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Non-local multi-class traffic flow models
A model for a network of conveyor belts with discontinuous speed and capacity
1. | Institut National des sciences appliquées (INSA) Rouen, Laboratoire de Mathématiques, 685 Avenue de l'Université, 76800 Saint-Étienne-du-Rouvray, France |
2. | University of Mannheim, Department of Mathematics, A5-6, 68131 Mannheim, Germany |
We introduce a macroscopic model for a network of conveyor belts with various speeds and capacities. In a different way from traffic flow models, the product densities are forced to move with a constant velocity unless they reach a maximal capacity and start to queue. This kind of dynamics is governed by scalar conservation laws consisting of a discontinuous flux function. We define appropriate coupling conditions to get well-posed solutions at intersections and provide a detailed description of the solution. Some numerical simulations are presented to illustrate and confirm the theoretical results for different network configurations.
References:
[1] |
D. Armbruster, S. Göttlich and M. Herty,
A scalar conservation law with discontinuous flux for supply chains with finite buffers, SIAM J. Appl. Math., 71 (2011), 1070-1087.
doi: 10.1137/100809374. |
[2] |
F. Camilli, A. Festa and S. Tozza,
A discrete hughes model for pedestrian flow on graphs, Netw. Heterog. Media, 12 (2017), 93-112.
doi: 10.3934/nhm.2017004. |
[3] |
C. d'Apice, S. Göttlich, M. Herty and B. Piccoli, Modeling, Simulation, and Optimization of Supply Chains: A Continuous Approach, SIAM, 2010.
doi: 10.1137/1.9780898717600. |
[4] |
J.-P. Dias and M. Figueira,
On the riemann problem for some discontinuous systems of conservation laws describing phase transitions, Commun. Pure Appl. Math., 3 (2004), 53-58.
doi: 10.3934/cpaa.2004.3.53. |
[5] |
J.-P. Dias, M. Figueira and J.-F. Rodrigues,
Solutions to a scalar discontinuous conservation law in a limit case of phase transitions, J. Math. Fluid Mech., 7 (2005), 153-163.
doi: 10.1007/s00021-004-0113-y. |
[6] |
U. S. Fjordholm, S. Mishra and E. Tadmor,
Arbitrarily high-order accurate entropy stable essentially nonoscillatory schemes for systems of conservation laws, SIAM J. Numer. Anal., 50 (2012), 544-573.
doi: 10.1137/110836961. |
[7] |
M. Garavello, K. Han and B. Piccoli, Models for Vehicular Traffic on Networks, volume 9, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2016. |
[8] |
M. Garavello, R. Natalini, B. Piccoli and A. Terracina,
Conservation laws with discontinuous flux, Netw. Heterog. Media, 2 (2007), 159-179.
doi: 10.3934/nhm.2007.2.159. |
[9] |
M. Garavello and B. Piccoli, Traffic Flow on Networks, volume 1, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2006. |
[10] |
S. Göttlich, A. Klar and P. Schindler,
Discontinuous conservation laws for production networks with finite buffers, SIAM J. Appl. Math., 73 (2013), 1117-1138.
doi: 10.1137/120882573. |
[11] |
M. Herty, C. Joerres and B. Piccoli,
Existence of solution to supply chain models based on partial differential equation with discontinuous flux function, J. Math. Anal. Appl., 401 (2013), 510-517.
doi: 10.1016/j.jmaa.2012.12.002. |
[12] |
J. D. Towers,
Convergence of a difference scheme for conservation laws with a discontinuous flux, SIAM J. Numer. Anal., 38 (2000), 681-698.
doi: 10.1137/S0036142999363668. |
[13] |
J. K. Wiens, J. M. Stockie and J. F. Williams,
Riemann solver for a kinematic wave traffic model with discontinuous flux, J. Comput. Phys., 242 (2013), 1-23.
doi: 10.1016/j.jcp.2013.02.024. |
show all references
References:
[1] |
D. Armbruster, S. Göttlich and M. Herty,
A scalar conservation law with discontinuous flux for supply chains with finite buffers, SIAM J. Appl. Math., 71 (2011), 1070-1087.
doi: 10.1137/100809374. |
[2] |
F. Camilli, A. Festa and S. Tozza,
A discrete hughes model for pedestrian flow on graphs, Netw. Heterog. Media, 12 (2017), 93-112.
doi: 10.3934/nhm.2017004. |
[3] |
C. d'Apice, S. Göttlich, M. Herty and B. Piccoli, Modeling, Simulation, and Optimization of Supply Chains: A Continuous Approach, SIAM, 2010.
doi: 10.1137/1.9780898717600. |
[4] |
J.-P. Dias and M. Figueira,
On the riemann problem for some discontinuous systems of conservation laws describing phase transitions, Commun. Pure Appl. Math., 3 (2004), 53-58.
doi: 10.3934/cpaa.2004.3.53. |
[5] |
J.-P. Dias, M. Figueira and J.-F. Rodrigues,
Solutions to a scalar discontinuous conservation law in a limit case of phase transitions, J. Math. Fluid Mech., 7 (2005), 153-163.
doi: 10.1007/s00021-004-0113-y. |
[6] |
U. S. Fjordholm, S. Mishra and E. Tadmor,
Arbitrarily high-order accurate entropy stable essentially nonoscillatory schemes for systems of conservation laws, SIAM J. Numer. Anal., 50 (2012), 544-573.
doi: 10.1137/110836961. |
[7] |
M. Garavello, K. Han and B. Piccoli, Models for Vehicular Traffic on Networks, volume 9, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2016. |
[8] |
M. Garavello, R. Natalini, B. Piccoli and A. Terracina,
Conservation laws with discontinuous flux, Netw. Heterog. Media, 2 (2007), 159-179.
doi: 10.3934/nhm.2007.2.159. |
[9] |
M. Garavello and B. Piccoli, Traffic Flow on Networks, volume 1, American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2006. |
[10] |
S. Göttlich, A. Klar and P. Schindler,
Discontinuous conservation laws for production networks with finite buffers, SIAM J. Appl. Math., 73 (2013), 1117-1138.
doi: 10.1137/120882573. |
[11] |
M. Herty, C. Joerres and B. Piccoli,
Existence of solution to supply chain models based on partial differential equation with discontinuous flux function, J. Math. Anal. Appl., 401 (2013), 510-517.
doi: 10.1016/j.jmaa.2012.12.002. |
[12] |
J. D. Towers,
Convergence of a difference scheme for conservation laws with a discontinuous flux, SIAM J. Numer. Anal., 38 (2000), 681-698.
doi: 10.1137/S0036142999363668. |
[13] |
J. K. Wiens, J. M. Stockie and J. F. Williams,
Riemann solver for a kinematic wave traffic model with discontinuous flux, J. Comput. Phys., 242 (2013), 1-23.
doi: 10.1016/j.jcp.2013.02.024. |











error | error | |||
0.1 | 0.0842 | 0.0051 | ||
0.05 | 0.0381 |
| 0.0042 | |
0.01 | 0.0184 |
| 0.0039 | |
0.01 | 0.0073 |
| 0.0037 | |
0.005 | 0.0057 |
| 0.0035 |
error | error | |||
0.1 | 0.0842 | 0.0051 | ||
0.05 | 0.0381 |
| 0.0042 | |
0.01 | 0.0184 |
| 0.0039 | |
0.01 | 0.0073 |
| 0.0037 | |
0.005 | 0.0057 |
| 0.0035 |
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