Information Technology Reference
In-Depth Information
7.4.2
Incorporating Dirichlet Boundary Conditions
The observant reader will notice that it is difficult to use (7.91) for computing new
values
u
`C1
1
and
u
`C1
n
at the boundary, because (7.91)fori
D
1 and i
D
n involves
values
u
`
1
and
u
nC1
outside
the grid. The solution is to use the boundary conditions
(
7.83
)and(
7.84
)fori
D
0 and i
D
n. These must be combined with scheme (7.91).
Suppose we have the boundary conditions
u
.0; t /
D
0 and
u
.1; t /
D
0,previ-
ously categorized as Dirichlet conditions. We can then use (7.91) to update
u
`C1
i
at
all
internal
grid points, i
D
1;:::;n
1.Fori
D
1,(7.91) involves the known values
u
0
,
u
1
,and
u
2
,and
u
0
is zero from the boundary condition. Similarly, for i
D
n
1,
(7.91) involves the known values
u
n2
,
u
n1
, and the boundary condition
u
n
D
0.
The
computational
procedure
can
be
summarized
as
we
have
done
in
Algorithm
7.1
.
Let us generalize the simple boundary conditions
u
.0; t /
D
u
.1; t /
D
0 a bit to
the choices
u
.0; t /
D
D
0
.t /;
u
.1; t /
D
D
1
.t /;
(7.92)
where D
0
.t / and D
1
.t / are prescribed functions. The computational procedure is
hardly affected by this slight generalization. We use scheme (7.91) for the internal
points i
D
2;:::;n
1, and at the boundary points we just insert the correct values:
u
`C1
0
u
`C1
n
D
D
0
.t
`C1
/;
D
D
1
.t
`C1
/:
Algorithm
7.2
incorporates the non-homogeneous Dirichlet conditions (7.92).
7.4.3
Incorporating Neumann Boundary Conditions
Implementing Neumann boundary conditions, such as (
7.84
),
@
@x
u
.1; t /
D
N
1
.t /;
Algorithm 7.1
Diffusion Equation with u
D
0
at the Boundary.
˛
t =x
2
SET INITIAL CONDITIONS
:
u
i
D
I.x
i
/; for i
D
0;:::;n
for `
D
0; 1; : : : ; m
1
UPDATE ALL INNER POINTS
:
for i
D
D
1;:::;n
1
˛
u
i
1
2
u
i
C
u
i
C
1
u
`
C
1
i
D
u
i
C
C
tf
`
i
INSERT BOUNDARY CONDITIONS
:
u
`
C
1
0
u
`
C
1
n
D
0;
D
0
