MATH 572
Numerical Methods for Scientific Computing II
Winter 2005
Assignment #4
due : Thursday , March 10
1. Consider the 2step BDF scheme,
∇
u
n
+
1
2
∇
2
u
n
=
hf
(
u
n
).
a) Find the characteristic roots
ζ
1
(
h
)
, ζ
2
(
h
) for the test equation and plot them using
Matlab over the interval

10
≤
hλ
≤
0.
b) Show analytically that the negative real axis is contained in the region of absolute
stability. (Note: the scheme is actually Astable, but it is not required to show that here.)
2. The Lorenz system
is defined by
y
1
y
2
y
3
=
σ
(
y
2

y
1
)
ry
1

y
2

y
1
y
3
y
1
y
2

by
3
.
These equations were originally derived as a model for thermal convection; the variables
represent the temperature, density, and velocity in a certain fluid flow. It was discovered by
numerical computations that the parameters
σ
= 10
, b
= 8
/
3
, r
= 28 yield a system with
chaotic dynamics. Solve this system with initial conditions
y
1
(0) = 0
, y
2
(0) = 1
, y
3
(0) = 0
up to time
t
= 100 using Matlab’s
ode45
solver. (Part of the exercise is to read the online
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 Fall '08
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 Math, Numerical Analysis, heat equation vt

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