Networks and Spatial Economics

, Volume 15, Issue 3, pp 419–441 | Cite as

Submission to the DTA2012 Special Issue: Convergence of Time Discretization Schemes for Continuous-Time Dynamic Network Loading Models

  • Rui Ma
  • Xuegang (Jeff) Ban
  • Jong-Shi Pang
  • Henry X. Liu


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 α 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 α 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.


Discretization scheme Dynamic traffic assignment Dynamic network loading Point-queue model Ordinary differential equation Convergence analysis 



The authors would like to thank the anonymous referees for their helpful suggestions on an earlier version of the paper. The work of Rui Ma and Xuegang (Jeff) Ban is based on research supported by the National Science Foundation under Grant EFRI 1024647. The work of Jong-Shi Pang is based on research supported by the National Science Foundation under Grants EFRI 1024984 and CMMI 0969600. The work of Henry X. Liu is based on research supported by the National Science Foundation under Grant EFRI 1024604. Any opinions, findings, and conclusions or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of the National Science Foundation.


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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Rui Ma
    • 1
  • Xuegang (Jeff) Ban
    • 2
  • Jong-Shi Pang
    • 3
  • Henry X. Liu
    • 4
  1. 1.Department of Civil and Environmental EngineeringUniversity of CaliforniaDavisUSA
  2. 2.Department of Civil and Environmental EngineeringRensselaer Polytechnic Institute (RPI)TroyUSA
  3. 3.Department of Industrial and Systems EngineeringUniversity of Southern CaliforniaLos AngelesUSA
  4. 4.Department of Civil EngineeringUniversity of MinnesotaMinneapolisUSA

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