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
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. Burger, A. Garcia, 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 preprint gave rise to the following definitive publication(s):
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, vol. 5, no. 3, pp. 461-485, (2010)