Information Technology Reference
In-Depth Information
We have in previous parts of the chapter seen that the ratio t=x
2
appears fre-
quently, so we use our ˛ parameter to represent this ratio. The equation above then
reads
˛
u
i1
C
.1
C
2˛/
u
i
˛
u
iC1
D
u
`1
i
C
t f
`
i
:
(7.119)
Let us write down what (7.119) implies in the case we have five grid points
(n
D
4)onŒ0; 1 and f
D
0:
˛
u
0
C
.1
C
2˛/
u
1
˛
u
2
D
u
`1
1
;
˛
u
1
C
.1
C
2˛/
u
2
˛
u
3
D
u
`1
2
;
˛
u
2
C
.1
C
2˛/
u
3
˛
u
4
D
u
`1
3
:
We can assume Dirichlet boundary conditions, say,
u
0
D
u
4
D
0 for simplicity.
Then we have
.1
C
2˛/
u
1
˛
u
2
D
u
`1
1
;
(7.120)
˛
u
1
C
.1
C
2˛/
u
2
˛
u
3
D
u
`1
2
;
(7.121)
˛
u
2
C
.1
C
2˛/
u
3
D
u
`1
3
:
(7.122)
This is nothing but a linear system of three equations for
u
1
,
u
2
,and
u
3
. We can
write this system in matrix form,
A
u
D b
;
where
0
1
1
C
2˛
˛ 0
˛1
C
2˛
˛
0
˛1
C
2˛
@
A
;
A D
0
1
0
1
u
`1
1
u
`1
2
u
`1
3
u
1
u
2
u
3
@
A
;
@
A
:
b D
u
D
To find the new
u
i
values we need to solve such a 3
3 system of linear equations,
which can be done by Gaussian elimination, hopefully known from a linear algebra
course.
As an alternative to forming a linear system where only the
u
i
values at the inner
grid are unknowns in the system, we can include the boundary points as well. That
is, we treat
u
i
, i
D
0; 1; 2; 3; 4, as unknowns and add the boundary conditions as
equations:
u
0
D
0;
˛
u
0
C
.1
C
2˛/
u
1
˛
u
2
D
u
`1
;
1
˛
u
1
C
.1
C
2˛/
u
2
˛
u
3
D
u
`1
2
;
˛
u
2
C
.1
C
2˛/
u
3
˛
u
4
D
u
`1
3
;
u
4
D
0:
