Problemi numerici per il flusso di traffico su reti
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1 Problemi numerici per il flusso di traffico su reti Benedetto Piccoli Istituto per le Applicazioni del Calcolo Mauro Picone Consiglio Nazionale delle Ricerche Roma Joint work with G. Bretti, M. Caramia, N. Cascone, A. Cutolo, C. D Apice, Y. Chitour, M. Garavello, M. Herty, A. Klar, A. Marigo, R. Manzo, R. Natalini, L. Rarità, A. Sgalambro, A. Terracina
2 Flows on networks Car Traffic Irrigation Channels Tlc and data networks Blood circulation Supply chains Gas pipelines
3 Fluidodynamical models for traffic flow LWR model M.J. Lighthill, G.B. Whitham, Richards 1955 Example
4 Traffic lights and Viale del Muro Torto Data reconstruction error: 9% free phase, 19% congested phase
5 Fluid dynamic models for SC (Armbruster-Degond-Ringhofer et alii) Conservation law for part density: processing rate part density
6 Processor with queue model (Goettlich-Herty-Klar) Queue Processor j Queue Queuebuffer occupancy change changeisis given givenby by the the difference between incoming and and outgoing flux flux
7 Other fluidodynamical models for car traffic flow Aw Rascle model (modified gas dynamics) Aw Rascle model solves typical problems of second order models: Cars going backwards! Other models: Greenberg, Zhang, Helbing, Klar, Rascle, Benzoni - Colombo, etc. Similar models for : irrigation channels, blood circulation, gas pipelines
8 Dynamics at junctions Highway Mountain Sea
9 Dynamics at junctions Highway Mountain Rule 1 : Traffic distribution coefficients Rule 2 : Through flux Sea
10 Solutions at junctions : Riemann problems (A) There are prescribed preference of drivers, i.e. traffic from incoming roads distribute on outgoing roads according to fixed (probabilistic) coefficients (B) Respecting rule (A) drivers behave so as to maximize flow (entropy). REMARKS: - The only conservation of cars does not give uniqueness - Rule (A) implies conservation of cars -The only rule (A) does not give uniqueness - Rules (A) and (B) determine a LP problem Previous work: Holden-Risebro, Lebacque, Daganzo
11 Another Riemann solver at junctions (tlc) Maximize the fluxes over incoming and outgoing lines: then apply rule (A) and priority rule
12 Numerical schemes Approximation schemes (explicit schemes): Godunov s scheme (first order) Kinetic scheme (kinetic scheme with 2 or 3 velocities) of first order (Aregba-Driollet Natalini) Kinetic scheme with 3 velocities of second order (Aregba-Driollet Natalini) Godunov scheme reads:, then the numerical flux :
13 LP solvers
14 Bilinear model Simple model with reasonable properties Two characteristic velocities Respect phenomenon of backward moving clusters
15 Modified Godunov IDEA: Use bilinear model to have simplified choices of numerical fluxes
16 FSF scheme 1. Use simplified flux function with two characteristic speeds 2. Make use of theoretical results to bound the number of regimes changes 3. Track exactly regimes changes or separating shocks and use simple dynamics for one-sided zones
17 Comparison of schemes
18 Problems : Real data 1. Data: measurements and elaboration 2. Dimensionality: big networks Radars Manual counting Satellite data Videocameras 1500 arcs network Plates reading
19 Car trajectory on network Determine the trajectory of a car on a loaded network Convergence of WFT in papers by Colombo and Marson
20
21 Simulation results
22 Lipschitz continuous dependence (tlc and GHK supply chain model) Lemma (tlc)
23 Google : Networks Heterogeneous Media Editor in Chief : Benedetto Piccoli (IAC, Rome) Henri Berestycki (Ehess, Paris) Jean-Patrick Lebacque (INRETS) Leonid Berlyand (PennState) Claude Le Bris (ENPC, Paris) Andrea Braides (Roma, Tor Vergata) Pierre-Louis Lions (Paris Dauphine) Alberto Bressan (PennState) Dag Lukkassen (Narvik) Suncica Canic (Houston) David MacDonald (Ottawa) Jennifer Chayes (Microsoft) Masayasu Mimura (Meiji) Zhangxin Chen (Dallas) Roberto Natalini (IAC, Rome) Camillo De Lellis (Zurich) Charles Newman (Courant) Antonio DeSimone (SISSA, Trieste) George Papanicolaou (Stanford) Emmanuele Di Benedetto (Vanderbilt) Luigi Preziosi (Torino) Wilfrid Gangbo (GeorgiaTech) Alfio Quarteroni (Lausanne) Dirk Helbing (Dresden) Angela Stevens (Heidelberg) Thomas Hillen (Alberta) Eitan Tadmor (Maryland) Shi Jin (Madison) Hans Van Duijn (Eindhoven) Kenneth Karlsen (Oslo) Jean Luis Vazquez (Madrid) Axel Klar (Kaiserlautern) Shih-Hsien Yu (Hong Kong)
24 Thank you for your attention! 1. G. Bretti and B. Piccoli, A tracking algorithm for a single car moving in a road network, SIAM. Appl. Dyn. Syst. to apper 2. G. Bretti, R. Natalini and B. Piccoli, Numerical Approximations of Traffic Flow Models on Networks, Networks and Heterogeneous Media, vol. 1 n. 1, (2006), pp G. Bretti, R. Natalini and B. Piccoli, Fast algorithms for the approximation of a traffic flow model on networks, Discrete and Continuous Dynamical Systems - Series B (DCDS-B), v. 6 n. 3, (2006) pp A. Cascone, A. Marigo, B. Piccoli, L. Rarità, Decentralized optimal routing for packets flow on data networks, Siam J. Control and Optimization, submitted. 5. Y. Chitour, B. Piccoli, Traffic circles and timing of traffic lights for cars flow, Discrete and Continuous Dynamical Systems-Series B 5 (2005), G.M. Coclite, M. Garavello, B. Piccoli, Traffic Flow on a Road Network, Siam Math. Anal 36 (2005), C. D Apice, R. Manzo, B. Piccoli, Packets flow on telecommunication networks, preprint, SIAM J. Math. Anal. 37 (2006) M. Garavello and B. Piccoli, Traffic Flow on Networks, AIMS Series on Applied Mathematics, vol. 1, American Institute of Mathematical Sciences, 2006, ISBN-13: M. Garavello, B. Piccoli, Source-Destination Flow on a Road Network, Communications Mathematical Sciences 3 (2005), M. Garavello, B. Piccoli, Traffic flow on a road network using the Aw-Rascle model. Comm. Partial Differential Equations 31 (2006), no. 1-3, M. Garavello, R. Natalini, B. Piccoli and A. Terracina, Conservation laws with discontinuous flux', Network Heterogeneous Media, vol. 2 no. 1, (2007) pp M. Herty, A. Klar, B. Piccoli, Existence of solutions for supply chain models based on partial differential equations, SIAM J. Math. Anal. 39 (2007), A. Marigo and B. Piccoli, A fluid-dynamic model for T-junctions, Siam J. Math. An. to appear
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