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In-Depth Information
F
1
D
f.t
n
;
u
n
/;
D
f.t
nC1=2
;
u
n
C
t
2
F
2
F
1
/;
D
f.t
nC1=2
;
u
n
C
t
2
F
3
F
2
/;
F
4
D
f.t
nC1
;
u
n
C
tF
3
/;
t
6
u
nC1
D
u
n
C
ŒF
1
C
2F
2
C
2F
3
C
F
4
:
Implement this scheme and perform the experiments of (g) and (h) above. Try,
using experiments, to figure out the accuracy of this scheme.
(m) Consider the initial value problem
2.1
t/
"
2
u
0
.t /
D
u
u
.0/
D
e
1="
2
;
where " is given. Show that the solution of this problem is given by
/
2
:
u
.t /
D
e
.
1t
"
(n) Set "
D
1=4: We want to compute the solution at time t
D
1:1: Compare
the accuracy of all the schemes above (explicit Euler, implicit Euler, Crank-
Nicolson, Heun, and Runge-Kutta) for this problem. Say we want the absolute
error to be less than 10
5
: Make a ranking of the schemes based on how much
CPU time they need to achieve this accuracy.
