Information Technology Reference
In-Depth Information
Exercise 8.8.
Consider the problem
v
t
C
K
v
D
v
xx
for x
2
.0; 1/; t > 0;
v
.0; t /
D
v
.1; t /
D
0
for t>0;
v
.x; 0/
D
f.x/
for x
2
.0; 1/;
where K is a constant and f is a given initial condition. Let
u
denote the solution
of (
8.1
)-(
8.3
). Show that
v
.x; t /
D
e
Kt
u
.x; t /:
˘
Exercise 8.9.
In the text above we showed that the numerical approximations
generated by the explicit scheme (7.91) satisfy the bound
j
u
i
j
max
i
max
i
j
f.x
i
/
j
for `
D
0;:::;m:
(8.96)
The purpose of the present exercise is to prove a somewhat stronger result.
We will assume throughout this exercise that the discretization parameters have
been chosen such that
t
x
2
1
2
:
˛
D
(8.97)
Let the function G
W
IR
3
!
IR be defined by
G.U
;U;U
C
/
D
˛U
C
.1
2˛/U
C
˛U
C
:
(a) Show that the approximations generated by the scheme (7.91) satisfy
u
`
C
1
i
D
G.
u
i
1
;
u
i
;
u
i
C
1
/:
(b) Prove that
@G
@U
@G
@U
@G
@U
C
0;
0;
0;
provided that the discretization parameters satisfy (8.97).
For notational purposes we introduce the symbols
u
max
and
u
min
for the maxi-
mum and minimum values, respectively, of the discrete approximations at time step
t
`
, i.e.,
u
max
D
max
i
u
i
for `
D
0;:::;m;
and
u
min
D
min
i
u
i
for `
D
0;:::;m:
(c) Apply the monotonicity property derived in (b) to prove that
u
`
C
1
i
u
max
for i
D
2;:::;n
1 and `
D
0;:::;m;
