Modeling traffic flow

Modeling traffic flow
Tord Oscarsson
October 2001
1 Introduction
One of the simplest non-trivial applications of the fluid equations is to combine
the continuity equation
∂ρ · (ρv)
= −∂ (1)
∂t ∂x
with a linear relationship between the density ρ and the velocity v. To simplify
further, let us consider the one dimensional case and let
v(ρ) = v m (1 − ρ
ρ m
). (2)
Here v m and ρ m denote the maximum velocity and density attainable in the
system. With this relation between density and velocity, the continuity equation
can be rewritten in a general form as
∂ρ
∂t = − ∂
∂x [(α + 1 βρ)ρ], (3)
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where α = v m and β = −2v m /ρ m . Equation (3) is the generalized in-viscid
Burger’s equation, which occurs frequently in physics. The solution to Burger’s
equation is generally very complex. However, the evolution of ρ from the initial
conditions
{
ρm x ≤ 0
ρ =
(4)
0 otherwise
can be determined analytically, and is shown in figure 1. The solution is given
by
⎧
⎨ ρ m
x < −v m t
− ρm 2
⎩
( x
v − 1) −v mt mt ≤ x ≤ v m t
(5)
0 x > v m t
To solve (1) and (2) numerically, we can use either the Lax or the Lax-
Wendroff method. The Lax iteration scheme becomes
ρ n+1
j
= 1 2 (ρn j+1 + ρn j−1 ) − ∆t
2∆x (F n j+1 − F n j−1 ) (6)
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ρ(x,t)
t > 0
1
t 2
> t 1
x
.
Figure 1: Solution to Burger’s equation for the initial conditionsin (4)
where
F (ρ) = ρv(ρ) (7)
is the mass flux, and Fj
n = F (ρ(x j , t n )). The Lax-Wendroff iteration scheme is
given by
ρ n+1
j = ρ n j − ∆t
+ 2
( ∆t
2∆x
2∆x (F j+1 n − F j−1 n )
) 2 [ ]
qj+1/2 n (F j+1 n − F j n ) − qn j−1/2 (F j n − F j−1 n )
(8)
where
Exercise 1
q n j
= ( dF
dρ )n j . (9)
Write a computer program for solving Burger’s equation along the lines described
above. Implement both the Lax and the Lax-Wendroff scheme, and
employ periodic boundary conditions.
Test your program by solving for the initial conditions
{
ρm for L/4 ≤ x ≤ 3L/4
ρ(x, t = 0) =
0 otherwise
(10)
where L is the length of the system. The solution obtained for these initial
conditions is similar to the solution in figure 1. The density will decrease linearly
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etween two points moving along the x-axis. This continues until the density
at x = L/4 begins to change. After this, the solution is difficult to predict
analytically. Use L = 400 m, v m = 25 m/s, ρ m = 1 arbitrary units, and the
number of grid points NG = 40.
Question 1:
What time step should you use in this case?
Question 2:
The above initial condition has a peculiar effect in the Lax-Wendroff scheme.
What happens and why? Suggest a way of altering the initial conditions slightly
that will get rid of this effect.
Run the program until the density at x = L/4 is beginning to change. Plot
ρ, v, and F as functions of x for various times. Does the solution evolve as
in figure 1 for a) the Lax scheme? b) the Lax-Wendroff scheme? Can you see
any difference in the solutions obtained with the two methods? Try increasing
the step length ∆t, keeping NG fixed. What happens? Also, try increasing NG
while keeping ∆t fixed. What happens?
Exercise 2
Burger’s equation is often used as a first approximation of high-way traffic flow
(really the flow on a one-way road without car-passing capabilities). Which
traffic situation can be modeled by the solution obtained in exercise 1?
Run your program as in exercise 1. Use only the Lax-Wendroff method and
use a Gaussian initial density
ρ(x, t = 0) = ηρ m exp[−((x − L/4)/(L/8)) 2 ]. (11)
where η is a constant 0 < η ≤ 1. In traffic this type of perturbation can occur
where a slip road is feeding traffic onto the high-way. The cars already on the
high-way break to allow cars to move onto the road, creating a concentration of
cars close to the slip road. Use the same parameters as in exercise 1, and choose
η = 0.1. Run the simulation with NT = 30 and the same ∆t as in the previous
exercise. What is the duration in seconds of this simulation. What happens to
the density perturbation in this time? Also, examine the velocity profile. Are
these realistic, considering the traffic application?
Run the program again, but with η = 0.9, corresponding to a car density
close to maximum at x = L/4. How does the result differ from the η = 0.1 case.
What traffic situation could this solution correspond to?
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2 Presentation of results
Collect your results in a brief report where you present your figures, and comments
to the figures, answers to questions in the instructions, and a listing of
your computer program. Do not hand in more then about 10 figures, but try to
chose figures that illustrate interesting features.
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