Amath-Math 586/Atm S 581 Spring 2005
Homework 4 Solutions
1) The suggested nondimensional rescaling of the traffic flow equation gives:
0( , 0 < x < BL.
0
)(/)(/)(12/
TX t
QCQQ CAqUBA qAQq=+ = + − 0
)
x
The desired form can be obtained by setting 0
1/AQ
=
, B = 1/L, and C = U0B = U0/L.
2) The characteristic form of the nondimensional traffic flow equation is:
0
dq
dt
= on () 1 2
dx cq q
dt
==−. (*)
Each characteristic
Γ
τ
can be labelled by the time
τ
at which it intersects x = 0. Then (*)
implies
q is constant along
Γ
τ
: (1 2 )( )xqt
τ
=− − . (*’)
If
τ
> 0, the left boundary condition tells us the value q2
( ) 0.12sin
τ
πτ
= along
Γ
τ
. If
τ
< 0,
the initial condition implies () 0q
τ
=
. Treating
τ
as a parameter which varies from -1 to
∞
,
this together with (*’)determines the solution for all 0 ≤ x ≤ 1, t ≥ 0 and in particular at x = 1.
This is implemented and plotted as part of the Matlab script hw4p234.m (Fig. 1).
3) The Asselin-filtered leapfrog/centered-in-space method is implemented in hw4p234.m. The
method qualitatively tracks the exact solution, but a spurious overshoot is visible for t
slightly larger than 2 and an erroneous ‘shoulder’ appears for t near 2.4. Its max-norm error
is 0.0591.
4) The Lax-Wendroff (LW) and MC methods are also implemented with the specified ∆x and ∆t
in hw4p234.m The LW method also produces a spurious overshoot, but smaller than the
leapfrog method, and has a max-norm error of 0.0445. The MC method, as expected,
removes the undershoot and has the smallest max-norm error of 0.0258.
0 1 2
3
0
0.05
0.1
t
q(1,t)
Exact
Leap
LW
MC
Fig. 1. The exact and finite volume solutions to the traffic flow problem at the right boundary.