A front tracking method for a strongly coupled PDE-ODE system with moving density constraints in traffic flow

In this paper we introduce a numerical method for tracking a bus trajectory on a road network. The mathematical model taken into consideration is a strongly coupled PDE-ODE system: the PDE is a scalar hyperbolic conservation law describing the traffic flow while the ODE, that describes the bus trajectory, needs to be intended in a Caratheodory sense. The moving constraint is given by an inequality on the flux which accounts for the bottleneck created by the bus on the road. The finite volume algorithm uses a locally non-uniform moving mesh which tracks the bus position. Some numerical tests are shown to describe the behavior of the solution.

[1]  Paola Goatin,et al.  Finite volume schemes for locally constrained conservation laws , 2010, Numerische Mathematik.

[2]  J. Nédélec,et al.  First order quasilinear equations with boundary conditions , 1979 .

[3]  Thomas Y. Hou,et al.  Computational Methods for Propagating Phase Boundaries , 1996 .

[4]  Rinaldo M. Colombo,et al.  A Hölder continuous ODE related to traffic flow , 2003, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[5]  Benedetto Piccoli,et al.  Moving Bottlenecks in Car Traffic Flow: A PDE-ODE Coupled Model , 2011, SIAM J. Math. Anal..

[6]  M J Lighthill,et al.  On kinematic waves II. A theory of traffic flow on long crowded roads , 1955, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[7]  Carlos F. Daganzo,et al.  Moving Bottlenecks: A Numerical Method that Converges in Flows , 2003 .

[8]  Benedetto Piccoli,et al.  A Tracking Algorithm for Car Paths on Road Networks , 2008, SIAM J. Appl. Dyn. Syst..

[9]  Rinaldo M. Colombo,et al.  A well posed conservation law with a variable unilateral constraint , 2007 .

[10]  I. Bohachevsky,et al.  Finite difference method for numerical computation of discontinuous solutions of the equations of fluid dynamics , 1959 .

[11]  P. I. Richards.  Shock Waves on the Highway , 1956 .

[12]  Carlos F. Daganzo,et al.  On the Numerical Treatment of Moving Bottlenecks , 2003 .

[13]  Benjamin Boutin,et al.  Convergent and conservative schemes for nonclassical solutions based on kinetic relations , 2007, 0712.3766.

[14]  R. Colombo,et al.  Mixed systems: ODEs - balance laws , 2012 .

[15]  Christophe Chalons,et al.  General constrained conservation laws. Application to pedestrian flow modeling , 2013, Networks Heterog. Media.