# American Institute of Mathematical Sciences

June 2019, 14(2): 389-410. doi: 10.3934/nhm.2019016

## 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

Received  October 2018 Revised  January 2019 Published  April 2019

Fund Project: This work was partially supported by the Haute-Normandie Regional Council via the M2NUM project and the project GO 1920/7-1 by the German Research Foundation (DFG)

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.

Citation: Adriano Festa, Simone Göttlich, Marion Pfirsching. A model for a network of conveyor belts with discontinuous speed and capacity. Networks & Heterogeneous Media, 2019, 14 (2) : 389-410. doi: 10.3934/nhm.2019016
##### 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.

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##### 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.
A conveyor belt in a brewery. Image courtesy of Sidel Blowing & Services SAS
Characteristics in the non-congested case
Trajectories in the congested case
Solution in the congested case: evolution of three shock waves
Scheme of the two cases considered of one-to-two junction: passive (left) and active (right)
Choice of the merging parameter $q$
Regularized flux function $f_{ \xi, i}$
Test 1: non-congested case with $a_{{1}} = 1$ and $a_{{2}} = 2$
Test 2: congested case with $a_{{1}} = 2$ and $a_{{2}} = 1$
Test 2: space-time diagram for the congested case
Test 3: "passive" junction with distribution parameter $\mu = 0.5$
Test 4: "active" junction with distribution parameter $\mu = 0.5$
Test 5: merging junction with parameter $q = 0.3$
Decreasing step sizes (left), decreasing smoothing parameter $\xi$ (right)
 $\Delta x$ $\Delta t$ error $\xi$ error 0.1 $2 \cdot 10^{-4}$ 0.0842 $5 \cdot 10^{-2}$ 0.0051 0.05 $\phantom{2 \cdot }10^{-4}$ 0.0381 $2 \cdot 10^{-2}$ 0.0042 0.01 $5 \cdot 10^{-5}$ 0.0184 $\phantom{1 \cdot} 10^{-2}$ 0.0039 0.01 $2 \cdot 10^{-5}$ 0.0073 $5 \cdot 10^{-3}$ 0.0037 0.005 $\phantom{2 \cdot }10^{-5}$ 0.0057 $2 \cdot 10^{-3}$ 0.0035
 $\Delta x$ $\Delta t$ error $\xi$ error 0.1 $2 \cdot 10^{-4}$ 0.0842 $5 \cdot 10^{-2}$ 0.0051 0.05 $\phantom{2 \cdot }10^{-4}$ 0.0381 $2 \cdot 10^{-2}$ 0.0042 0.01 $5 \cdot 10^{-5}$ 0.0184 $\phantom{1 \cdot} 10^{-2}$ 0.0039 0.01 $2 \cdot 10^{-5}$ 0.0073 $5 \cdot 10^{-3}$ 0.0037 0.005 $\phantom{2 \cdot }10^{-5}$ 0.0057 $2 \cdot 10^{-3}$ 0.0035
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