AME 521: Numerical Methods
FALL 1999, J. M. Powers. Prepared by S. V. Shepel
Problem Set 2
Due 7 October 1999
1. The differential equation describing the ice build-up (on a lake for example)
is given by
yd
2
y
dt
2
+2
dy
dt
2
+4
L
1+
y
2+
y
dy
dt
0
4
L
2+
y
=0
where
y
is the thermal resistance of the ice slab (thickness divided by conductiv-
ity),
L
is a dimensionless latent heat for which 0.75 is a reasonable value, and
t
is the nondimensional time. Appropriate initial conditions are
y
(0)=0
:
02
and
y
0
(0) = 0
:
5
. Solve the problem for all values of
t
until
y
=2
by using finite
difference schemes of first and second order.
In the case of a problem when the analytical solution does not exist, one can
use a very fine mesh and/or a higher order method to obtain a solution of high
accuracy. Obtain such a solution, and use it to verify convergence properties of
both the 1st and 2nd schemes by measuring the norm of the error. Plot the error
norm vs. discretization time interval on a logarithmic scale.
2. (a) Find the necessary and sufficient conditions on the coefficients
a
,
b
,
and
c
of the difference equation
ay
n
0
1
+
by
n
+
cy
n
+1
=0
;n
=0
;
6
1
;
6
2
;
...
so that the solutions of it are bounded. Find the explicit form of all the possible
solutions.
(b) Consider the eigenvalue problem
y
00
=
y ; y
(0) =
y
(1)=0
(1)
Find the eigenvalues
and eigenfunctions analytically.
Now consider a finite difference analog of the problem given by
1
h
2
(
y
n
0
1
0
2
y
n
+
y
n
+1
)=
y
n
;
y
0
=
y
N
=0
;Nh
=1
(2)
where
h
is the discretization interval and
N
is the number of the intervals.
Find analytically all the eigenfunctions
and the eigenvalues
of the
difference equation (2). How many eigenvalues can you find? For
N
=10
,
calculate the first 10 eigenvalues and eigenfunctions of Eq. (2) and compare them
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