Two algorithms for a fully coupled and consistently macroscopic PDE-ODE system modeling a moving bottleneck on a road

Gabriella Bretti; Emiliano Cristiani; Corrado Lattanzio; Amelio Maurizi; Benedetto Piccoli; Gabriella Bretti; Emiliano Cristiani; Corrado Lattanzio; Amelio Maurizi; Benedetto Piccoli

doi:10.3934/Mine.2018.1.55

Mathematics in Engineering

2019, Volume 1, Issue 1: 55-83. doi: 10.3934/Mine.2018.1.55

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Research article

Two algorithms for a fully coupled and consistently macroscopic PDE-ODE system modeling a moving bottleneck on a road

1.
Istituto per le Applicazioni del Calcolo “M. Picone”, Consiglio Nazionale delle Ricerche, Via deiTaurini 19, 00185 - Rome, Italy
2.
Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica, Università degli Studidell’Aquila, Via Vetoio, 67100 - Coppito (AQ), Italy
3.
Department of Mathematical Sciences, Rutgers University, 311 N 5th Street, Camden, NJ 08102,USA

Received: 19 June 2018 Accepted: 25 July 2018 Published: 14 September 2018

In this paper we propose two numerical algorithms to solve a coupled PDE-ODE system which models a slow vehicle (bottleneck) moving on a road together with other cars. The resulting system is fully coupled because the dynamics of the slow vehicle depends on the density of cars and, at the same time, it causes a capacity drop in the road, thus limiting the car flux. The first algorithm, based on the Wave Front Tracking method, is suitable for theoretical investigations and convergence results. The second one, based on the Godunov scheme, is used for numerical simulations. The case of multiple bottlenecks is also investigated.
- conservation laws,
- discontinuous ODEs,
- fluid dynamic models,
- LWR model
Citation: Gabriella Bretti, Emiliano Cristiani, Corrado Lattanzio, Amelio Maurizi, Benedetto Piccoli. Two algorithms for a fully coupled and consistently macroscopic PDE-ODE system modeling a moving bottleneck on a road[J]. Mathematics in Engineering, 2019, 1(1): 55-83. doi: 10.3934/Mine.2018.1.55

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Abstract

In this paper we propose two numerical algorithms to solve a coupled PDE-ODE system which models a slow vehicle (bottleneck) moving on a road together with other cars. The resulting system is fully coupled because the dynamics of the slow vehicle depends on the density of cars and, at the same time, it causes a capacity drop in the road, thus limiting the car flux. The first algorithm, based on the Wave Front Tracking method, is suitable for theoretical investigations and convergence results. The second one, based on the Godunov scheme, is used for numerical simulations. The case of multiple bottlenecks is also investigated.

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