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Arthur van Dam. A Moving Mesh Finite Volume Solver for Macroscopic Traffic Flow Models. MSc Thesis. Mathematical Institute, Utrecht University, May 2002. Internal report TNO 02-7N-152-I500. (PDF (6.1MB))


This report presents the research done as graduation project for Computational Science at the Department of Mathematics of Utrecht University. The research was conducted within the project "Modelling of traffic flow and driver behaviour in congestion for the analysis of Automated Driver Assistance Systems", which is part of a joint research program of the Netherlands TRAIL Research School and the Dutch Organization of Applied Science TNO. One of the main research interests in this project is to develop insight into the mechanisms of individual and collective driving behaviour in congested conditions.

The research presented in this report consists of three parts. The report presents an introduction to traffic flow models, especially macroscopic formulations as nonlinear systems of Partial Differential Equations. A moving mesh finite volume solver for PDE systems of this kind is studied, implemented and investigated. For one specific model, traffic flow simulations are performed in various scenarios.

Traffic Flow Models

Modelling traffic flow can be done at various levels of detail: submicroscopic, microscopic, mesoscopic and macroscopic. (Sub-)microscopic models distinguish the individual drivers and describe their behaviour in great detail. Macroscopic models describe the traffic flow by continuous aggregate functions like average density, velocity and flow in the space-time domain. The dynamics of traffic flow is modelled by a nonlinear system of two or three PDEs. Mesoscopic models form a huge 'transition' class between micro- an macroscopic models.

Kerner [22, 23, 24] has presented a model of two PDEs, describing the dynamics of traffic density r and average velocity V . Hoogendoorn [15, 16, 17, 18, 19, 20, 21] has done much research in the late nineties on 2- and 3-PDE formulations, including more realistic description of macroscopic flow dynamics by introducing multiple userclasses, like person cars and trucks.

Moving Mesh Solver

In many fields, the technique of adaptive meshing has become increasingly important in the last decade. Adaptive meshing is a technique to concentrate meshpoints in a discretized DE solver at the locations where they are needed the most. Often, this means placing the meshpoints near regions where the solution has a large gradient. To use a time-dependent adaptive mesh, i.e. a moving mesh, monitor functions are used to automatically 'monitor' the importance of the various parts of the domain, by assigning a 'weight'-value to each location.

Using moving meshes in combination with an advanced finite volume PDE solver can yield serious gain in accuracy at relatively low extrafficosts. The choice of a suitable monitor function is very important.

Implementing a Traffic Flow Simulator

Using the formulations from macroscopic traffic flow models and the solving techniques like moving meshes and a finite volume PDE solver, a powerful traffic flow simulator can be implemented in a matlab program. By using a model- and problem-based description of the traffic flow PDEs to be used, a flexible simulator is the result.

By entering the desired traffic flow PDEs and domain specifications as separate modules, the simulator is a powerful tool in investigating all kinds of traffic flow models and during the development of new macroscopic formulations.

Traffic Flow Characteristics

Although it has been criticised for not always being so realistic, Kerners model shows various interesting characteristics in the formation of traffic jams. These can often be intuitively explained using fundamental diagrams. The effectiveness of velocity management systems can also be investigated by running macroscopic traffic flow simulations, albeit in a qualitative way.

Outline for further Research

Based upon this research and its conclusions, some directions for future research can be given:
  • Since the solver is aimed at a rather general class of PDEs, no problem specific tricks can be implemented, whereas these tricks can generally yield faster runs and sometimes more accurate results. Possibly, the PDE-solving part of the moving mesh solver could be extended with some additional PDE solvers that are slightly more aimed at certain more specific PDEs from the general class of nonlinear hyperbolic PDEs. For each problem to be solved, the most suitable solver is chosen.
  • The monitor function that is finally used in this research is a quite powerful one, but maybe even better ones exist. This could make the mesh-moving even more e cient and more accurate.
  • For the investigated models, more advanced simulations could be performed, like multi-lane formulations and a more detailed study of multiple userclass characteristics.
  • Models that could not directly be solved with the eventual solver should be further investigated. If it is possible to identify the term(s) that cause the instability with the solver, measures could be provided to overcome the problems.
  • In case the developed software in TraFlowPACK is used often, a graphical shell could be built around it, to make it more userfriendly.


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Arthur.Dam02MscAbstract r1.3 - 11 May 2005 - 11:15 - ArthurVanDam
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