A distributed model of traffic flows on extended regions
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1.
DEI, Politecnico di Milano, V. Ponzio 35/5, 20133 Milano
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2.
MOX, Dipartimento di Matematica "F. Brioschi”, Politecnico di Milano, P. L. da Vinci 32, 20133 Milano
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3.
IACS/CMCS, Chair of Modeling and Scientific Computing, EPFL, Station 8, CH-1015 Lausanne
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Received:
01 January 2010
Revised:
01 April 2010
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Primary: 35Q70, 76S05; Secondary: 76M50, 82C80.
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This work deals with the modelling of traffic flows in complex
networks, spanning two-dimensional regions whose size
( macroscale ) is much greater than the characteristic size of
the network arcs ( microscale). A typical example is the
modelling of traffic flow in large urbanized areas with diameter of
hundreds of kilometers, where standard models of traffic flows on
networks resolving all the streets are computationally too
expensive. Starting from a stochastic lattice gas model with simple
constitutive laws, we derive a distributed two-dimensional model of
traffic flow, in the form of a non-linear diffusion-advection
equation for the particle density. The equation is formally
equivalent to a (non-linear) Darcy's filtration law. In particular,
it contains two parameters that can be seen as the porosity and the
permeability tensor of the network. We provide suitable algorithms
to extract these parameters starting from the geometry of the
network and a given microscale model of traffic flow (for instance
based on cellular automata). Finally, we compare the fully
microscopic simulation with the finite element solution of
our upscaled model in realistic cases, showing that our model is
able to capture the large-scale feature of the flow.
Citation: Fabio Della Rossa, Carlo D’Angelo, Alfio Quarteroni. A distributed model of traffic flows on extended regions[J]. Networks and Heterogeneous Media, 2010, 5(3): 525-544. doi: 10.3934/nhm.2010.5.525
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Abstract
This work deals with the modelling of traffic flows in complex
networks, spanning two-dimensional regions whose size
( macroscale ) is much greater than the characteristic size of
the network arcs ( microscale). A typical example is the
modelling of traffic flow in large urbanized areas with diameter of
hundreds of kilometers, where standard models of traffic flows on
networks resolving all the streets are computationally too
expensive. Starting from a stochastic lattice gas model with simple
constitutive laws, we derive a distributed two-dimensional model of
traffic flow, in the form of a non-linear diffusion-advection
equation for the particle density. The equation is formally
equivalent to a (non-linear) Darcy's filtration law. In particular,
it contains two parameters that can be seen as the porosity and the
permeability tensor of the network. We provide suitable algorithms
to extract these parameters starting from the geometry of the
network and a given microscale model of traffic flow (for instance
based on cellular automata). Finally, we compare the fully
microscopic simulation with the finite element solution of
our upscaled model in realistic cases, showing that our model is
able to capture the large-scale feature of the flow.
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