Car path tracking in traffic flow networks with bounded buffers at junctions. (English) Zbl 1446.90061
Summary: This article deals with the modeling for an individual car path through a road network, where the dynamics is driven by a coupled system of ordinary and partial differential equations. The network is characterized by bounded buffers at junctions that allow for the interpretation of roundabouts or on-ramps while the traffic dynamics is based on first-order macroscopic equations of Lighthill-Whitham-Richards (LWR) type. Trajectories for single drivers are then influenced by the surrounding traffic and can be tracked by appropriate numerical algorithms. The computational experiments show how the modeling framework can be used as navigation device.
MSC:
| 90B20 | Traffic problems in operations research |
| 90B10 | Deterministic network models in operations research |
| 34B45 | Boundary value problems on graphs and networks for ordinary differential equations |
Keywords:
boundary value problems on graphs and networks; finite difference schemes; traffic flow on networks; traffic problemsReferences:
| [1] | TreiberM, KestingA. Traffic Flow Dynamics: Data, Models and Simulation. Berlin, Heidelberg: Springer; 2013. |
| [2] | GaravelloM, HanK, PiccoliB. Models for Vehicular Traffic on Networks. AIMS series on applied mathematics. 9Springfield, MO: American Institute of Mathematical Sciences; 2016. · Zbl 1351.90045 |
| [3] | GaravelloM, PiccoliB. Traffic Flow on Networks: Conservation Law Models. AIMS series on applied mathematics. 1Springfield, MO: American Inst of Mathematical Sciences; 2006. · Zbl 1136.90012 |
| [4] | PiccoliB, NataliniR, BrettiG. Numerical approximations of a traffic flow model on networks. Networks and Heterogeneous Media. 2006;1(1):57‐84. https://doi.org/10.3934/nhm.2006.1.57 · Zbl 1124.90005 · doi:10.3934/nhm.2006.1.57 |
| [5] | GoatinP, GaravelloM. The cauchy problem at a node with buffer. Discrete and Continuous Dynamical Systems‐Series A. 2012;32(6):1915‐1938. · Zbl 1238.90033 |
| [6] | HertyM, LebacqueJ‐P, MoutariS. A novel model for intersections of vehicular traffic flow. Networks and Heterogeneous Media. 2009;4(4):813‐826. https://doi.org/10.3934/nhm.2009.4.813 · Zbl 1187.35137 · doi:10.3934/nhm.2009.4.813 |
| [7] | D’ApiceC, GöttlichS, HertyM, PiccoliB. Modeling, simulation, and optimization of supply chains: a continuous approach. Philadelphia: Society for Industrial and Applied Mathematics; 2010. · Zbl 1205.90003 |
| [8] | BrettiG, PiccoliB. A tracking algorithm for car paths on road networks. SIAM Journal on Applied Dynamical Systems. 2008;7(2):510‐531. https://doi.org/10.1137/070697768 · Zbl 1168.35389 · doi:10.1137/070697768 |
| [9] | Delle MonacheML, ReillyJ, SamaranayakeS, KricheneW, GoatinP, BayenA. A pde‐ode model for a junction with ramp buffer. SIAM Journal on Applied Mathematics. 2014;74(1):22‐39. https://doi.org/10.1137/130908993 · Zbl 1292.90077 · doi:10.1137/130908993 |
| [10] | GöttlichS, HarterC. A weakly coupled model of differential equations for thief tracking. Networks and Heterogeneous Media. 2016;11(3):447‐469. https://doi.org/10.3934/nhm.2016004 · Zbl 1346.90239 · doi:10.3934/nhm.2016004 |
| [11] | R. M.Colombo (ed.) and A.Marson (ed.), Conservation laws and O.D.E.s. a traffic problem, in Hyperbolic Problems: Theory, Numerics, Applications, T. Y.Hou (ed.) and E.Tadmor (ed.), eds., Springer, Berlin, Heidelberg, 2003, pp. 455-461. · Zbl 1058.90015 |
| [12] | ColomboRM, MarsonA. A hölder continuous ode related to traffic flow. Proceedings of the Royal Society of Edinburgh: Section A Mathematics. 2003;133(4):759‐772. https://doi.org/10.1017/S0308210500002663 · Zbl 1052.34007 · doi:10.1017/S0308210500002663 |
| [13] | LighthillMJ, WhithamGB. On kinematic waves. ii. a theory of traffic flow on long crowded roads. Proceedings of the Royal Society of London, Series A: Mathematical, Physical and Engineering Sciences. 1955;229(1178):317‐345. https://doi.org/10.1098/rspa.1955.0089 · Zbl 0064.20906 · doi:10.1098/rspa.1955.0089 |
| [14] | RichardsPI. Shock waves on the highway. Operations Research. 1956;4(1):42‐51. https://doi.org/10.1287/opre.4.1.42 · Zbl 1414.90094 · doi:10.1287/opre.4.1.42 |
| [15] | Dal SantoE, DonadelloC, PellegrinoSF, RosiniMD. Representation of capacity drop at a road merge via point constraints in a first order traffic model. ESAIM: Mathematical Modelling and Numerical Analysis. 2019;53(1):1‐34. https://doi.org/10.1051/m2an/2019002 · Zbl 1420.35156 · doi:10.1051/m2an/2019002 |
| [16] | HautB, BastinG, ChitourY. A macroscopic traffic model for road networks with a representation of the capacity drop phenomenon at the junctions. IFAC Proceedings Volumes. 2005;38(1):114‐119. https://doi.org/10.3182/20050703-6-CZ-1902.02042 · doi:10.3182/20050703-6-CZ-1902.02042 |
| [17] | FilippovA. F., Differential equations with discontinuous righthand sides, 18 of Mathematics and its applications Soviet series, Kluwer, Dordrecht, 1988. · Zbl 0664.34001 |
| [18] | DijkstraEW. A note on two problems in connexion with graphs. Numerische Mathematik. 1959;1(1):269‐271. https://doi.org/10.1007/BF01386390 · Zbl 0092.16002 · doi:10.1007/BF01386390 |
| [19] | CookeKL, HalseyE. The shortest route through a network with time‐dependent internodal transit times. Journal of Mathematical Analysis and Applications. 1966;14(3):493‐498. https://doi.org/10.1016/0022-247X(66)90009-6 · Zbl 0173.47601 · doi:10.1016/0022-247X(66)90009-6 |
| [20] | DreyfusSE. An appraisal of some shortest‐path algorithms. Operations Research. 1969;17(3):395‐412. · Zbl 0172.44202 |
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