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TRAFFIC FLOW SIMULATION BY USING A

MATHEMATICAL MODEL BASED ON A

NONLINEAR VELOCITY-DENSITY FUNCTION

Thesis submitted in partial fulfillment of the requirements

for the degree of theMasters of Science

in

Mathematics

By-

Muhammad Humayun Kabir

Exam Roll: Math 060654

Reg. No: 17606

Session: 2005-06

Supervisor:

Dr. Laek Sazzad Andallah

Department of Mathematics

Jahangirnagar University

Savar Dhaka-1342

Bangladesh


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December 31, 2008

Introduction

Nowadays traffic flow and congestion is one of the main societal and economical

problems related to transportation in industrialized countries. Traffic congestion is one

of the greatest problems in Bangladesh like some other countries of the world. In this

respect, countries managing traffic in congested networks requires a clear

understanding of traffic flow operations. Traffic problems on highways and in urban

areas attract considerable attention. Therefore, an efficient traffic control and

management is essential in order to get rid of such huge traffic congestion.

The study of traffic flow aims to understand traffic behavior in urban context in order to

several questions:

a) Where to install traffic lights or stop signs

b) How long the cycle of traffic lights should be

c) Where to build up accesses, exits, overpasses or underpasses

d) How many lanes for a highway to construct

e) Where to develop alternative outline of transportation like monorails or trams.

The aims of this analysis are principally represented by the maximization of cars flow,

and the minimization of traffic congestions, accidents and pollutions etc.

Many scientists have been working to develop various mathematical models in order to

describe and subsequently, optimize traffic flow, such as

o The microscopic car following model.

o The macroscopic fluid-dynamic model, and

o The kinetic (Boltzmann) model.

These models describe diverse situations with different assumptions and

simplifications. We focus on macroscopic fluid-dynamic model because it is more

efficient and easy to implement than other modeling approaches. The macroscopic

approach is analogous to theories of fluid dynamics or continuum hypothesis.

Introduction

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Macroscopic traffic flow models are characterized by representations of traffic flow in

terms of aggregate measures such as flux, space mean speed, and density.

Unlike microscopic models which represent individual vehicle movements, macroscopic

models sacrifice a great deal of detail but gain by way of efficiency an ability to deal

with problems of much larger scope.

The macroscopic traffic model developed first by Lighthill and Whitham (1955) and

Richard (1956) shortly called LWR model was most suitable for correct description of

traffic flow. In this model, vehicles in traffic flow are considered as particles in fluid:

further the behavior of traffic flow is modeled by the method of Fluid dynamics and

formulated by hyperbolic partial differential equation (PDE).

The macroscopic traffic flow model is used to study traffic flow by collective variables

such as traffic flow rate (flux) ( )txq , , traffic speed ( )txV , and traffic density ( )tx, ,

all of which are functions of space, Rx and time, +Rt . The most well-known

LWR model is formulated by employing the conservation equation

( )

(*)0 =

+

x

q

t

The LWR model comes from two facts and one assumption. The two facts are

a) On a homogeneous road without sources and sinks, the number of vehicles onthe road is conserved and

b) The flux, q is a product of density, and speed V .

The assumption is about the existence of a unique relation between speed and density.

A numerical study for linear density-velocity relationship has been performed in [2].

In this thesis, we use a non-linear velocity-density relationship of the form

( )

=

2

max

max 1

VV

then the flux is of the form

( )

=

2

max

max 1.

Vq

In chapter 1, we derive the macroscopic traffic flow model with corresponding variables

based on [2], [5]. In chapter 2, we present the analytic solution of the nonlinear PDE (*)

which is in implicit form. It is very difficult to incorporate the realistic data in the analytic

solution of this PDE. As a result it is almost obligatory to use the numerical solution

Introduction

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technique for the solution of this PDE in which the initial and boundary data can be

incorporated in a much more efficient way.

In chapter 3, we develop the numerical finite difference schemes for our problem which

formulates an explicit upwind difference scheme based on [10], [12]. We also establish

the stability condition and well-posed-ness of the explicit upwind scheme and those

works has been presented in the international conference [3].

In chapter 4, we present the results of the traffic flow simulation for various traffic flow

parameters. First we develop a computer program for the explicit upwind difference

scheme for linear advection equation and estimate the error of numerical solution which

is very close to the numerical solution. This result guarantees the implementation of the

scheme and the correctness of the computer program. Then we develop a computer

program for the explicit upwind difference scheme for our model and implement the

scheme for artificially generated initial and boundary data and verify the well-known

qualitative behaviors of different flow variables. Finally we try to apply our model for the

traffic flow analysis in a 10 (ten) km range of Dhaka-Aricha highway. Chapter 5

contains the conclusion.

Introduction

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