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

Raimund Bürger; Kenneth H. Karlsen; John D. Towers; Raimund Bürger; Kenneth H. Karlsen; John D. Towers

doi:10.3934/nhm.2010.5.461

Networks and Heterogeneous Media

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

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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.
- Difference schemes,
- entropy condition,
- systems of conservation laws,
- discontinuous flux,
- multi-species kinematric flow model
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

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Abstract

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.
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