Search Advanced

Networks and Heterogeneous Media

2010, Volume 5, Issue 3: 461-485. doi: 10.3934/nhm.2010.5.461
Previous Article Next Article

On some difference schemes and entropy conditions for a class of multi-species kinematic flow models with discontinuous flux

  • 1.
    CI2MA and Departamento de Ingeniería Matemática, Facultad de Ciencias Físicas y Matemáticas, Universidad de Concepción, Casilla 160-C, Concepción
  • 2.
    Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053, Blindern, N–0316 Oslo
  • 3.
    MiraCosta College, 3333 Manchester Avenue, Cardiff-by-the-Sea, CA 92007-1516
  • Received: 01 January 2010 Revised: 01 April 2010

  • Primary: 76M25; Secondary: 65M06, 76T99, 90B20.

  • We study a system of conservation laws that describes multi-species kinematic flows with an emphasis on models of multiclass traffic flow and of the creaming of oil-in-water dispersions. The flux can have a spatial discontinuity which models abrupt changes of road surface conditions or of the cross-sectional area in a settling vessel. For this system, an entropy inequality is proposed that singles out a relevant solution at the interface. It is shown that "piecewise smooth" limit solutions generated by the semi-discrete version of a numerical scheme the authors recently proposed [R. Bürger, A. García, K.H. Karlsen and J.D. Towers, J. Engrg. Math. 60:387-425, 2008] satisfy this entropy inequality. We present an improvement to this scheme by means of a special interface flux that is activated only at a few grid points where the discontinuity is located. While an entropy inequality is established for the semi-discrete versions of the scheme only, numerical experiments support that the fully discrete scheme are equally entropy-admissible.

    Citation: Raimund Bürger, Kenneth H. Karlsen, John D. Towers. On some difference schemes and entropy conditions for a class ofmulti-species kinematic flow models with discontinuous flux[J]. Networks and Heterogeneous Media, 2010, 5(3): 461-485. doi: 10.3934/nhm.2010.5.461

    Related Papers:

    [1] Raimund Bürger, Kenneth H. Karlsen, John D. Towers . On some difference schemes and entropy conditions for a class of multi-species kinematic flow models with discontinuous flux. Networks and Heterogeneous Media, 2010, 5(3): 461-485. doi: 10.3934/nhm.2010.5.461
    [2] Raimund Bürger, Christophe Chalons, Rafael Ordoñez, Luis Miguel Villada . A multiclass Lighthill-Whitham-Richards traffic model with a discontinuous velocity function. Networks and Heterogeneous Media, 2021, 16(2): 187-219. doi: 10.3934/nhm.2021004
    [3] Raimund Bürger, Stefan Diehl, M. Carmen Martí, Yolanda Vásquez . A difference scheme for a triangular system of conservation laws with discontinuous flux modeling three-phase flows. Networks and Heterogeneous Media, 2023, 18(1): 140-190. doi: 10.3934/nhm.2023006
    [4] Raimund Bürger, Harold Deivi Contreras, Luis Miguel Villada . A Hilliges-Weidlich-type scheme for a one-dimensional scalar conservation law with nonlocal flux. Networks and Heterogeneous Media, 2023, 18(2): 664-693. doi: 10.3934/nhm.2023029
    [5] . Adimurthi, Siddhartha Mishra, G.D. Veerappa Gowda . Existence and stability of entropy solutions for a conservation law with discontinuous non-convex fluxes. Networks and Heterogeneous Media, 2007, 2(1): 127-157. doi: 10.3934/nhm.2007.2.127
    [6] Raimund Bürger, Stefan Diehl, María Carmen Martí . A conservation law with multiply discontinuous flux modelling a flotation column. Networks and Heterogeneous Media, 2018, 13(2): 339-371. doi: 10.3934/nhm.2018015
    [7] Clément Cancès . On the effects of discontinuous capillarities for immiscible two-phase flows in porous media made of several rock-types. Networks and Heterogeneous Media, 2010, 5(3): 635-647. doi: 10.3934/nhm.2010.5.635
    [8] Raimund Bürger, Antonio García, Kenneth H. Karlsen, John D. Towers . Difference schemes, entropy solutions, and speedup impulse for an inhomogeneous kinematic traffic flow model. Networks and Heterogeneous Media, 2008, 3(1): 1-41. doi: 10.3934/nhm.2008.3.1
    [9] 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
    [10] Christophe Chalons, Paola Goatin, Nicolas Seguin . General constrained conservation laws. Application to pedestrian flow modeling. Networks and Heterogeneous Media, 2013, 8(2): 433-463. doi: 10.3934/nhm.2013.8.433
  • We study a system of conservation laws that describes multi-species kinematic flows with an emphasis on models of multiclass traffic flow and of the creaming of oil-in-water dispersions. The flux can have a spatial discontinuity which models abrupt changes of road surface conditions or of the cross-sectional area in a settling vessel. For this system, an entropy inequality is proposed that singles out a relevant solution at the interface. It is shown that "piecewise smooth" limit solutions generated by the semi-discrete version of a numerical scheme the authors recently proposed [R. Bürger, A. García, K.H. Karlsen and J.D. Towers, J. Engrg. Math. 60:387-425, 2008] satisfy this entropy inequality. We present an improvement to this scheme by means of a special interface flux that is activated only at a few grid points where the discontinuity is located. While an entropy inequality is established for the semi-discrete versions of the scheme only, numerical experiments support that the fully discrete scheme are equally entropy-admissible.


