Information Technology Reference
In-Depth Information
Algorithm 7.7
The
Scheme for the Diffusion Equation with u
.0; t /
D
D
0
.t /
and u
.1; t /
D
D
1
.t /
.
SET INITIAL CONDITION
:
u
i
D
I.x
i
/;
for
i
D
0;:::;n
t =x
2
˛
D
t
0
D
0
COMPUTE CONSTANT COEFFICIENT MATRIX
:
A
i;j
D
0 for i; j
D
0; 1; : : : ; n
A
0;0
D
1
A
n;n
D
1
for
i
D
1;:::;n
1
A
i;i
1
D
˛
,
A
i;i
C
1
D
˛
2˛
FACTORIZE MATRIX
:
c
,
d
=
factorize
(
p
,
q
,
r
)
for `
D
1; 1; : : : ; m
t
`
D
t
`
1
C
t
FILL RIGHT
-
HAND SIDE
:
b
0
D
A
i;i
D
1
C
D
0
.t
`
/
b
n
D
D
1
.t
`
/
D
for i
1;:::;n
1
/
u
`
1
i
//
u
`
1
i
/
u
`
1
i
C
1
b
i
D
˛.1
C
.1
2˛.1
C
˛.1
C
1
t .f
`
i
/f
`
1
i
C
.1
/
SOLVE LINEAR SYSTEM
:
u=
backsubstitution
(
b
,
r
,
c
,
d
)
u
i
u
Looking at this scheme, we realize that if we set
D
0, we recover the explicit
forward Euler scheme.
19
Setting
D
1, we get the implicit backward Euler scheme,
and setting
D
1=2, we arrive at the Crank-Nicolson scheme. Therefore, in a
program it can be wise to implement the scheme and then just choose a value to
recover one of the three specialized schemes.
The value
D
0 is a special case, since then we have a diagonal matrix system
and there is hence no need to invoke a factorization and back-substitution procedure.
In programs we might use the scheme for all implicit schemes (
¤
0) and offer
a separate and simpler implementation for the explicit scheme (
D
0).
Finally, we summarize our numerical methods for the diffusion PDE by listing
the most general computational procedure: Algorithm
7.7
, expressing the scheme.
19
Scheme (
7.147
) with
D
0 equals (7.90) if in the latter we exchange ` with `
1 and `
C
1
with `.
