Traffic Flow Simulation By Using A Mathematical Model Based On

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 the Masters of Science in Mathematics

ByMuhammad 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

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 2

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) q ( x, t ) , traffic speed V ( x, t ) and traffic density ρ( x, t ) , all of which are functions of space, x ∈ R and time, t ∈R + . The most well-known LWR model is formulated by employing the conservation equation

∂ρ ∂q ( ρ ) + =0 ∂t ∂x

− − − −(*)

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 on the road is conserved and

b) The flux, q is a product of density, ρ 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   ρ V ( ρ ) = V max 1 −    ρ max 

  

2

   

then the flux is of the form   ρ q ( ρ ) = ρ.V max 1 −    ρ max 

  

2

   

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 3

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 4