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where N
t
D
5 for t
D
1=100, 1=200, 1=300, 1=400,and1=500. Compute
also
e
t
t
and show that
e
t
ct;
where c is a constant independent of t .
(j) Derive the Crank-Nicolson scheme for system (
3.108
). Show that the system
can be written in the form
1
t
2
=4
u
n
t
v
n
;
1
1C
t
2
4
u
nC1
D
t
u
n
C
1
t
2
=4
v
n
:
(3.111)
1
1C
t
2
4
v
nC1
D
(k) Implement scheme (
3.111
) and redo the computations in (i) using this scheme.
Compute
e
t
t
2
,wheree
t
isgivenby(
3.110
), and conclude that
dt
2
;
e
t
where d is independent of t .
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