Information Technology Reference
In-Depth Information
The first-order derivative at x
can be approximated by a centered difference
iC
2
using the point x
iC1
to the right and the point x
i
to the left:
@
u
@x
`
u
iC1
u
i
x
:
iC
2
Similarly, the first-order derivative at x
can be approximated by a centered
i
2
difference using the point x
i
to the right and the point x
i1
to the left:
@
u
@x
`
u
i
u
i1
x
:
i
2
Combining the differences gives (
7.89
), a calculation that you should verify. The
derivation of (
7.89
) shows that we used only centered difference approximations.
Therefore, (
7.89
) is also a centered difference. Centered differences are known to
be more accurate than one-sided differences. You may thus wonder why we do not
use a centered difference for @
u
=@t. The reason is because our one-sided difference
gives a computational algorithm that is much simpler to implement than if we use a
centered difference in time.
The Finite Difference Scheme
Inserting the difference approximations (
7.88
)and(
7.89
)in(7.87) results in
u
`C1
i
D
u
i1
2
u
i
C
u
iC1
u
i
t
C
f
`
i
:
(7.90)
x
2
This is our discrete version of the PDE (7.87). Alternative versions will be derived
later.
The computational algorithm consists of computing
u
along the t
D
t
`
lines, one
line at a time. That is, we compute
u
i
values, for i
D
0;:::;n,firstfort
D
0,
which is trivial since
u
is known from the initial condition (
7.85
)att
D
0, then for
t
D
t
1
D
t , then for t
D
t
2
D
2t , and so forth. Suppose that the
u
i
values at
time level t
D
t
`
are known for all the spatial points (i
D
0;:::;n). We can then use
(7.90) to find values at the next time level t
D
t
`C1
.Wesolve(7.90) with respect to
u
`C1
i
, yielding a simple formula for the solution at the new time level:
u
i1
C
tf
`
i
C
t
x
2
u
`C1
i
D
u
i
2
u
i
C
u
iC1
:
(7.91)
We refer to (7.90)or(7.91)asa
finite difference scheme
for the PDE (7.86). At a gen-
eral point .x
i
;t
`C1
/ we can compute the function value
u
`C1
i
from the known values
u
i1
,
u
i
,and
u
iC1
. This procedure is graphically illustrated in Fig.
7.12
. Many refer
