Information Technology Reference
In-Depth Information
Algorithm 7.2
Diffusion Equation with u
.0; t /
D
D
0
.t /
and u
.1; t /
D
D
1
.t /
.
˛
D
t =x
2
SET INITIAL CONDITION
:
u
i
D
I.x
i
/;
for i
D
0;:::;n
t
0
D
0
for `
D
0; 1; : : : ; m
1
t
`
C
1
D
t
UPDATE ALL INNER POINTS
:
for i
t
`
C
D
1;:::;n
1
˛
u
`
u
i
C
1
u
`
C
1
i
u
i
C
2
u
i
C
tf
`
i
D
C
INSERT BOUNDARY CONDITIONS
:
u
`
C
1
0
D
u
`
C
1
n
D
D
0
.t
`
C
1
/;
D
1
.t
`
C
1
/
where N
1
is known, is more difficult than implementing the conditions where
u
is
known (Dirichlet conditions). The recipe goes as follows. First, we discretize the
derivative at the boundary, i.e., replace @
u
=@x by a finite difference. Since we use
a centered spatial difference in the PDE, we should use a centered difference at the
boundary too:
@x
u
.1; t
`
/
u
nC1
u
n1
2x
@
:
(7.93)
The discrete version of the boundary condition then reads
u
nC1
u
n1
2x
D
N
1
.t
`
/:
(7.94)
Unfortunately, this formula involves a
u
value outside the grid,
u
nC1
.However,if
we use scheme (7.91) at the boundary point x
D
x
n
,
u
n1
C
tf
n
C
t
x
2
u
`C1
n
D
u
n
2
u
n
C
u
nC1
;
(7.95)
we have two equations, both involving the fictitious value
u
nC1
. This enables us to
eliminate this value. Solving (7.94) with respect to
u
nC1
,
u
nC1
D
u
n1
C
2N
1
x :
Inserting this expression in (7.95) yields a modified scheme at the boundary:
u
n1
x
C
tf
n
t
x
2
u
`C1
n
D
u
n
u
n
C
N
1
C
2
:
(7.96)
