American Institute of Mathematical Sciences

Advanced
September  2010, 5(3): 565-581. doi: 10.3934/nhm.2010.5.565

A review of conservation laws on networks

Mauro Garavello 1,

1. 

DiSTA, University of Piemonte Orientale "A. Avogadro”, Viale T. Michel 11, 15121 Alessandria, Italy

Received  January 2010 Revised  April 2010 Published  July 2010

This paper deals with various applications of conservation laws on networks. In particular we consider the car traffic, described by the Lighthill-Whitham-Richards model and by the Aw-Rascle-Zhang model, the telecommunication case, by using the model introduced by D'Apice-Manzo-Piccoli and, finally, the case of a gas pipeline, modeled by the classical $p$-system. For each of these models we present a review of some results about Riemann and Cauchy problems in the case of a network, formed by a single vertex with $n$ incoming and $m$ outgoing arcs.
Keywords: car traffic, Conservation laws, gas pipeline, telecommunication., networks.
Mathematics Subject Classification: Primary: 35L65; Secondary: 76N10, 90B2.
Citation: Mauro Garavello. A review of conservation laws on networks. Networks and Heterogeneous Media, 2010, 5 (3) : 565-581. doi: 10.3934/nhm.2010.5.565
[1]

Mapundi K. Banda, Michael Herty, Axel Klar. Gas flow in pipeline networks. Networks and Heterogeneous Media, 2006, 1 (1) : 41-56. doi: 10.3934/nhm.2006.1.41
[2]

Alessia Marigo. Optimal traffic distribution and priority coefficients for telecommunication networks. Networks and Heterogeneous Media, 2006, 1 (2) : 315-336. doi: 10.3934/nhm.2006.1.315
[3]

Yogiraj Mantri, Michael Herty, Sebastian Noelle. Well-balanced scheme for gas-flow in pipeline networks. Networks and Heterogeneous Media, 2019, 14 (4) : 659-676. doi: 10.3934/nhm.2019026
[4]

Georges Bastin, B. Haut, Jean-Michel Coron, Brigitte d'Andréa-Novel. Lyapunov stability analysis of networks of scalar conservation laws. Networks and Heterogeneous Media, 2007, 2 (4) : 751-759. doi: 10.3934/nhm.2007.2.751
[5]

Wen Shen. Traveling waves for conservation laws with nonlocal flux for traffic flow on rough roads. Networks and Heterogeneous Media, 2019, 14 (4) : 709-732. doi: 10.3934/nhm.2019028
[6]

Martin Gugat, Alexander Keimer, Günter Leugering, Zhiqiang Wang. Analysis of a system of nonlocal conservation laws for multi-commodity flow on networks. Networks and Heterogeneous Media, 2015, 10 (4) : 749-785. doi: 10.3934/nhm.2015.10.749
[7]

Markus Musch, Ulrik Skre Fjordholm, Nils Henrik Risebro. Well-posedness theory for nonlinear scalar conservation laws on networks. Networks and Heterogeneous Media, 2022, 17 (1) : 101-128. doi: 10.3934/nhm.2021025
[8]

Guillaume Costeseque, Jean-Patrick Lebacque. Discussion about traffic junction modelling: Conservation laws VS Hamilton-Jacobi equations. Discrete and Continuous Dynamical Systems - S, 2014, 7 (3) : 411-433. doi: 10.3934/dcdss.2014.7.411
[9]

Avner Friedman. Conservation laws in mathematical biology. Discrete and Continuous Dynamical Systems, 2012, 32 (9) : 3081-3097. doi: 10.3934/dcds.2012.32.3081
[10]

Len G. Margolin, Roy S. Baty. Conservation laws in discrete geometry. Journal of Geometric Mechanics, 2019, 11 (2) : 187-203. doi: 10.3934/jgm.2019010
[11]

Mauro Garavello, Roberto Natalini, Benedetto Piccoli, Andrea Terracina. Conservation laws with discontinuous flux. Networks and Heterogeneous Media, 2007, 2 (1) : 159-179. doi: 10.3934/nhm.2007.2.159
[12]

Martin Gugat, Falk M. Hante, Markus Hirsch-Dick, Günter Leugering. Stationary states in gas networks. Networks and Heterogeneous Media, 2015, 10 (2) : 295-320. doi: 10.3934/nhm.2015.10.295
[13]

Jingmei Zhou, Xiangmo Zhao, Xin Cheng, Zhigang Xu. Visualization analysis of traffic congestion based on floating car data. Discrete and Continuous Dynamical Systems - S, 2015, 8 (6) : 1423-1433. doi: 10.3934/dcdss.2015.8.1423
[14]

Marte Godvik, Harald Hanche-Olsen. Car-following and the macroscopic Aw-Rascle traffic flow model. Discrete and Continuous Dynamical Systems - B, 2010, 13 (2) : 279-303. doi: 10.3934/dcdsb.2010.13.279
[15]

Wen-Xiu Ma. Conservation laws by symmetries and adjoint symmetries. Discrete and Continuous Dynamical Systems - S, 2018, 11 (4) : 707-721. doi: 10.3934/dcdss.2018044
[16]

Tai-Ping Liu, Shih-Hsien Yu. Hyperbolic conservation laws and dynamic systems. Discrete and Continuous Dynamical Systems, 2000, 6 (1) : 143-145. doi: 10.3934/dcds.2000.6.143
[17]

Yanbo Hu, Wancheng Sheng. The Riemann problem of conservation laws in magnetogasdynamics. Communications on Pure and Applied Analysis, 2013, 12 (2) : 755-769. doi: 10.3934/cpaa.2013.12.755
[18]

Stefano Bianchini, Elio Marconi. On the concentration of entropy for scalar conservation laws. Discrete and Continuous Dynamical Systems - S, 2016, 9 (1) : 73-88. doi: 10.3934/dcdss.2016.9.73
[19]

Michael Herty, Veronika Sachers. Adjoint calculus for optimization of gas networks. Networks and Heterogeneous Media, 2007, 2 (4) : 733-750. doi: 10.3934/nhm.2007.2.733
[20]

Christophe Prieur. Control of systems of conservation laws with boundary errors. Networks and Heterogeneous Media, 2009, 4 (2) : 393-407. doi: 10.3934/nhm.2009.4.393

2020 Impact Factor: 1.213

Tools

Metrics

  • PDF downloads (103)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy