T = 0.05$ for the two models with non-linear velocity function $v(\rho) = 1-\rho^5$
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Figure 7.  Approximate solutions to the LWR and non-local model (4) to (6) for different $\eta$ at $T = 0.05$
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Table 1.  $L^1$ errors for $v(\rho) = 1-\rho$ at $T = 0.1$
$n$GodunovLxF
09.38e-031.99e-02
16.97e-031.30e-02
24.29e-039.31e-03
33.00e-036.41e-03
41.96e-034.27e-03
51.33e-032.71e-03
69.05e-041.64e-03
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$n$GodunovLxF
09.38e-031.99e-02
16.97e-031.30e-02
24.29e-039.31e-03
33.00e-036.41e-03
41.96e-034.27e-03
51.33e-032.71e-03
69.05e-041.64e-03
Table 2.  $L^1$ errors for $v(\rho) = 1-\rho^5$ at $T = 0.05$
$n$GodunovLxF
01.77e-023.13e-02
11.24e-022.20e-02
28.49e-031.41e-02
35.18e-038.67e-03
43.29e-035.45e-03
52.02e-033.47e-03
61.21e-032.06e-03
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$n$GodunovLxF
01.77e-023.13e-02
11.24e-022.20e-02
28.49e-031.41e-02
35.18e-038.67e-03
43.29e-035.45e-03
52.02e-033.47e-03
61.21e-032.06e-03
Table 3.  $L^1$ distances between the approximate solutions to the local LWR model and the non-local model for different $\eta$ at $T = 0.05$
$\eta$ $10^{-1}$ $10^{-2}$ $10^{-3}$ $10^{-4}$
$L^1$ distance4.46e-026.85e-039.90e-041.60e-04
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$\eta$ $10^{-1}$ $10^{-2}$ $10^{-3}$ $10^{-4}$
$L^1$ distance4.46e-026.85e-039.90e-041.60e-04
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