Figure 1.
Illustration of a non-local traffic flow model either given by (1)-(3) or (4)-(6)
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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
University of Mannheim, Department of Mathematics, 68131 Mannheim, Germany |
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 and 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. |
| [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. |
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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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [13] |
S. N. Kružkov, First order quasilinear equations in several independent variables,
Mathematics of the USSR-Sbornik, 10 (1970), 217.
doi: 10.1070/SM1970v010n02ABEH002156. |
| [14] |
R. J. LeVeque,
Numerical Methods for Conservation Laws, Birkhäuser Verlag, Basel, 1990.
doi: 10.1007/978-3-0348-5116-9. |
| [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. |
| [16] |
P. I. Richards,
Shock waves on the highway, Oper. Res., 4 (1956), 42-51.
doi: 10.1287/opre.4.1.42. |
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [13] |
S. N. Kružkov, First order quasilinear equations in several independent variables,
Mathematics of the USSR-Sbornik, 10 (1970), 217.
doi: 10.1070/SM1970v010n02ABEH002156. |
| [14] |
R. J. LeVeque,
Numerical Methods for Conservation Laws, Birkhäuser Verlag, Basel, 1990.
doi: 10.1007/978-3-0348-5116-9. |
| [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. |
| [16] |
P. I. Richards,
Shock waves on the highway, Oper. Res., 4 (1956), 42-51.
doi: 10.1287/opre.4.1.42. |
Figure 5.
Comparison of the Godunov and LxF scheme for $v(\rho) = 1-\rho^5$ , $h = 0.01$ at $T = 0.05$
Figure 6.
Approximate solutions at $T = 0.05$ for the two models with non-linear velocity function $v(\rho) = 1-\rho^5$
Figure 7.
Approximate solutions to the LWR and non-local model (4) to (6) for different $\eta$ at $T = 0.05$
|
| Godunov | LxF |
| 0 | 9.38e-03 | 1.99e-02 |
| 1 | 6.97e-03 | 1.30e-02 |
| 2 | 4.29e-03 | 9.31e-03 |
| 3 | 3.00e-03 | 6.41e-03 |
| 4 | 1.96e-03 | 4.27e-03 |
| 5 | 1.33e-03 | 2.71e-03 |
| 6 | 9.05e-04 | 1.64e-03 |
|
| Godunov | LxF |
| 0 | 9.38e-03 | 1.99e-02 |
| 1 | 6.97e-03 | 1.30e-02 |
| 2 | 4.29e-03 | 9.31e-03 |
| 3 | 3.00e-03 | 6.41e-03 |
| 4 | 1.96e-03 | 4.27e-03 |
| 5 | 1.33e-03 | 2.71e-03 |
| 6 | 9.05e-04 | 1.64e-03 |
|
| Godunov | LxF |
| 0 | 1.77e-02 | 3.13e-02 |
| 1 | 1.24e-02 | 2.20e-02 |
| 2 | 8.49e-03 | 1.41e-02 |
| 3 | 5.18e-03 | 8.67e-03 |
| 4 | 3.29e-03 | 5.45e-03 |
| 5 | 2.02e-03 | 3.47e-03 |
| 6 | 1.21e-03 | 2.06e-03 |
|
| Godunov | LxF |
| 0 | 1.77e-02 | 3.13e-02 |
| 1 | 1.24e-02 | 2.20e-02 |
| 2 | 8.49e-03 | 1.41e-02 |
| 3 | 5.18e-03 | 8.67e-03 |
| 4 | 3.29e-03 | 5.45e-03 |
| 5 | 2.02e-03 | 3.47e-03 |
| 6 | 1.21e-03 | 2.06e-03 |
Table 3.
$L^1$ distances between the approximate solutions to the local LWR model and the non-local model for different $\eta$ at $T = 0.05$
|
| | | | |
| | 4.46e-02 | 6.85e-03 | 9.90e-04 | 1.60e-04 |
|
| | | | |
| | 4.46e-02 | 6.85e-03 | 9.90e-04 | 1.60e-04 |
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