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the (explicit) Forward Euler scheme, the implicit Euler scheme, and the (implicit)
Backward Euler scheme. The two schemes can be summarized by
u
nC1
D
u
n
C
tf .
u
n
/
and
u
nC1
tf .
u
nC1
/
D
u
n
:
Here we use the standard notation with t
n
D
nt; where t
D
T=N;thatis,we
want to compute the solution from t
D
0 to t
D
T using N time steps. We note that
a possibly nonlinear equation has to be solved in order to go from time t
n
to time
t
nC1
in the implicit scheme. The numerical experiments reported on page
60
clearly
indicates that the accuracy of both schemes is O.t/: The purpose of this project is
to study a few schemes that are more accurate.
(a) Suppose
u
is a sufficiently smooth function. Use the Taylor series to show that
1
2
k
2
u
00
.t /
C
1
6
k
3
u
000
.t /
C
O.k
4
/:
u
.t
C
k/
D
u
.t /
C
k
u
0
.t /
C
(b) Set t
D
t
nC1=2
D
.n
C
1=2/t and k
D
t=2: Show that
t
2
1
2
.
t
u
0
.t
nC1=2
/
C
/
2
u
00
.t
nC1=2
/
u
.t
nC1
/
D
u
.t
nC1=2
/
C
2
C
1
6
.
t
/
3
u
000
.t
nC1=2
/
C
O.t
4
/:
2
Set k
D
t=2; to show that
t
2
1
2
.
t
u
0
.t
nC1=2
/
C
/
2
u
00
.t
nC1=2
/
u
.t
n
/
D
u
.t
nC1=2
/
2
1
6
.
t
/
3
u
000
.t
nC1=2
/
C
O.t
4
/:
2
(c) Use the two equations in (b) to show that
u
.t
nC1
/
u
.t
n
/
t
D
u
0
.t
nC1=2
/
C
O.t
2
/:
(d) Use the differential equation (
2.69
) to show that
u
.t
nC1
/
u
.t
n
/
t
D
f.
u
.t
nC1=2
//
C
O.t
2
/:
(e) Use the Taylor series to show that
1
2
.f .
u
.t
n
/
C
f.
u
.t
nC1
//
C
O.t
3
/;
f.
u
.t
nC1=2
//
D
