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

*(English)*Zbl 1263.76048Summary: 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. Eng. Math. 60, No. 3–4, 387–425 (2008; Zbl 1200.76126)] 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.

##### MSC:

76M25 | Other numerical methods (fluid mechanics) (MSC2010) |

65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |

76T99 | Multiphase and multicomponent flows |

90B20 | Traffic problems in operations research |