Traffic flow on networks

Benedetto PiccoliIstituto per le Applicazioni del Calcolo “Mauro Picone” –

CNR – Roma

Joint work with G. Bretti, Y. Chitour, M. Garavello, R.Natalini , A. Sgalambro

Macroscopic modelsVehicular traffic can be treated in different ways with microscopic,

mesoscopic or macroscopic models.

Macroscopic models mimic some phenomena such as the creation ofshocksshocks and their propagation backwards along the road, since they candevelop discontinuities in a finite time even starting from smooth data.

Representation of intersections: - Backward propagation of queues- Distribution of flow capacity resource on the downstream links of a node to its upstream links

LWR model

Fluidodynamical models for traffic flow

Example

M.J. Lighthill, G.B. Whitham, Richards 1955

Traffic features difficult to reproduce:

traffic jamsEmpirical Evidences:

1) Once created, jams are stable and can move for hours against the flow of traffic

2) The flow out of a jam is a stable, reproducible quantity

Bilinear model

• Simple model with reasonable properties• Two characteristic velocities

• Respect phenomenon of backward moving clusters

Fluidodynamical models for traffic flow Aw Rascle model

Aw Rascle model solves typical problems of second order models: Cars going backwards!

Other models: Greenberg, Helbing, Klar, Rascle, Benzoni - Colombo, etc.

Road networks• Road networds consists of a finite set of roads with junctions connecting roads :

Problem: how to define a solution at junctions.

Solutions at junctions Solve the Riemann problem at junctions

(A) There are prescribed preference of drivers, i.e. traffic from incoming roads distribute on outgoing roads according to fixed (probabilistic) coefficients

(B) Respecting rule (A) drivers behave so as to maximize flow.

REMARK:-The only conservation of cars does not give uniqueness- Rule (A) implies conservation of cars- The only rule (A) does not give uniqueness

Rules (A) and (B)Rule (A) corresponds to fix a traffic distribution matrix

Using rule (A) and (B) (under generic assumption on the matrix A), it is possible to define a unique solution to Riemann problems at junctions.

Remark: other definition given by Holden-Risebro ’95.

LP problem at junctions

It is enough to solve a LP problem at junctions for incoming fluxes!Then other fluxes and densities are determined.

Other way of looking: Demand – Supply of J.P. Lebacque

Solutions via WFT for 2x2 junctions

Continuous dependence Lipschitz continuous dependence does not

hold: explicit counterexample.

Open problems: control the BV norm of density, extension to networks with any junction.

Lipschitz continuous dependence holds only for:

1. Single crossing with assumptions on initial data.

2. Small BV perturbations of generic equilibria.

Open problem: continuous dependence.

Numerical approximation

Approximation schemes (explicit schemes): - Godunov’s scheme (first order)

- Kinetic scheme (kinetic scheme with 2 or 3 velocities) of first order (Aregba-Driollet – NataliniAregba-Driollet – Natalini)

- Kinetic scheme with 3 velocities of second order (Aregba-Driollet – NataliniAregba-Driollet – Natalini)

Godunov schemeConstruct piecewise constant approximation of the initial data :

The scheme defines recursively

starting from .

CFL-like condition:

The projection of the exact solution on a piecewise constant function is:

These values are computed with the Gauss-Green formula.

Godunov’s schemeThe scheme reads:

with the numerical flux (associated to flux function ):

Kinetic schemeAdvantages of kinetic scheme: Extension to High order No instability at the boundary

Drawback: Diffusivity at first order for 2 velocities (Lax-Friedrichs

scheme).

Godunov scheme: Kinetic scheme:

Simulation on Rome road network

Simulation on Rome road network

LP solvers

Running times

Large simulations Necessity of simulating networks with

thousands of arcs and nodes Fast simulation to apply for optimization

problems Elaboration of big data bases for

network characteristic Visualization time

Modified Godunov

IDEA: Use bilinear model to have simplified choices of Numerical fluxed

FVST scheme

2. Make use of theoretical results to bound the number of regimes changes

3. Track exactly regimes changes or separating shocks and use simple dynamics for one-sided zones

1. Use simplified flux function with two characteristic speed

FVST schemeFinite volumes shock tracking scheme: Strongly bounded computational times Error due only to initial data rounding

and junction data rounding Bounded to solutions for empty initial

network

Comparison of schemes

Salerno network simulation

Networks and Heterogeneous Media

Packets flow on telecommunication networks

Benedetto PiccoliIstituto per le Applicazioni del Calcolo “Mauro Picone” –

CNR – Roma

Joint work with C. D’Apice, R. Manzo, A. Marigo

Packets flow on telecommunication networks

Telecommunication networks as Internet: no conservation of packets at small time scales.

Assume there exists a loss probability function and packets are re-sent if lost.

Then at 1st step: (1-p) packets sent, p lost at 2nd step: p(1-p) packets sent, p^2 lost…. at kth step: p^(k-1) (1-p) sent, p^k lost …

Finally the average transmission time and velocity are:

Riemann problems at junctionsMaximize the fluxes over incoming and outgoing lines: remove rule (A)

Riemann problems at junctions

BV EstimatesFor interactions with a junction we get:

For special flux fuction we get BV estimates on the densities.

Lipschitz continuous dependence

Lemma