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u
1
D
u
n
D
0
for `
D
1;:::;m;
u
i
D
f.x
i
/
for i
D
1;:::;n:
Write a computer program that implements this scheme.
(b) Recapture the argument leading to the bound (8.88). Prove that the numerical
approximations generated by (8.115) satisfy
j
u
i
j
max
i
max
i
j
f.x
i
/
j
for `
D
0;:::;m;
provided that
1
2M
x
2
:
t
(8.116)
(c) Our goal now is to investigate whether or not the energy bound (8.7), valid for
the model problem (
8.1
)-(
8.3
), holds in the present case. Let
k.x/
D
1
C
x;
f.x/
D
x.1
x/;
and T
D
1. For several choices of the discretization parameters x and t ,
use the scheme (8.115) to compute approximate solutions of the model prob-
lem (
8.108
)-(
8.110
). Make sure that the condition (8.116) is satisfied. In each
experiment you should compute
x
1
2
.
u
n
/
2
!
2
.
u
1
/
2
C
n
X
i
D
2
/
2
C
1
.
u
i
(8.117)
and
x
1
!
:
2
.f .x
1
//
2
C
n
X
i
D
2
1
2
.f .x
n
//
2
.f .x
i
//
2
C
(8.118)
Explain why we can use (8.117)and(8.118) to test experimentally whether or
not an inequality of the form (8.7) seems to be valid in the present case. What
are your experiments indicating?
(d) Modify the argument leading to (8.7) to the present problem and show that
Z
1
u
2
.x; t /
dx
Z
1
0
u
2
.x; 0/
dx
D
Z
1
0
f
2
.x/
dx
for t
0:
0
(e) In Sect.
8.1.1
we established a bound for the first order derivative of the solution
of the heat equation (
8.1
)-(
8.3
), cf. inequality (8.11). Is a similar property valid
for the problem (
8.108
)-(
8.110
)? More precisely, we will analyze whether the
bound
