Sharmin Report

8/2/2019 Sharmin Report

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Chapter 1Mathematical Modeling

In this chapter, we present an outline of mathematical modeling, a partial differential equation

as a mathematical model for traffic flow. Also we discuss about basic variables of traffic

model,relationship among velocity, flux and density. Also we represent specific speed-density

relationship and specific flow-density relationship. Our discussions on this chapter follow from

[1], [5], [7] and [8].

1.1 Outline of Mathematical Modeling

Mathematical modeling is an inevitable component of scientific and technical progress. The

formulation of the problem of mathematical modeling, an object leads to a precise plan of

actions. It can be conditionally split into three stages: model-algorithm code (refer to the

diagram).

Figure 1.1: Diagram of Mathematical Modeling

At the first stage, the equivalent of the object is chosen reflecting its major properties in a

mathematical form- the lows, controlling it, connections peculiar to the components and so on.

Object

Computercode

Mathematical

modeling

Numerical

algorithm

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The mathematical model is investigated using theoretical methods, enabling one to obtain

important preliminary knowledge about the object. The second stage is the choice or

development of the algorithm for the realization of a model on the computer. The model is can

call them the electronic equivalents of the investigated objects, already suitable fir direct tests

on the experimental facility the computer.

1.2 Mathematical Model of Traffic Flow

Let us consider a single-line highway.

Let [ ]( )21 ,xxMt = Number of cars in [ ]21 ,xx at time t. We assume [ ]( )21 ,xxMt = ( )dxxtx

x

2

1

,

where ( )xt, =car density distribution function which is continuously differentiable.Then by the law of conservation of number of cars (i.e. by the mass conservation law) we have

[ ]( ) [ ]( ) =+ 2121 ,, xxMxxM ttt (Number of cars entering per unit time at 1x ) t - (Number of

cars leaving per unit time at 2x ) ( )2tot +

[ ]( ) [ ]( ) =+ 2121 ,, xxMxxM ttt ( ) ( )[ ] ( )221 ,, totxtqxtq +

2

1

x

x

( ) ( )dx

t

xtxtt

+,,

= ( ) ( )[ ] ( )toxtqxtq + 21 ,,

where flux ( )xtq , :=Number of cars which pass the point x per unit time at time t.

2

1

x

x

( ) ( )dx

t

xtxtt

+ ,,

= ( )todxx

qx

x

+

2

1

For 0t we have

02

1

=

+

dx

x

q

t

x

x

As 1x 2x then

0=+

x

q

t

(1.2)(

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Equation (1.2) is a first order partial differential equation called the equation of continuity with

two unknowns ( )xt, and ( )xtq , in one equation and is not solvable. Therefore, one needs to

model the flux ( )xtq , as a function ( )xt, in order to obtain the equation (1.2) in a closed

form. We may have the following two cases.

1. Linear Model (Case 1):

We assume all cars have some constant velocity v >0. Then from the relationship among

velocity, flux and density, the flux vq = yields the equation of continuity (1.2) in the form

0=+

xv

x

(1.3)

a linear first order partial differential equation(PDE) called linear advection equation, which in

closed form i.e. solvable.

2. Non-linear Model (Case 2):

In this case we consider velocity

)(vv = (1.4)

as a function of density then we have the equation (1.2) as the form

( )( )

0=+

x

v

t

(1.5)

Which is a first order PDE and non linear in

This yield

0

0

=

++

=

+

+

xd

dvv

t

xd

dv

x

v

t

(1.6)

Which is linear in derivatives but non-linear in , termed as quasi-linearequation.e

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1.3 Basic Variables of Traffic Model

1.3.1 Velocity Field of Traffic

Consider a car moving along a highway. If ( )tx the position of the car at time t, positionx .

The position of the car might designate. In a highway each car (vehicle) is designated by )(txi

as shown in figure.

Figure 1.2: Position of cars ix in a highway

There are two ways to measure velocity. The most common is to measure velocity iv of each

car,dt

dxv ii = . With n cars there are n different velocities, each depending on time .In another

situation, if the number of cars is too large that it is very hard to record the velocities ofeach

car, we associate to each point in spacex and at time t, a unique velocity ( )xtv , , called a

velocity field. The velocity field ( )xtv , at the cars position )(txi must be the cars velocity

( )tvi , =i 1,2,,n

( )( ) ( )tvttxv ii =,

2x

3x

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Thus the existence of a velocity field ( )xtv , implies that at each space x and at time t, there is

one velocity and this model does not allow cars to pass each other.

1.3.2 Flux of TrafficFlow (flux) is one of the most common traffic parameters. In addition to the car velocities, an

observer fixed at a given position could easily record the number of cars that passed in a given

length of time. The quantity, which measures the averagenumber of cars passing perhour, is

called the traffic flux (flow) which depends on time t and positionx . The traffic flux is

symbolized as ( )xtq , .

If measurements are make on extremely short time interval, for example, 10 second intervals,

then computed flow fluctuates widely as a function of time.

To avoid these difficulties, we assume that there exists a measuring interval such that it is long

so that many cars pass the observer in the measuring interval (eliminating the wild

fluctuations), it is short enough so that the variations in the traffic flow are not smoothed over

averaging over too long a period of time.

If such a measuring time exists, then the step-like curve for the traffic flow can be

approximated by a continuous function of time.

1.3.3 Density of Traffic

The traffic density on the road associated with a given location x and time t, the average

number of vehicles per unit length of road at the location and time specified. Clearly to

measure a density we need a stretch of road with enough cars on it to allow a reasonable

arithmetic mean. At the same time, we want to talk about the spatial variation of traffic density

along the road, so the length over which we average should not be too long either, or else we

will be getting to the natural scale of variation of the density.

We will use the traditional symbol for fluid density, namely , for the traffic density. Thus

( )xt, is the average number of cars per unit length at the location x and time t.

If all vehicles have length L (or elseL is a good average length) and the spacing (or average

spacing) between the cars is d , then each vehicle takes up dL + units of road, so the

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approximatesdL +

1vehicles will be present per unit length of road. Thus the constant density

of the traffic in this case is =dL +

1. In this case, is restricted to a certain range, 0

max , where max is the value at which cars are bumper to bumper.

1.4 Relationship among Velocity, Flux and Density

Let us consider the number of cars that pass a fixed position 0xx = in a very small time t ,

i.e. between the time 0t and tt +0 .In this small time the cars have not moved far and hence

),( xtv and ( )xt, can be approximated by constants, their values at 0xx = and 0tt= .In a

small time ,t the cars that occupy a short space, approximately txtv ),( , will pass the

observer at 0xx = and the corresponding number of cars passing is approximately

( ) ( )xttxtv ,, . Thus the flux ( )xtq , , i.e. the number of cars passing through 0xx = per unit

time, is given by

( )xtxtvxtq ,),(),( = (1.7)

Where

( )xtq , =Traffic flow (vehicles/hour)

( )xtv , =Traffic velocity (miles/hour, kilometers/hour)

( )xt, =Traffic density (vehicles/mile, vehicles/kilometer)

Flow is the product of speed and density, the flow is equal to zero when one or both of these

terms is zero. It is also possible to deduce that the flow is maximized at some critical

combination of speed and density.

Two common traffic conditions illustrate these points. The first is the modern traffic jam,

where traffic densities are very high and speeds are very low. This combination produces a

very low flow. The second condition occurs when traffic densities are very low and drivers can

obtain free flow speed without any undue stress caused by other vehicles on the roadway. The

extremely low density compensates for the high speeds and the resulting flow is very low.

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The discussion of the velocity -flux-density relationship mentioned several velocity-density

conditions. Two of these conditions are extremely significant and have been given special

names.

Free Flow Speed

This is the mean speed that vehicles will travel on a roadway when the density of vehicles is

low. Under low-density conditions, drivers no longer worry other vehicles. They subsequently

proceed at speeds that are controlled by the performance of their vehicles, the conditions of the

roadway and the posted speed limit.

Jam Density

Extremely high densities can bring traffic on a roadway to a complete stop. The density

at which traffic stops is called the jam density.

1.5 Specific Speed-Density Relationship

We now focus on how to solve the non-linear equation (1.6) once ( )v is given. For the

moment our concern is whether or not this assumption is justified, and then what the function

( )v should be.

On a single-lane open road this assumption seems to be fairly reasonable. An isolated car tends

to have a maximum velocity of travel, either the result of speed limits or road conditions or

driver cautions, call it maxv . Then for our function ( )v , we should take ( ) max0 vv = . We know

that traffic speeds tend to go down with increasing traffic density, so we should assume that

0,0 >< d

dv. Also there is surely a density, bumper to bumper traffic say, where the speed is

essentially zero. We denote thisdensity max . If L is theaverage car length, we could take

max .

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The first steady-state speed-density relation is introduced by Greenshields, who proposed a

linear relationship between speed and density that is

( ) = maxmax 1

vv

Various relationships were developed following the Greenshields direction. We use the basic

linear speed-density relation (linear function) which is of the form

( )

=

max

max 1

vv (1.8)

Where maxv = maximum speed (free flow speed)

max = maximum density (jam density)

With the aid of equation (1.8), equation (1.7) gives a relationship for the traffic flux (flow) as a

function of density that is presented in the following form:

( )

=

max

2

max vq (1.9)

Now if we put the velocity-density function ( )

=

max

max 1

vv of equation (1.8) into the

general non-linear model PDE (1.5), then the explicit non-linear PDE is obtained as in the form

01.max

max =

+

v

xt (1.10)

1.6 Specific Flow-Density Relationship

The speed-density relation of freeways interpreted as being linear by Greenschilds (1935)

exerts dominant influences on the relation between the flow q (vehicles/sec.) and the density

(vehicles/km.). Lighthill and Whitham conjectured a flow density relation from the

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information that at low density values speed

qv = was regarded as a function of the flow and

at high density values the headway

N(N is the number of vehicle at certain location) was

regarded as a function of speed

qv = .

The homogeneous velocity is considered that means it does not depend on the highway

location x or at time t explicitly. It follows that ( )vv = and the equation (1.7) gives the rate

of flow (flux) is a function of density that is

( ) ( ) vq = (1.11)

It is obvious that the maximum speed, maxv is achieved at the smallest density 0min then

the equation (1.11) gives the results

( ) 00 =q

( ) 0max =v

( ) ( ) 0maxmaxmax == vq (1.12)

The rate of traffic flow must be positive for all values of density, max0


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Figure 1.4: Fundamental diagram of traffic flow

Since our velocity-density relationship (1.8) is linear. So the flow-density relation (1.9) is

parabolic and concave. The highest flow occurs when its slop vanishes, i.e.

02

1max

max =

=

v

d

dq (1.14)

It gives that the maximum traffic flow is achieved at a point of the fundamental diagram for

2

max= and in this case the maximum flow is

maxmaxmax4

1vq =

q

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Chapter 2

Analytical Solution for the Traffic Flow Model

In this chapter, we present the analytic solution of the traffic flow model with linear and non-

linear PDEs by the method of characteristics. We also discuss about the existence and

uniqueness of the analytic solution of the traffic flow model based on [7] and [10].

2.1 Method of Characteristics

The method of characteristics is a method that can be used to solve the initial value problem for

general first order partial differential equation (PDE). We consider the first order linear PDE

( ) ( ) ( ) 0,,, =+

+

uxtCt

utxB

x

utxA (2.1)

in two variables along with the initial condition ( ) ( )xuxu 00, = .The goal of the method of characteristics, when applied to the equation (2.1) is to change co-

ordinates from ( )tx, to a new co-ordinate system ( )sx ,0 in which the PDE becomes an

ordinary differential equation (ODE) along certain curves in the tx plane. Such curves,

( ) ( )[ ]{ }


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Then we have

( ) ( )t

utxB

x

utxA

ds

dt

t

u

ds

dx

x

u

ds

du

+

=

+

= ,,

and along the characteristics, the PDE becomes an ODE

0),( =+ utxCds

du (2.4)

The equation (2.2) and (2.3) will be referred to as the characteristic equations.

2.1.1 General Strategy

The general strategy for applying the method of characteristics to a PDE of the form (2.1) is

presented bellow

Step-1: At first we solve the two characteristic equations (2.2) and (2.3). We find the constant

of integration by setting ( )0

0 xx = . Now we have the transformation from ( )tx, to ( )sx ,0 and

xx = ( )sx ,0 , tt= ( )sx ,0 .

Step-2: We solve the ODE (2.4) with initial condition ( ) ( )00 xuu = where 0x are the initial

points on the characteristic curves along the 0=t axis in the tx plane.

Step-3: Now we have a solution ( )sxu ,0 . We solve for 0x and s in terms of x and t by using

step-1 and put these values in ( )sxu ,0 to get the solution to the original PDE (2.1) as ( )txu , .

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2.2 Analytical Solution of the linear PDE (linear

Advection Equation)

We consider the linear PDE (linear advection equation), which is written as,

0=+

xv

t

The Cauchy problem is defined by this equation on the domain 0, 0). The solution

( )tx, is constant along each ray 0xvtx = which is known as characteristicsof the equation.

The characteristic curves are in the ( )tx plane satisfying the ordinary differentiableequations

( ) ,vtx = ( ) 00 xx = . If we differentiate ( )tx, along one of these curves to find the rate of

change ofalong the characteristic,

we find that

( )( ) ( )( ) ( )( )ttxx

xttxt

ttxdt

d,,,

+

=

0=

+

=x

vt

Confirming that is constant along theses characteristics. More generally, we might consider

a variable co-efficient advection equation of the form

( ) 0)( =

+

xxv

t

Where ( )xv is a smooth function.

