1. A Traffic Flow Problem: Macroscopic View
A DISSERTATION
submitted by
ARINDAM MANDAL
(MA13C008)
in partial fulfilment
for the award of the degree of
MASTER OF SCIENCE
IN
MATHEMATICS
under the supervision of
Prof. S. SUNDAR
DEPARTMENT OF MATHEMATICS
INDIAN INSTITUTE OF TECHNOLOGY MADRAS
CHENNAI – 600 036
April 2015
3. CERTIFICATE
This is to certify that the project report entitled “A Traffic Flow Problem: Macro-
scopic View”, submitted by ‘Mr. Arindam Mandal’ , to the Indian Institute of
Technology, Madras, in the partial fulfilment for the award of the degree of “Mas-
ter of Science in Mathematics”, is a bonafide record of the project work done by
him under my supervision.
Place: Chennai
Date: 28 April 2015
Prof. S. Sundar
Department of Mathematics
IIT Madras, 600 036
4. ACKNOWLEDGEMENTS
I wish to express my profound gratitude to my supervisor Prof. S. Sundar, De-
partment of Mathematics, Indian Institute of Technology Madras and Mr. S. K.
Sharma, Research Scholar, Department of Mathematics, IIT Madras for their
invaluable guidance and constant encouragement throughout the tenure of this
project.
I would like to express my sincere thanks to Prof. M Thamban Nair, the
Head, Department of Mathematics, for providing suitable atmosphere in the de-
partment and use of the library during my project work. I am also thankful to the
authorities of the Institute for providing me with the necessary facilities to accom-
plish this work. Finally, I thank all my friends and classmates who made my stay
in the campus so blissful.
(Arindam Mandal)
i
5. ABSTRACT
The work presented in this report deals with the mathematical model of some traffic
flow problems which have received a wide importance in further study of compli-
cated traffic flow problems. We have formulated only deterministic mathematical
models. The outline of the work presented in the report is as follows:
In Chapter 1, we have introduced the fundamental traffic variables viz. traffic
velocity, traffic density and rate of traffic flow. In Chapter 2, we relate these fun-
damental variables for macroscopic traffic flow models. This leads us to the for-
mulation of traffic problems in terms of partial differential equation as described in
Chapter 5. We then solve this equation with the help of method of characteristics.
In Chapter 4, we have introduced the essential parts of this method. In Chapter 3, a
linear car following model is presented. The work is concluded in Chapter 6 with
a remark on the effect of traffic lights in the traffic flow.
ii
8. NOTATIONS
The set of all real numbers.
t Time parameter.
x Space parameter.
Kp A sensitivity parameter.
v(x, t) Speed of the traffic flow.
vmax Maximum speed.
q(x, t) Rate of traffic flow.
qmax Maximum traffic flow.
ρ(x, t) Density of the traffic flow.
ρjam Jam density.
ρmax Maximum traffic density.
ρcrit Critical traffic density
v
9. CHAPTER 1
Introduction
Science attempts to establish an understanding of all types of phenomena. To agree
with the experiments or observations qualitatively, different explanations can some-
times be given. However, when theory and experiments quantitatively agree, then
we can usually be more confident in the validity of the theory. In this manner,
mathematics becomes an integral part of the scientific methods [4].
Mathematical models involve three steps:
1. The formulation of the problem - the approximations and assumptions, based
on experiments or observations, that are necessary to develop, simplify, and
understand the mathematical model.
2. To solve the formulated realistic problems like traffic problems etc.
3. The interpretation of the mathematical results in the context of the non-
mathematical problem.
It may be impossible in an experiment to entirely eliminate certain undesirable
effects. Furthermore, one is never sure which effects may be negligible in nature. A
mathematical model has an advantage in that we are able to consider only certain
effects, the object being to see which effects account for given observations and
which effects are immaterial.
Now we will introduce our model which deals with traffic flow problems. Trans-
porting problems have worried man long years back. However, nowadays, these
problems can be solved. Traffic flow problems which may be liable to a scien-
tific analysis include : where to install traffic lights; where to install the signals to
control the traffic; how long the cycle of traffic signal lights should be; whether to
change a two-way street to a one-way street; where to construct entrances, exits
and over passes; how many lanes to build a new highway.
The ultimate aim is to understand the traffic flow phenomena in order to make
decisions which may moderate traffic congestion, maximize flow of traffic, elim-
inate accidents, minimize automobile exhaust pollution and other desirable traf-
fic situations [4]. Of all these kinds of traffic flow problems, we will study only
10. some simple problems which have received a mathematical formulation; how traf-
fic flows along a unidirectional road. We will study traffic situations resulting from
the complex interaction of many vehicles rather than analyzing the behavior of in-
dividual cars. We will only formulate mathematical models that are deterministic.
We define three fundamental traffic variables: velocity field, traffic density and
traffic flow, for investigating traffic flow problems.
1.1 Velocity Field
Let us imagine a highway with many cars, which are moving. We used to measure
the velocity ui of car, ui = dxi
dt
, where xi(t) is the position of i-th car. In many
situations, it is very difficult to keep the track of each car since the number of cars
may be very large. We define another way to measure the velocity. We associate
to each point in x − t plane a unique velocity, u(x, t), is called velocity field. This
velocity at x, time t, is the velocity of a car at the place if the car is there at that
time. Mathematically, the velocity field u(x, t) at the car’s position xi(t) must be
the car’s velocity ui(t). The existence of a velocity field implies that at each x and
t, there is one velocity. Thus this model does not allow cars to pass each other.
e.g. ,
u(x, t) =
30X + 30L
15t + L
.
We determine the motion of the car which starts at X = L
2
at t = 0.
Now,
dX
dt
=
30X + 30L
15t + L
.
By variable separable method,
dX
30X + 30L
=
dt
15t + L
.
Then,
30X + 30L = (15t + L)2
K,
2
11. where K is constant. Using initial condition at t = 0, x = L
2
, we find K=45
L
.
Therefore,
X + L =
3
2L
(15t + L)2
.
1.2 Traffic Flow
Consider an observer fixed at a certain position along a highway could measure the
number of cars that passed in a given length of time. The observer could compute
the average number of cars per unit time (say per hour). This quantity is called
Traffic flow, q. Suppose the following measurements were taken at one place over
half-hour intervals.
Time (A.M.) No. of cars passing No. of cars passing per hour
7.00-7.30 433 866
7.30-8.00 652 1304
8.00-8.30 594 1188
8.30-9.00 551 1102
In this example, the largest flow occurred during the period 7.30 to 8.00. Thus the
flow depends on time, i.e. q = q(t). At different positions along the road, the flow
might be different. Thus the flow also depends on x and we have q = q(x, t).
By measuring the traffic flow over half-hour intervals, we are not able to find the
variations in the flow which might occur over shorter lengths of time. For example,
the flow may be heavier in 7.45-8.00 than in the period 7.30-7.45. If we take the
measurements over very small interval of time, for example (10 seconds),
Time (in seconds after 7.00) No. of cars No. of cars per hour
0-9 0 0
10-19 2 720
20-29 1 360
30-39 4 1440
40-49 1 360
In these measurements, the flow fluctuates wildly as a function of time. We assume
that there exists a measuring interval such that,
3
12. 1. It is long enough so that many cars pass the observer in the measuring interval
to eliminate the wild fluctuations.
2. It is short enough so that variations in the traffic flow can be distinguished.
If such a measuring time exists, the traffic flow can be approximated by a con-
tinuous function of time.
1.3 Traffic Density
The number of cars at a fixed time between two position can be counted. Perhaps a
car is counted only if its center is in the region. The number of cars in a given length
of roadway which might be converted into number of cars per mile, a quantity is
called the density of cars, ρ . The word car is used loosely to represent any vehicle.
We imagine a situation in which all cars are equally spaced. It is now assumed
that all vehicles have same length, L. L is measured in miles for convenience. If
the distance between the cars is d, then the density of cars per mile, is
ρ =
1
L + d
.
As with traffic flow, there are difficulties with traffic density if measurements are
made on too short interval or too long interval [4]. If we expect to approximate the
density as a continuous function of x, the measuring distance must be large enough
so that many cars are contained in the region and small enough so that variations in
densities can be measured.
The flow of traffic on the traffic arteries are predicted with Traffic flow theory,
which is detailed in two levels given below,
The Macroscopic modeling of traffic assumes a sufficiently large number of
cars in a lane or, on a road such that each stream of autos can be treated as would
treat fluid flowing in a tube or, stream [1].
The second level of traffic modeling, Microscopic modeling, addresses the in-
teraction of individual cars in a line of traffic. Microscopic models describe how an
individual follower car responds to an individual leader car by modeling it’s accel-
4
13. eration as a function of various perceived stimuli which might be distance between
the leader and follower cars, the relative speed of two cars or, the reaction time of
the operator of the follower car. The microscopic models are also used to support
the modeling of vehicular control, i.e. to implement control strategies that enable
lines of traffic to maintain high flow rates at high speeds.
5
14. CHAPTER 2
Macroscopic Traffic Flow
We assert the validity that the flow of a stream of cars can be modeled as a field,
much as we would model the flow of fluid [1]. We want to relate the speed of a line
of traffic to the amount of traffic in that line or lane. We use the three variables to
describe such traffic flows:
1. The rate of traffic flow q(x, t),
2. The density of the flow ρ(x, t),
3. The speed of the flow v(x, t).
Consider traffic moving in one direction along an arbitrary stretch of a road.
Then the conservation principle states that the change in the number of cars within
that stretch of road results from the flow of traffic into and out of that road interval
and from the generation or, consumption of cars within the interval. We assume
that the cars are neither generated nor consumed within that road interval.
Thus imagine a co-ordinate x along a particular stretch or, interval of road un-
der consideration that has endpoints defined by x and x + ∆x. The number of cars
within this road interval of length ∆x is given by ∆N(x, t). Then by the conserva-
tion principle, the change in the number of cars within the interval ∆N(x, t) during
a time interval ∆t is, in the limit, equal to the rate of traffic flow q(x, t), i.e.
q(x, t) = lim
∆t→0
∆N(x, t)
∆t
. (1)
The change in the number of cars within the road interval ∆N(x, t), is simply
the difference between the number of cars going in and out of that stretch of road
at each end, N(x, t) and N(x + ∆x, t) respectively, i.e.
∆N(x, t) = N(x, t) − N(x + ∆x, t). (2)
15. If ∆x denotes the length of road interval that is traveled during the time ∆t, the
statement of conservation of cars (1), can also be written as,
q(x, t) = lim
∆t→0
∆N(x, t)
∆x
∆x
∆t
(3)
where ∆x
∆t
= v(x, t), the speed of the traffic. (4)
Now we substitute (2) and (4) in (3) and we get,
q(x, t) = lim
∆x→0
N(x, t) − N(x + ∆x, t)
∆x
v(x, t). (5)
Note that the limit in (5) is taken as ∆x tends to 0 and that it’s dimensions
correspond to the number of vehicles per unit length of road, which we define as
the "density of the traffic flow",
ρ(x, t) = lim
∆x→0
N(x, t) − N(x + ∆x, t)
∆x
.
Then from (5) we get,
q(x, t) = ρ(x, t)v(x, t). (6)
This equation is dimensionally correct and consistent as,
NT−1
= NL−1
LT−1
= NT−1
.
Also the equation (6) can be shown to make physical sense by a rather simple
argument derived by looking at two different ways to counting the number of cars
passing a specified point on the road during a very small time interval.
One measure of the traffic count is the number of cars, ∆N, passing a point
during a time interval ∆t is simply the product of flow rate q and the time interval,
i.e. ∆N = q∆t.
The second measure count assumes that during the same small interval of time
a car moving with a speed v will cover a distance ∆x = v∆t. The number of
vehicles passing through that distance is found from another product, the density ρ
7
16. times the distance, i.e. ∆N = ρ∆x.
Hence equating we get, q∆t = ρ∆x, which is clearly an averaged version of
(6) that accords well with this elementary physical reasoning.
We also observe that the equation (6) is expressed in three variables q, ρ and v.
Therefore, it is of very limited use in this form without substantial further informa-
tion. However it is clear that traffic density ρ and speed v are the two fundamental
traffic variables because we can determine the rate q at which traffic flows by in-
serting them into (6).
Further, if we could relate speed directly to the density i.e. v = v(ρ), then we
could write a direct relationship between the traffic flow rate q and the density ρ,
q(ρ) = ρv(ρ). (7)
2.1 Relating traffic speed to traffic density
Any driver would agree that traffic speed and traffic density are related. Drivers
speed up when traffic is sparse and they slow down to clog up the arteries when
traffic is thick. Thus we are tempted to postulate that there is a direct relationship
between traffic speed and traffic density, v = v(ρ). Then we expect that a driver
will drive fastest vmax when the density is at it’s smallest value, ρ → 0. The speed
decreases as the density increases which is a statement about the slope of the v
versus ρ curve. Finally v = 0 at some maximum or, jam density ρjam, presumably
when the traffic is bumper to bumper. Then,
v(ρ = 0) = vmax (8)
dv
dρ
≤ 0 (9)
and
v(ρ = ρjam) = 0. (10)
The precise shape of this curve is unknown. Only the endpoint values and the sign
of the slope are specified at this point.
8
17. Figure 2.1: Variation of traffic velocity with traffic density
2.2 Relating traffic flow to traffic density: the funda-
mental diagram
For macroscopic models we can take the speed to be homogeneous, which means
it does not explicitly depends on the road co-ordinate x or, on time t. Then we can
write v = v(ρ), anticipating as in equation (7) that traffic flow ultimately depends
only on the density ρ.
Therefore as a driver’s fastest speed vmax occurs when the density is at it’s
smallest ρ = 0, equation (7) tells us that
q(ρ = 0) = 0
i.e. the traffic flow rate is "0".
Similarly, when the traffic slows to a halt at it’s maximum density v(ρjam) = 0,
equation (7) tells us once again that the traffic flow rate is "0";
q(ρjam) = ρjamv(ρjam) = 0
The traffic flow rate must be positive for all values of the density 0 < ρ < ρjam and
must attain it’s maximum value somewhere in the interval. Further the slope of the
traffic flow is given by,
dq
dρ
= v(ρ) + ρ
dv
dρ
. [From(7)]
The qualitative results just found are embodied in the generic curve shown in the
9
18. next figure, which is called the "Fundamental Diagram Of Traffic Flow". Also the
precise shape of this curve is unknown; the endpoint values are specified and the
variation of the slope can be inferred. To make some of these qualitative ideas more
Figure 2.2: Variation of traffic flow rate with traffic density
specific, consider the following linear speed density relationship,
v(ρ) = vmax[1 −
ρ
ρjam
].
This relationship clearly satisfies all the conditions required by equations (8), (9),
(10). Moreover, as the simplest mathematical expression that satisfies these con-
ditions, it is particularly attractive as a "building block" for further modeling, pro-
vides that it adequately models reality.
When substituted into (7), it produces a relationship for the traffic flow rate as
a function of density that is parabolic,
q(ρ) =ρvmax[1 −
ρ
ρjam
]
=vmax[ρ −
ρ2
ρjam
].
Now the maximum flow rate occurs when its slope vanishes, i.e.
dq(ρ)
dρ
= vmax[1 −
2ρ
ρjam
] = 0 (11)
⇒ ρ =
ρjam
2
i.e. equation (11) shows that the maximum traffic flow rate under these assumptions
occurs at the mid point of the fundamental diagram, when ρ =
ρjam
2
i.e. ρ
ρjam
= 1
2
10
19. and so,
qmax =
ρjam
2
vmax[1 −
1
2
]
=
1
4
ρjamvmax.
We can also take, v(ρ) = vmax[1 − [ ρ
ρjam
]2
]. Then it also satisfies equations (8), (9)
and (10). Therefore,
q =vmax[ρ −
ρ3
ρ2
jam
]
⇒
dq
dρ
=vmax[1 −
3ρ2
ρ2
jam
].
For maximum flow rate,
dq
dρ
=0
⇒
ρ
ρjam
=
1
√
3
.
Therefore,
qmax =
1
√
3
ρjamvmax[1 −
1
3
]
=
2
3
√
3
ρjamvmax
So, basically we can take,
v(ρ) = vmax[1 − [
ρ
ρjam
]m
].
Now to look at individual cars we move to the microscopic model. Our interest
is in using the microscopic models to develop the traffic speed density relations
that we need to do macroscopic evaluations of capacity, which we require if we are
going to design highway systems.
11
20. CHAPTER 3
Microscopic Traffic Models
In this chapter we are looking for models that describe how drivers respond to
the stimuli of their traffic situations. The driver will perceive a variety of stimuli,
including the distance between vehicles, their relative speed and their perceived
relative acceleration. We thus seek psychological, not mechanical, models in order
to model human behavior. The driver’s response will depend on the responder’s
sensitivity to the given stimuli as well as on the speed with which the response
is undertaken [1]. Thus some times delay should also be incorporated into such
models.
3.1 Linear Car Following Model
Imagine a line of cars traversing a given road as shown in the adjacent figure.
Figure 3.1: The nomenclature for a line (or lane) of cars on a highway
Each car is identified by a discreet co-ordinate that varies in time, so that the
location of the n-th car is given by xn(t). We also assume that the line has a rea-
sonable value of local density and does not permit passing or, overtaking. Then the
basic equation of car following for such a single lane of traffic is the psychological
one,
Response = Sensitivity × Stimulus.
The response will generally be modeled as the acceleration of the (n+1) th follower
car d2xn+1(t)
dt2 as it moves behind the n th leader car. The stimulus will be modeled in
terms of the co-ordinate of the follower car relative to the leader car, which can in
21. turn be written in terms of the traffic density ρ. The acceleration is then integrated
to determine the speed of that car as a function of the traffic density which is the
input we require for our macroscopic modeling.
Consider a simple linear car following model in which the driver of the follower
car responds to the speed of the leader car relative to the follower car,
d2
xn+1(t)
dt2
= −Kp[
dxn+1(t)
dt
−
dxn(t)
dt
].
The coefficient Kp introduced here is a sensitivity parameter that has dimensions
of per unit time. Note that with Kp > 0 the follower car will decelerate to avoid
hitting the car in front if it is slowing down, relatively speaking. We can model the
time it takes the following driver to respond to events by building in a reaction time
that slows the follower’s acceleration by the delay time T,
d2
xn+1(t + T)
dt2
= −Kp[
dxn+1(t)
dt
−
dxn(t)
dt
]. (12)
Assuming that the sensitivity parameter, Kp is a constant, equation (12) is a linear
ordinary differential equation with constant coefficients that can be integrated once
to yield,
dxn+1(t + T)
dt
= −Kp[xn+1(t) − xn(t)] + Cn+1, (13)
where Cn+1 is the arbitrary constant, with dimensions of speed, that results from the
integration just performed. Note that (13) clearly relates the speed of the follower
car to the distance maintained between the follower and leader cars.
Thus it is a natural precursor of the speed density relationship that we seek. Let
us further assume that all of the cars have the same length L and that the spacing
between common points on any pair of cars is given by d(t). i.e.
d(t) = xn(t) − L − xn+1(t). (14)
It then follows that the number of cars, NR, found in a stretch of road of length LR
is,
NR =
LR
L + d(t)
. (15)
13
22. Which means the density of cars on that road, by using (14) and (15), is,
ρ =
NR
LR
=
1
L + d(t)
=
1
xn(t) − xn+1(t)
. (16)
Thus we have in equation (16) a relationship between the macroscopic traffic den-
sity ρ and the microscopic coordinates of the leader and follower cars.
Let us still further assume, for now atleast, that the traffic flow is in a steady
state by which we mean that all of the cars traveling at the same speed. Then,
dxn+1(t + T)
dt
=
dxn+1(t)
dt
= v. (17)
Now substituting (16) and (17) into (13) we get,
v =
Kp
ρ
+ c.
Now from (10) the speed is zero when the density is at it’s maximum or, jam value.
Hence,
0 =
Kp
ρjam
+ c
⇒ c = −
Kp
ρjam
i.e.,
v = Kp[
1
ρ
−
1
ρjam
].
This adjacent curve seems reasonable enough except for the fact that it shows an
Figure 3.2: A curve illustrating the traffic speed-density relationship corresponding
to a linear car following model
14
23. infinite speed as the density goes to zero, a result that hardly seems credible. So we
have a model that seems reasonable and credible over a good portion of the relevant
domain, but that crashes in some region. So, we need to fix this.
Fixing the high (infinite at ρ = 0) speed at small values of ρ is straightforward
enough. All we need to do is stipulate that a maximum speed applies for all values
of density below some specified critical density. This seems like a reasonable fix
that roughly accords with our everyday driving experience. This fix is shown in the
next figure and the equations (18), (19), and (20).
v(ρ) = vmax; ρ < ρcrit (18)
and
v(ρ) = Kp[
1
ρ
−
1
ρjam
]; ρ ≥ ρcrit (19)
where,
ρcrit = [
vmax
Kp
+
1
ρjam
]−1
. (20)
Then the traffic flow rate corresponding to this fixed speed density relationship is
Figure 3.3: A curve illustrating the traffic speed-density relationship corresponding
to a fixed linear car following model
found as,
q(ρ) = ρvmax; ρ < ρcrit
q(ρ) = Kp[1 −
ρ
ρjam
]; ρ ≥ ρcrit.
The traffic flow rate pictured in the next figure increases linearly with density from
15
24. 0 and reaches it’s maximum value, the capacity, when ρ = ρcrit. Then,
qmax = q(ρcrit) = ρcritvmax = Kp[1 −
ρcrit
ρjam
].
For density values ρ ≥ ρcrit, the traffic flow rate decreases linearly with ρ from it’s
Figure 3.4: A curve illustrating the traffic flow rate and traffic density relationship
corresponding to a fixed linear car following model
maximum value at ρ = ρcrit until it vanishes altogether at ρ = ρjam.
Now to look what happens in the case of conservation of cars and traffic density
waves to see the behavior of the traffic in real time we need to know about method
of characteristics, which is followed in the next chapter.
16
25. CHAPTER 4
Method Of Characteristics
The fundamental idea associated with PDE is the notion of a characteristic, a curve
in spacetime along which information is carried[5]. In this chapter we shall go
through some PDEs which can be dealt with method of characteristics.
Example 1- Consider the linear PDE,
ut + cux = 0; x ∈ , t > 0
u(x, 0) = u0(x), x ∈ (1)
Now by Lagrange Method, the char. equation is, dt
1
= dx
c
= du
0
. This gives du
dt
= 0
and dx
dt
= c.
i.e. u =constant along x = ct + c1, where c1 is an arbitrary constant.
Figure 4.1: Characteristics x − ct =constant
Now consider a point (ξ, 0), then c1 = ξ. So we get, x = ct+ξ, i.e., ξ = x−ct.
Then the PDE (1) is reduced to an ODE which is integrated along the family of
curves x − ct = ξ. If we draw one of these curves in the x − t plane that passes
through an arbitrary point (x, t), it intersects the x-axis at (ξ, 0) and speed is c and
slope is 1
c
because we are graphing t versus x. Since u is constant along this curve,
then,
u(x, t) = u(ξ, 0) = u0(ξ) = u0(x − ct),
26. which is the solution of (1).
Example 2- Now consider,
ut + uux = 0; x ∈ , t > 0,
u(x, 0) = 1; x < 0
= 0; x > 0.
The characteristic system is, dt
1
= dx
u
= du
0
, which gives u is constant along
dx
dt
= u
or, x = u(x, t)t + c.
Now consider a point (ξ, 0). This gives, ξ = u(ξ, 0).0 + c ⇒ c = ξ. Thus we get,
x = u0(ξ)t + ξ.
Therefore, u is constant along x = u0(ξ)t + ξ.
Now, for ξ > 0, u0(ξ) = 0, then, x = 0 + ξ ⇒ x = ξ. For ξ < 0, u0(ξ) = 1, then,
x = 1.t + ξ ⇒ x = t + ξ. We see that the characteristics are intersecting at t = 0
Figure 4.2: Characteristics of the problem
itself. The characteristics are intersecting at x = 0, in where u should be 0 or, 1? To
avoid this we insert a straight line x = mt along which the discontinuity at x = 0
is carried. Now we will use Rankiv-Hugnoid (RH) condition[see APPENDIXB]
to
find that m.
Consider,
ut + [φ(u)]x = 0.
18
27. Then,
ds(t)
dt
=
φ(u)+
− φ(u)−
u+ − u−
.
Here
ut + uux = ut + [
u2
2
]x = 0.
So, φ(u) = u2
2
, then,
ds(t)
dt
=
1
2
[
(u+
)2
− (u−
)2
u+ − u−
]
=
1
2
[u+
+ u−
]
=
1
2
[0 + 1]
=
1
2
.
Thus we get,
s(t) =
t
2
+ c.
Now, s(0) = 0, since at the x-axis the value of u is 0. i.e. c = 0. So, s(t) = t
2
is
the shock path.
Figure 4.3: Insertion of a line x = mt along which the discontinuity is carried
Therefore the solution is given by,
u(x, t) = 1, x <
t
2
= 0, x >
t
2
.
19
28. 4.1 Rarefaction Waves
Another difficulty can occur with non-linear equations having discontinuities in
initial or, boundary data [5].
Example 3- For this consider the equation,
ut + uux = 0; x ∈ , t > 0
u(x, 0) = 0, x < 0
= 1, x > 0
The characteristic system is, dt
1
= dx
u
= du
0
, which gives u = constant along
x = u(x, t)t + ξ.
When ξ < 0, u(ξ, 0) = 0, then, x = ξ. And when ξ > 0, u(ξ, 0) = 1, then,
x = t + ξ, i.e. ξ = x − t. Thus no point in the void region is reached by the
Figure 4.4: Spacetime diagram showing a characteristic void
characteristics. So we take a special solution u = x
t
and x = ct, where 0 < c < 1.
Then,
u(x, t) = 0, ifx < 0
=
x
t
, if0 < x < t
= 1, ifx > t
20
29. Figure 4.5: Insertion of a characteristic fan in the void in Figure 4.4
Example 4- Consider,
ut + uux = 0; x ∈ , t > 0
u(x, 0) = 1, x < 0
= −1, 0 < x < 1
= 0, x > 1
The characteristic system is dt
1
= dx
u
= du
0
, which gives, u =constant along x =
u(x, t)t + ξ.
When ξ < 0, u(ξ, 0) = 1, we get, x = t + ξ.
When 0 < ξ < 1, u(ξ, 0) = −1, we get, x = −t + ξ.
And when ξ > 1, u(ξ, 0) = 0, we get, x = ξ.
Therefore the characteristics are given by,
x = t + ξ, ξ < 0
= −t + ξ, 0 < ξ < 1
= ξ, ξ > 1.
Now x = t + ξ and x = −t + ξ intersect. Then,
ds
dt
=
u+
+ u−
2
⇒
ds
dt
=
−1 + 1
2
⇒
ds
dt
=0
⇒ s =c.
21
30. Figure 4.6: Characteristic diagram with intersecting characteristics
Now s(0) = 0 gives c = 0. Thus x = 0 is the shock path for 0 ≤ t ≤ 1. Now the
Figure 4.7: Insertion of a shock for 0 < t < 1
void area is reached by no points of the characteristics. So we fix, u = x−1
t−0
= x−1
t
and x−1
t
= c, where 0 < c < 1. This characteristics intersects with x = t + ξ. For
this case,
dx
dt
=
u+
+ u−
2
=
x−1
t
+ 1
2
=
x − 1 + t
2t
.
Solving this differential equation we get,
x = t + 1 + c
√
t.
The intersection point is (0, 1), i.e. when x = 0, then t = 1. This gives c = −2.
i.e.,
x = t + 1 − 2
√
t
22
31. is the shock path. This shock path intersects with x = 1 and for the point of
intersection,
1 = t + 1 − 2
√
t
⇒ t = 4
Then (1, 4) is the intersecting point and the shock path is, x = t + 1 − 2
√
t for
1 ≤ t ≤ 4.
Figure 4.8: Insertion of a fan in the void region and the resulting shock for
1 < t < 4
Again x = t + ξ and x = ξ intersect at (1, 4). For this intersection we get a shock
path as,
ds
dt
=
0 + 1
2
or, s(t) =
t
2
+ c.
We have when x = 1, t = 4, which gives us c = −1. Hence,
x =
t
2
− 1.
23
32. Figure 4.9: Continuation of the shock for t > 4
Then the solution is given by, for 0 < t < 1,
u = 1; x < 0
= −1; 0 < x < 1 + t
=
x − 1
t
; 1 + t < x < 1
= 0; x > 1
and for 1 ≤ t ≤ 4,
u = 1; x < t + 1 − 2
√
t
=
x − 1
t
; t + 1 − 2
√
t < x < 1
= 0; x > 1
For t > 4,
u = 1; x <
t
2
− 1
= 0; x >
t
2
− 1.
24
33. Now we shall use this method of characteristics to solve the partial differential
equation generated by the conservation of cars.
25
34. CHAPTER 5
Traffic Density Waves
Consider a one way road on where the cars are going in one direction with same ve-
locity v(x, t) and equal distance between them. Then consider an interval (x1, x2)
on the road. Then the number of cars in the interval (x1, x2) is =
x2
x1
ρ(x, t)dx,
where ρ(x, t) is the density of cars.
Now, q(x, t) = ρ(x, t)v(x, t) is the number of cars which pass through x at time
t. Then by conservation law, the number of cars in the interval (x1, x2) changes
according to the number of cars which enter or, leave this interval. i.e.,
d
dt
x2
x1
ρ(x, t)dx = ρ(x1, t)v(x1, t) − ρ(x2, t)v(x2, t)
i.e.,
x2
x1
ρtdx = −
x2
x1
(ρv)xdx
i.e.,
x2
x1
(ρt + (ρv)x)dx = 0
Therefore,
ρt + (ρv)x = 0; x ∈ , t > 0
This is an equation of the type,
ut + f(u)x = 0; x ∈ , t > 0
u(x, 0) = u0(x); x ∈ ,
where f : → and f(ρ) = ρv.
This equation can be solved by method of characteristics and the solution may
develop discontinuities after a finite time.
35. 5.1 Riemann Problem
At first we consider three properties [2] of f as follows
1. f is a C2
function,
2. f is a strictly concave function[see APPENDIXA]
,
3. f(0) = f(ρmax) = 0.
In a traffic jam situation, the flow of traffic at one place is limited and the flow
at some other part of the road is different, which means we have different density
functions at different points of the road. We consider a point x = 0 on the road and
let the initial datum is given by,
ρ(x, 0) = ρ1, ifx < 0
= ρ2, ifx > 0.
i.e. before the point x on the road the initial density is different than the density
after that point. Then we get the equation as,
ρt + f(ρ)x = 0
ρ(x, 0) = ρ1, ifx < 0
= ρ2, ifx > 0.
where f(ρ, v) = ρv and v ≡ v(ρ).
CASE 1- When ρ1 < ρ2, then 1 and 2 imply that df(ρ1)
dρ
> df(ρ2)
dρ
, since a differen-
tiable function f(x) is concave if df
dx
is monotonically decreasing.
Then,
ds
dt
=
[f(ρ)]
[ρ]
=
f(ρ+
) − f(ρ−
)
ρ+ − ρ−
= λ(let)
i.e., x = λt + c, where c is a constant. Now at (ξ, 0); x = ξ, t = 0, then c = ξ.
27
36. Therefore, x − λt = ξ.
Thus the solution is given by,
ρ(x, t) = ρ1, ifx < λt
= ρ2, ifx > λt
Figure 5.1: The solution to the Riemann problem when ρ1 < ρ2, where ρ−
= ρ1
and ρ+
= ρ2
CASE 2- When ρ1 > ρ2, then df(ρ1)
dρ
< df(ρ2)
dρ
because of the same reason as CASE
1. Thus we get,
ρt + f(ρ)x = 0
or, ρt +
df
dρ
ρx = 0
Now the characteristic system is, dt
1
= dx
df
dρ
= dρ
0
.
Then ρ =constant along dx
dt
= f (ρ)
i.e. ρ =constant along x = f (ρ)t + c, where c is a constant and f (ρ) = df
dρ
.
At O, t = 0, x = 0, then c = 0. Therefore, x = f (ρ)t. Now consider a point
(ξ, 0), where ξ is x-coordinate.
When ξ < 0, ρ = ρ1, we get, x = f (ρ1)t.
When ξ > 0, ρ = ρ2, we get, x = f (ρ2)t.
28
37. Figure 5.2: The solution to the Riemann problem when ρ1 > ρ2, where ρ−
= ρ1
and ρ+
= ρ2
For the void region we take ρ = (f )−1
(x
t
) since,
f (ρ1)t < x < f (ρ2)t
or, f (ρ1) <
x
t
< f (ρ2)
or, ρ1 < (f )−1
(
x
t
) < ρ2
Then the solution is given by the rarefaction wave,
ρ(x, t) = ρ1, ifx < f (ρ1)t
= (f )−1
(
x
t
), iff (ρ1)t < x < f (ρ2)t
= ρ2, ifx > f (ρ2)t
Thus if we know the initial densities of different places on the road then we can
calculate the traffic density of different places in any time [2]. Now the remaining
case is what happens to the traffic when it is halted by a traffic light or, loosed by a
traffic light. To see this we go to the next chapter.
29
38. CHAPTER 6
Effects Of Traffic Lights In The Traffic
6.1 Behavior of traffic after the traffic light changes
to green
Now we formulate and solve the traffic problem what happens after a traffic light
turns green.
Suppose that traffic is lined up behind a red traffic light. We denote the position
of the light as x = 0. Since the cars are bumper to bumper behind the traffic light
we have, ρ = ρmax for x < 0. We assume that the cars are lined up indefinitely
and of course, are not moving for our analysis [2]. Also we assume that there is no
traffic ahead of the traffic light, i.e. ρ = 0 for x > 0.
Suppose at t = 0, the traffic light turns from red to green. Now we solve the
PDE,
∂ρ
∂t
+
∂q
∂x
= 0
or,
∂ρ
∂t
+
∂q
∂ρ
∂ρ
∂x
= 0
to find the density of cars at later times where q = q(ρ) and ρ = ρ(x, t). We have
the initial condition,
ρ(x, 0) = ρmax, x < 0
= 0, x > 0
Note that the initial condition is discontinuous function. Also, q
ρ
=
N
t
N
x
= x
t
Now,
dq
dρ
=
dx
dt
⇒
dρ
dt
=
∂ρ
∂t
+
dx
dt
∂ρ
∂x
= 0
39. Then ρ = constant along x = dq(ρ)
dρ
t + k, where each characteristics may have
different k as k is a constant. We analyze all characteristics that intersects the
initial data at the positive region of x (x > 0). There ρ(x, 0) = 0. Thus ρ = 0
along such lines.
Then,
dx
dt
=
dq
dρ
|ρ=0 = v(0) = vmax.
Therefore the characteristic curves which intersect the x−axis (x > 0) are all
straight lines with slope vmax. Hence the characteristic which starts from x = x0
at t = 0 is given by,
x = vmaxt + k,
where k is a constant.
For x = x0, t = 0, k = x0, we have, x = vmaxt + x0. The first characteristic in this
region starts at x = 0 and hence,
x = vmaxt.
Thus in the region, x > vmaxt, the density is zero; i.e. no cars have reached this
region. As soon as the light changes, the first car to reach the point x takes the
time, t = x
vmax
and thus there would be no cars at x for t < x
vmax
.
Now we analyze the characteristics that intersect the initial data for x < 0,
where ρ = ρmax. We determine the characteristic for ρ = ρmax from the equation,
dx
dt
=
dq
dρ
|ρ=ρmax
= [ρ
dv
dρ
+ v]|ρ=ρmax
= ρmax
dv
dρ
|ρ=ρmax + v(ρmax)
= ρmaxv (ρmax) < 0,
since dv(ρ)
dρ
= v (ρ) < 0 and ρmax > 0 and v(ρmax) = 0.
Now the maximum density is certainly in the region of "Heavy Traffic". Thus
31
40. the characteristic which intersect at x < 0,
x = ρmaxv (ρmax)t + x0,
where x0 > 0.
At t = 0, the characteristic starts from x = 0, then, x = ρmaxv (ρmax)t.
In the region x < ρmaxv (ρmax)t the cars cannot move, i.e. the cars stand still.
Then,
ρ = ρmax; x < ρmaxv (ρmax)t
= 0; x > vmaxt
Figure 6.1: Method of characteristics: regions of no traffic and bumper to bumper
traffic
Now we discuss the density for the region, ρmaxv (ρmax)t < x < vmaxt
Figure 6.2: Discontinuous model of initial traffic density
Actually cars pass through the green traffic light in this region. To discuss
this problem, we assume that the initial traffic density was not discontinuous but
smoothly varied between ρ = 0 and ρ = ρmax in a very small distance ∆x near the
traffic light.
32
41. Figure 6.3: Continuous model of initial traffic density
If ∆x = 0, the characteristic along which ρ = 0 and ρ = ρmax may be sketched
in a space time diagram as Figure 6.4. There must be characteristics which start
close to the origin. ρ is constant along the line,
x =
dq
dρ
t + x0,
where x0 is very small.
Figure 6.4: Space time diagram for rapid transition from no traffic to bumper to
bumper traffic
Now ρ varies continuously between ρ = 0 and ρ = ρmax, the velocity dq
dρ
varies
between vmax and ρmaxv (ρmax) respectively. We know that as density increases,
the velocity decreases. There is a value ρ = ρmax where velocity is zero, then
for heavy traffic the velocity is negative. Thus the straight line characteristics with
different slopes may be sketched in the Figure 6.5. Notice that the characteristics
fan out.
If the initial traffic density is in fact discontinuous, then we will obtain the den-
sity in the unknown region by considering the limit of continuous initial condition
problem as ∆x → 0.
33
42. Figure 6.5: Method of characteristics: ∆x is the initial distance over which density
changes from 0 to ρmax
ρ is constant along the characteristic dx
dt
= dq
dρ
,which are again straight lines,
x =
dq
dρ
t + x0.
The characteristic not corresponding to ρ = 0 or, ρ = ρmax go through x = 0 at
t = 0. This is the result of letting ∆x → 0. Thus x0 = 0 and x = dq
dρ
t. At the
discontinuity x = 0, all traffic densities between ρ = 0 and ρ = ρmax are observed.
The characteristics are called fan like characteristics. Along each characteristic the
density is constant.
Figure 6.6: Fan shaped characteristics due to discontinuous initial data
To obtain the density at a given (x, t), we must determine which characteristic
goes through that position at that time. At (x, t), the velocity is known,
dq
dρ
=
x
t
(1)
Since dq
dρ
depends only on ρ, we can solve for ρ as a function of x and t in the region
of fan like characteristic. At a given x and t in this region dq
dρ
is calculated using
(1). In the diagram (dq
dρ
versus ρ), the density is determined graphically.
34
43. Figure 6.7: Determination of traffic density from traffic density wave velocity
The maximum flow occurs where dq
dρ
= 0. Thus the density which is stationary
indicates positions at which the flow of cars is a maximum. As soon as the light
changes from red to green the maximum flow occurs at the light, x = 0, and stays
there for all the future time. This suggests a method to measure the maximum flow.
Consider an observer at a traffic light. He has to wait until the light turns red and
many cars line up. Then when the light change to green, simply measure the flow
at the light. If v = v(ρ) is correct, then this measured traffic flow of cars will be
constant and equal to the road’s capacity.
6.2 Uniform traffic stopped by a red light
We will discuss a uniform stream of moving traffic which is suddenly halted by a
red light at x = 0. Assume that there is no slowing down and cars immediately stop
when light turns red. The initial density is ρ = ρ0. The red light is mathematically
modeled in the following way. At x = 0, the traffic is stopped. Thus ρ = ρmax for
all time t > 0.
Characteristics that emanate from regions in which ρ = ρ0 move at dq(ρ0)
dρ
while
characteristics emanating from the position of the light x = 0, ρ = ρmax travel at a
velocity dq(ρmax)
dρ
[The density waves associated with lower densities travel faster].
Now,
ρ0 < ρmax ⇒
dq(ρ0)
dρ
>
dq(ρmax)
dρ
,
since dq
dρ
is the velocity and as density increases the velocity decreases.
Therefore the characteristics intersect each other whether the initial traffic is
35
44. light or, heavy.
Figure 6.8: Stopping of uniform traffic: Characteristics
The cross hatched region indicates that the method of characteristics yield a
multivalued solution to the PDE. The difficulty is remedied by considering a shock
wave. The shock waves are the propagating waves demarcating the path at which
densities and velocities abruptly change.
Figure 6.9: Unknown shock path
The cross hatched region is to be avoided. Let us suppose that there is a shock
wave. On one side of the shock wave the method of characteristics suggests that
the traffic density is uniform ρ = ρ0 and on the other side ρ = ρmax bumper to
bumper traffic. The theory for such a discontinuous condition implies that the path
for any shock wave must satisfy the shock condition,
ds
dt
=
[q]
[ρ]
,
where [q] = the jump in the traffic flow and [ρ] = the jump in the density.
The initial position of the shock is known, giving a condition for this ODE.
36
45. This shock must initiate s = 0 at t = 0. Then,
ds
dt
=
ρmaxv(ρmax) − ρ0v(ρ0)
ρmax − ρ0
⇒
ds
dt
= −
ρ0v(ρ0)
ρmax − ρ0
⇒ s = −
ρ0v(ρ0)
ρmax − ρ0
t + k,
since v(ρmax)) = 0 and k is a constant. Now, using the condition s = 0 when t = 0
we get k = 0. Therefore,
s = −
ρ0v(ρ0)
ρmax − ρ0
t < 0,
since ρmax > ρ0. Thus the shock wave moves at a constant negative velocity.
The shock separates cars standing still from cars moving forward at velocity v(ρ0).
The cars must decelerate from v(ρ0) to zero velocity instantaneously. As such
instantaneous deceleration are fatal, the theory predicts accidents at these shocks
[3].
In a realistic model, the dependence of velocity only on density v(ρ) must be
modified only in the regions very near shocks. In such regions, the good driver will
presumably see stopped traffic and slow down.
Here,
ρ = ρ0; x < −
ρ0v(ρ0)
ρmax − ρ0
t
= ρmax; −
ρ0v(ρ0)
ρmax − ρ0
t < x < 0
and the shock path will be a straight line.
Figure 6.10: Constant velocity shock wave
37
46. 6.3 Conclusion
When the red light turns green, the solution is obtained by the method of char-
acteristics. It indicates that characteristics fan out around the region x = 0. The
observer travel at a different constant velocities dq
dρ
depending on which density they
initially observe at x = 0. The maximum traffic flow occurs at bumper to bumper
traffic. When traffic is stopped by red light, the model yields multivalued solution
to the characteristics. So, the shock wave method rescues us in finding the solution.
The shock wave moves at constant negative velocity. In the region of near shocks
accidents occur. The shock wave solution is appreciable except in the region when
the density changes from ρ = ρ0 to ρ = ρmax.
38
47. APPENDIX A
1.Concave function- A real valued function f on an interval is said to be concave
if, for any x and y in the interval and for any t in [0, 1],
f((1 − t)x + ty) ≥ (1 − t)f(x) + tf(y)
A function f is called strictly concave if,
f((1 − t)x + ty) > (1 − t)f(x) + tf(y)
Figure 6.11: A concave function
Property- A differentiable function f is concave on an interval if it’s derivative
function f is monotonically decreasing on that interval, i.e., a concave function
has a decreasing slope.
39
48. APPENDIX B
2.Runkine-Hugoniot (R-H) condition [5]- Consider,
ut + (φ(u))x = 0; x ∈ , t > 0
u(x, 0) = u0(x)
where φ is continuously differentiable. Now if u suffers a simple discontinuity
along x = s(t), then
ds(t)
dt
=
[φ(u)]
[u]
,
where [u] and [φ(u)] defines the jump in u and φ(u) respectively.
40
49. REFERENCES
[1] Dym, C., Principles Of Mathematical Modeling. Academic Press, 2004.
[2] Garavello, M. and B. Piccoli, Traffic flow on networks. AIMS, 2006.
[3] Gazis, D., Traffic Science. New York: John Wiley and Sons, 1974.
[4] Haberman, R., Mathematical Models. SIAM, Philadelphia, 1998.
[5] Logan, J. D., An introduction to nonlinear partial differential equation. John
Wiley and Sons, 2008.
41
