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Submission to the DTA2012 special issue: Convergence of time discretization schemes for continuous-time dynamic network loading models. (English) Zbl 1338.90111
Summary: Dynamic Network Loading (DNL) is an essential component of Dynamic Traffic Assignment (DTA) and Dynamic User Equilibrium (DUE). Most DNL models are formulated in continuous time but solved in discrete time to obtain numerical solutions. This paper discusses the importance of choosing proper discretization schemes to numerically solve continuous-time DNL models and further to obtain convergence and other desirable properties of the discretization schemes. We use the recently developed \(\alpha\) point-queue model as an example. We first develop theoretical results to prove the consistency, stability and convergence of the implicit and explicit discretization schemes for solving the \(\alpha\) point-queue model. We then conduct numerical experiments to show such results accordingly. We also discuss the implications of the implicit and explicit discretization schemes to the developments of DNL and DTA/DUE solution algorithms.
MSC:
90B20 Traffic problems in operations research
65K05 Numerical mathematical programming methods
93B40 Computational methods in systems theory (MSC2010)
Software:
bvp4c; dde23
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[1] Ban, X; Liu, H; Ferris, M; Ran, B, A link-node complementarity model and solution algorithm for dynamic user equilibria with exact flow propagations, Transp Res Part B, 42, 823-842, (2008)
[2] Ban, X; Pang, JS; Liu, H; Ma, R, Continuous-time point-queue models in dynamic network loading, Transp Res Part B, 46, 360-380, (2012)
[3] Ban, X; Pang, JS; Liu, H; Ma, R, Modeling and solving continuous-time instantaneous dynamic user equilibria: A differential complementarity systems approach, Transp Res Part B, 46, 389-408, (2012)
[4] Carey, M, Dynamic traffic assignment with more flexible modelling with links, Netw Spat Econ, 1, 349-375, (2001)
[5] Carey M, Balijepalli C, Watling D (2013) Extending the cell transmission model to multiple lanes and lane-changing. Netw Spat Econ. doi:10.1007/s11067-013-9193-7 · Zbl 1338.90101
[6] Carey, M; Ge, Y, Comparing whole-link travel time models, Transp Res Part B, 37, 905-926, (2003)
[7] Carey, M; Ge, Y, Retaining desirable properties in discretising a travel-time model, Transp Res Part B, 41, 540-553, (2007)
[8] Carey, M; Ge, Y, Comparison of methods for path flow reassignment for dynamic user equilibrium, Netw Spat Econ, 12, 337-376, (2012) · Zbl 1332.90065
[9] Carey, M; McCartney, M, Behavior of a whole-link travel time model used in dynamic traffic assignment, Transp Res Part B, 36, 83-95, (2002)
[10] Carey, M; Subrahmanian, E, An approach to modelling time-varying flows on congested networks, Transp Res Part B, 34, 157-183, (2000)
[11] Daganzo, C, The cell transportation model: a dynamic representation of highway traffic consistent with the hydrodynamic theory, Transp Res Part B, 28, 269-287, (1994)
[12] Friesz, TL; Kim, T; Kwon, C; Rigdon, M, Approximate network loading and dual-time-scale dynamic user equilibrium, Transp Res Part B, 45, 176-207, (2010)
[13] Han, K; Friesz, TL; Yao, T, A partial differential equation formulation of vickrey’s bottleneck model, part I: methodology and theoretical analysis, Transp Res Part B, 49, 55-74, (2013)
[14] Han, K; Friesz, TL; Yao, T, A partial differential equation formulation of vickrey’s bottleneck model, part II: numerical analysis and computation, Transp Res Part B, 49, 75-93, (2013)
[15] Han, S, Dynamic traffic modeling and dynamic stochastic user equilibrium assignment for general road networks, Transp Res Part B, 37, 225-249, (2003)
[16] Heydecker B, Verlander N (1999) Calculation of dynamic traffic equilibrium assignments. In: Proceedings of the European transport conferences, Seminar F., pp 79-91 · Zbl 0983.65079
[17] Hoffman JD (1992) Numerical methods for engineers and scientists. McGraw-Hill, New York · Zbl 0823.65006
[18] Kuwahara, M; Akamatsu, T, Decomposition of the reactive dynamic assignments with queues for a many-to-many origin-destination pattern, Transp Res Part B, 31, 1-10, (1997)
[19] Li, J; Fujiwara, O; Kawakami, S, A reactive dynamic user equilibrium model in networks with queues, Transp Res Part B, 34, 605-624, (2000)
[20] Lighthill, M; Whitham, J, On kinematic waves. I: flow movement in long rivers; ii: a theory of traffic flow on long crowded roads, Proc Royal Soc A, 229, 281-345, (1955) · Zbl 0064.20905
[21] Nie, X; Zhang, H, A comparative study of some macroscopic link models used in dynamic traffic assignment, Netw Spat Econ, 5, 89-115, (2005) · Zbl 1081.90017
[22] Nie, X; Zhang, H, Delay-function-based link models: their properties and computational issues, Transp Res Part B, 39, 729-751, (2005)
[23] Pang, JS; Stewart, D, Differential variational inequalities, Math Prog, 113, 345-424, (2008) · Zbl 1139.58011
[24] Peeta, S; Ziliaskopoulos, A, Foundations of dynamic traffic assignment: the past, the present and the future, Netw Spat Econ, 1, 233-265, (2001)
[25] Ran B, Boyce D (1996) Modeling dynamic transportation networks: an intelligent transportation systems oriented approach. Springer-Verlag, New York · Zbl 0898.90004
[26] Richards, P, Shockwaves on the highway, Oper Res, 4, 42-51, (1956)
[27] Shampine L, Gladwell I, Thompson S (2003) Solving ODEs with MATLAB. Cambridge University Press, Cambridge · Zbl 1040.65058
[28] Shampine, L; Thompson, S, Solving ddes in MATLAB, Appl Numer Math, 37, 441-458, (2001) · Zbl 0983.65079
[29] Strikwerda J (1989) Finite difference schemes and partial differential equations. Chapman & Hall, London · Zbl 0681.65064
[30] Wie, B; Tobin, R; Carey, M, The existence, uniqueness and computation of an arc-based dynamic network user equilibrium formulation, Transp Res Part B, 36, 897-918, (2002)
[31] Xu, Y; Wu, J; Florian, M; Marcotte, P; Zhu, D, Advances in the continuous dynamic network loading problem, Transp Sci, 33, 341-353, (1999) · Zbl 0958.90005
[32] Zhang H (2001) Continuum flow models, traffic flow theory, a state-of-the-art report. Technical report, Transportation Research Board, original text by Reinhart Kuhne and Panos Michalopoulos
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