Then we have

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( ) ( )

( ) ( ) ( ) ( )xvtxtxx

xvt

xvx

xvt

=

+

=

+

,,

It follows that the evolution of ( )tx, along any curve ( )tx satisfying

( ) ( )( )txvtx =

( ) 00 xx = (A)

Satisfies a simple ODE

( )( ) ( )( ) ( )( )txvttxttxdt

d = ,,

The curves determined by (A) are called characteristics. In this case the solution is not constant

along these curves, but can be easily determined by solving two sets of ODE.

It can be shown that if ( )x0 is a smooth function, say ( ) ,0kv then the solution

( )tx, is equally smooth in space and time, ( ) ( )( ) ,0,kv

2.3 Analytical Solution of the Non-linear PDE

The non-linear PDE (1.10) mentioned at section 1.42 can be solved if we know the traffic

density at a given initial time, i.e. if we have the traffic density at a given initial time 0t , we

can predict the traffic density for all future time 0tt , in principle.

Then we are required to solve an initial value problem (IVP) of the form

01.max

max =

+

v

xt (2.5)

( ) ( )xxt 00 , =

The IVP (2.5) can be solved by the method of characteristics as follows:

The PDE in the IVP (2.5) may be written as

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( )

0=

+

x

q

t

Where ( )

= maxmax 1.

vq

02

1

0

max

max =

+

=+

xv

t

xd

dq

t

Now 0=

+

=

dt

dx

xtdt

d

By integrating

= 0d

( ) tconsxt tan, = (2.6)

Where

=

max

max

21

v

dt

dx (2.7)

Equation (2.6) and (2.7) give

( )0

max

max

21 xtvtx +

=

(2.8)

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Which are the characteristics of the IVP (2.5)

Figure 2.1: The Characteristics for Traffic Flow Model

Now from equation (2.6) we have

0=dt

d

( ) ctx = , (2.9)

Since the characteristics through ( )tx, also passes through ( )0,0x and ( ) ctx =, is constant on

this curve, so we use the initial condition to write

( ) ( ) ( )000 0,, xxtxc === (2.10)

Equation (2.9) and (2.10) yield

( ) ( )00, xtx = (2.11)

Using equation (2.8),(2.11) takes the form

0x

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( )

= tvxtx

max

max0

21,

(2.12)

This is the analytic solution of the IVP (2.5)

This solution is in implicit form because

also appears in right side. It is much more difficultto transform into explicit form. Therefore, there is a demand of some efficient numerical

techniques for solving the IVP (2.5).

The Cauchy-Kovaleveskaya theorem is the main local existence and uniqueness theorem for

analytic partial differential equation (PDE) associated with Cauchy initial value problem. This

idea was given by Augustin Cauchy (1842) and Sophie Kovaleveskaya (1875). Since the IVP

(2.5) is a Cauchy IVP, so to discuss the existence and uniqueness of the IVP (2.5), we can

readily apply the Cauchy-Kovaleveskaya theorem.

2.3.1Existenceand uniqueness of the traffic flowmodel

As mentioned at section 2.2 the IVP (2.5) may also be written as

02

1max

max =

+

xv

t

(2.13)

( ) ( )xxt 00 , =

In that case,

max

max

21

v is analytic. So the Cauchy-Kovaleveskaya theorem can be applied

for the IVP (2.13) (Cauchy problem).

The Cauchy-Kovaleveskaya theorem guarantees that the Cauchy problem (2.13) has a unique

analytic solution.

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Chapter 3Numerical Schemes for the Traffic Model

In this chapter, we consider the model with two sided boundary conditions and a suitable

numerical scheme for this is the Lax-Friedrichs scheme [1], [5].

3.1 Linear Finite Difference Scheme

3.1.1 Basics of Discretization

We present finite difference approximations to obtain solutions of general non-linear scalar

conservation laws

( )

0=+

x

f

t

for tx = f for all states thatoccur

in the solution of(3.1). Thus we also provide initial data

( ) ( )xx 00, = (3.2)

and data at left hand boundary

( ) ( )tta a =, (3.3)

The equations (3.1), (3.2) and (3.3) lead to an initial boundary value problem (IBVP) and the

IBVP is well-posed if( ) 0

>

= f

.In order to approximate solution to (3.1), we will discretize spatial domain ( )ba,

bxxxxa

ll=


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Ttttt NN =


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Figure 3.1: Spatial and Temporal Discretization