  • This article has been cited by:

    1. Dianliang Qiao, Zhiyang Lin, Mingmin Guo, Xiaoxia Yang, Xiaoyang Li, Peng Zhang, Xiaoning Zhang, Riemann solvers of a conserved high-order traffic flow model with discontinuous fluxes, 2022, 413, 00963003, 126648, 10.1016/j.amc.2021.126648
    2. Raimund Bürger, Christophe Chalons, Luis M. Villada, Antidiffusive and Random-Sampling Lagrangian-Remap Schemes for the Multiclass Lighthill--Whitham--Richards Traffic Model, 2013, 35, 1064-8275, B1341, 10.1137/130923877
    3. Raimund Bürger, Paul E. Méndez, Carlos Parés, On entropy stable schemes for degenerate parabolic multispecies kinematic flow models, 2019, 35, 0749-159X, 1847, 10.1002/num.22381
    4. Raimund Bürger, Stefan Diehl, M. Carmen Martí, Yolanda Vásquez, A difference scheme for a triangular system of conservation laws with discontinuous flux modeling three-phase flows, 2022, 18, 1556-1801, 140, 10.3934/nhm.2023006
    5. Raimund Bürger, Christophe Chalons, Rafael Ordoñez, Luis Miguel Villada, A multiclass Lighthill-Whitham-Richards traffic model with a discontinuous velocity function, 2021, 16, 1556-181X, 187, 10.3934/nhm.2021004
    6. Dian-Liang Qiao, Peng Zhang, Zhi-Yang Lin, S.C. Wong, Keechoo Choi, A Runge–Kutta discontinuous Galerkin scheme for hyperbolic conservation laws with discontinuous fluxes, 2017, 292, 00963003, 309, 10.1016/j.amc.2016.07.030
    7. Boris Andreianov, Massimiliano D. Rosini, 2020, Chapter 7, 978-3-030-46078-5, 113, 10.1007/978-3-030-46079-2_7
    8. Raimund Bürger, Sonia Valbuena, Carlos A. Vega, A well‐balanced and entropy stable scheme for a reduced blood flow model, 2023, 39, 0749-159X, 2491, 10.1002/num.22975
  • Reader Comments
  • © 2010 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索
Networks and Heterogeneous Media
1.2 1.8

Metrics

Article views(3933) PDF downloads(114) Cited by(8)

Preview PDF
Download XML
Export Citation
Article outline
Abstract

Other Articles By Authors

Related pages

Tools

Copyright © AIMS Press

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog