1. Modeling of Traffic Flow Problems
Prof. S. Sundar

2. Plan Introduction Traffic Flow Models
Traffic Flow for single lane (Lighthill-Whitham)
Traffic Flow for single lane with traffic jam
Traffic Flow for two lane with traffic jam
Numerical Approximations to Linear Scalar Conservation Laws
Central Difference Scheme
Lax-Friedrich’s Scheme
Down-Wind Scheme
Up-Wind Scheme
Numerical Approximations to Non-Linear Scalar Conservation Laws

3. Introduction

4. Introduction x1 x x2 The number of cars in the interval (x1,x2)
Traffic Flow
Cars
The density of cars (number of cars per km.)
High Way
The number of cars which pass through x at time t
The number of cars in the interval (x1,x2) changes according to the number of cars which enter or leave this interval.
On Simplification
Conservation Law

5. Scalar Conservation Law
And can be solved using the method of characteristics
The solutions may develop discontinuties after a finite time
Weak Solution
Riemann Problem

6. Traffic Flow Models

7. Lighthill-Whitham model for a single lane
Scalar Conservation Law – Equation of motion
With non-linear flux function
0< < max
Where umax is the maximum attainable velocity of cars in traffic
If the highway is empty ( = 0), we drive with maximal velocity
And in heavy traffic, we tend to slow down, and in a tailback the carks are bumper to bumper (= max )

8. Propagation on a single lane with a traffic jam
The Lighthill-Whitham model for a single lane
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 function at different parts of the road.

9. Propagation on two lanes with a traffic jam
We use the similar model but with a source term on the right to depict the merging of lines 1
Main Road
The source term would include the rate of change of cars within the region x1-x0, beyond which we assume that there are no other merging roads x0 x1 2
Merging Road
 Is the rate at which the cars change from one lane to another, 0, 1 are initial densities.

10. Numerical Schemes

11. Numerical Approximation of Linear Scalar Conservation Laws
A simple linear scalar conservation law
where
Now, we discretize the (x,t) plane
For simplicity we take a uniform mesh
with h and k constant
The simplest of approximations to the solution at these grid points is the finite difference approximation i.e., to replace partial derivatives by difference quotients.
For example Taylor series
expansion of the above equation

12. Central Difference Scheme
A simple linear scalar conservation law
(i,n)
(i+1,n)
(i-1,n)
(i-1,n-1)
(i,n-1)
(i+1,n-1)
(i+1,n+1)
(i,n+1)
(i-1,n+1) x t
Using the Central Difference Approximation
(in space only ??)
Which can be re-written as
As we can compute uin+1 from the data uin explicitly, this is known as explicit scheme
Equivalently,
This is an implicit scheme, where a linear system has to be solved

13. Boundary Conditions
In Practice, we compute on a finite grid say x in (0,a) and we require appropriate Boundary Conditions.
Periodic Boundary Conditions
Discretized version
Setting i=0 or i=N, we required to determine u-1n or uN+1n and we consider these points as artificial points with
By periodicity

14. Numerical Implementation
Discontinuous Initial Data
Continuous Initial Data
h = 0.01
k = 0.001
x = 0 to 1
t = 0 to 0.25

15. u x

16. Lax-Friedrich’s Scheme
The time derivative is approximated using
(i,n)
(i+1,n)
(i-1,n)
(i-1,n-1)
(i,n-1)
(i+1,n-1)
(i+1,n+1)
(i,n+1)
(i-1,n+1) x t
And the spatial derivative is approximated
using the central difference scheme
Hence, the scheme is
We will see that the solution is smeared out, and this approximation becomes better and better for smaller k>0

17. Numerical Implementation
Discontinuous Initial Data
Continuous Initial Data
h = 0.01
k = 0.001
x = 0 to 1
t = 0 to 0.25

18. Down-Wind Scheme
The Lax-Friedrich’s scheme gives accurate approximations only if k is sufficiently small.
(i,n)
(i+1,n)
(i-1,n)
(i-1,n-1)
(i,n-1)
(i+1,n-1)
(i+1,n+1)
(i,n+1)
(i-1,n+1) x t
The Down-Wind scheme is described by
We will see that the numerical solution is unstable
The solution describes a wave from left to right.

19. Numerical Implementation
Discontinuous Initial Data
Continuous Initial Data
h = 0.01
k = 0.001
x = 0 to 1
t = 0 to 0.25

20. Up-Wind Scheme
In the Down-Wind Scheme, the spatial derivative at xi uses the information at xi+1 where the wave will go in the next time step, which does not make sense.
(i,n)
(i+1,n)
(i-1,n)
(i-1,n-1)
(i,n-1)
(i+1,n-1)
(i+1,n+1)
(i,n+1)
(i-1,n+1) x t
It would be more reasonable to use the information at xi-1 where the wave comes from.
Hence, the Up-wind Scheme is described as
We will see that, the solution is almost exact

21. Numerical Implementation
Discontinuous Initial Data
Continuous Initial Data
h = 0.01
k = 0.001
x = 0 to 1
t = 0 to 0.25

22. Numerical Approximation of Non-Linear Scalar Conservation Laws
Up-Wind Scheme
Lax-Friedrich’s Scheme
Quasi-Linear equation
Conservation form