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where f represents a given initial temperature distribution.
For given positive integers n and m we define the discretization parameters x
and t by
1
n
1
x
D
and
t
D
T
m
;
where T represents the endpoint of the time interval Œ0; T in which we want to
discretize the model problem (
8.108
)-(
8.110
). As in the text above, the grid points
are given by
x
i
D
.i
1/x
for i
D
1;:::;n;
t
`
D
`t
for `
D
0;:::;m;
and in addition we introduce the notation
x
i
C
1=2
D
x
i
C
x=2
for i
D
1;:::;n
1:
Let
k
i
C
1=2
D
k.x
i
C
1=2
/;
u
i
u
.x
i
;t
`
/;
for i
D
1;:::;n
1 and `
D
0; 1; : : : ; m;
and consider the approximations
u
`
C
1
i
u
i
t
u
t
.x
i
;t
`
/
;
(8.111)
.k
u
x
/.x
i
C
1=2
;t
`
/
.k
u
x
/.x
i
1=2
;t
`
/
x
.k
u
x
/
x
.x
i
;t
`
/
;
(8.112)
u
i
C
1
u
i
x
.k
u
x
/.x
i
C
1=2
;t
`
/
k
i
C
1=2
;
(8.113)
u
i
u
i
1
x
.k
u
x
/.x
i
1=2
;t
`
/
k
i
1=2
:
(8.114)
(a) Use the approximations (
8.111
)-(
8.114
) to derive the scheme
D
˛k
i
1=2
u
i
1
C
1
˛.k
i
1=2
C
k
i
C
1=2
/
u
i
u
`
C
1
i
C
˛k
i
C
1=2
u
i
C
1
(8.115)
for i
D
2;:::;n
1 and `
D
0;:::;m
1. The boundary conditions (
8.109
)
and the initial condition (
8.110
) are handled by setting