We will discretize the conservation law by approximating the conserved densities

( )dxtxx

i

i

x

x

n

i

i +

2

1

2

1

,1

and

the flux function

dttxft

f

n

n

t

tin

n

i

+

++

+

+

1

,

1

21

21

2

1

21

We will require our discretization to satisfy the conservative difference

][ 21

2

12

1

2

1

2

1

1+

+

+

+

+ =

n

i

n

ii

n

n

i

n

i ffx

t

initially, we define

+

=

2

1

2

1

)(1

0

0

i

i

x

xi

i dxxx

Further, at the left-hand boundaryax =

2

1 we define

f 21

2

1

+

n

=2

1

1

+

n

t+1

))((

n

n

t

t

a dttf

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3.1.2 A Finite Difference Scheme for the IBVP

We consider our specific non-linear traffic model problem as an initial and two point boundary

value problems

(IBVP)

==

=

=+

TtttTttt

bxaxxt

bxaTttx

f

t

0b

0a

00

0

);(b)(t,);(a)(t,B.C.and

);(),(I.C.with

,,0)(

(3.4)

Where

( )

=

max

max 1.

vf (3.5)

As we consider that the cars are running only in the positive x-direction, so the speed must be

positive,

i.e. 0)21()(max

max = vf (3.6)

in the range of

.

To develop this scheme, we discretize the space and time. We discretize the time derivative

t

in the IBVP (3.4) at any discrete point ),( in xt for 1,,0;,,1 == NnMi ; by the

forward difference formula.

The discretization of( )t

tx

,

is obtained by first order forward difference in time and the

discretization of( )

x

tx

,

is obtained by first order central difference in space.

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Forward Difference in Time

From Taylors series we write

( ) ( )( ) ( )

...,

!2

,,,

2

22

+++=+

t

txk

t

txktxktx

( ) ( ) ( )( )ko

k

txktx

t

tx

+=

,,,

( ) ( ) ( )k

txktxttx ,,, +

tt

xt nin

iin

+ 1),( (3.7)

Central Difference in Space

From Taylors series we write

( ) ( )( ) ( )

+

+

+=+2

22 ,

!2

,,,

x

txfh

x

txfhtxfthxf

( ) ( )( ) ( )

+

=

2

22 ,

!2

,,,

x

txfh

x

txfhtxfthxf

( ) ( ) ( )

h

thxfthxf

x

txf

2

,,, +

x

ff

x

xtf nin

iin

+

2

),( 11 (3.8)

We assume the uniform grid spacing with step size k and h for time and space respectively

ktt nn +=+1 and hxx ii +=+1 .

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24/27


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3.1.4 Comparison of Analytical and Numerical

Solution

Figure 3.3 shows the comparison of the analytical and numerical solution in the bound (t,x)-

plane and t=[0,6] min, x=[0,10] km.

Analytical Solution Numerical Solution

Figure 3.3: Comparison of Analytical Solution and Numerical solution

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3.2 Conclusion

In this report paper we have considered the macroscopic traffic flow model. First we

have shown the fundamental of traffic variables, analytical solution of the traffic flow

model. We consider our specific non-linear traffic model problem as an initial and two

point boundary value problems and use a suitable numerical scheme for this, that is the

Lax-Friedrichs scheme.

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BIBLIOGRAPHY

[1] Roksanara Hemel, Analytical and Numerical Methods for PDE - lecture note,

Department of Mathematics,Jahangirnagar University,Savar, Dhaka.

[2] L.S. Andallah, Analytical and Numerical Methods for PDE - lecture note,

Department of Mathematics,Jahangirnagar University,Savar, Dhaka.

[3] L.S. Andallah, Shajib Ali, M.O.Gani, M.K. pandit and J.Akhter A finite

Difference Scheme for a Traffic Flow Model based on a linear Velocity-Density

Function Jahangirnagar University Journal of Science, volume 32,Number 1,June

2009.

[4] LeVeque, R.J.(1992) Numerical Methods for Conservation Laws, nd2

Edition,Springer.

[5] Klar, A.R.D.Kuhne and R.Wegener (1996) Mathematical Models for Vehicular

Traffic, Technical University of Kaiserlautern, Germany.

[6] Haberman,R.(1977) Mathematical models, Prentice-Hall,Inc.

[7] Kuhne,R. and P.Michalopoulous (1997) Continuum Flow Model.

[8] Bretti, G.R. Natalini and B.Piccoli,(2007) A Fluid Dynamic Traffic Model on Road

Networks,Comp. Methods Eng.,CIMNE, Barcelona, Spain.

[9] Wikipedia, the free Encyclopedia Traffic flow from retrieves online.

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