Information Technology Reference
In-Depth Information
@t
u
.x
i
;t
`
/
u
i
u
`1
i
@
:
(7.116)
t
For the second-order derivative in space we use (
7.89
). Inserting these approxima-
tions in the PDE yields
u
i
u
`1
i
D
u
i1
2
u
i
C
u
iC1
C
f
`
i
:
(7.117)
t
x
2
We assume that
u
`1
i
is known while
u
i
is unknown, i
D
0;:::;n. Note that here
we view values at time level ` as unknown, while we sought
u
values at time level
`
C
1 in Chap.
7
. The reason is because we used a forward difference (involving
u
i
and
u
`C1
i
) in time in Sect.
7.4.1
, while we use a backward difference here (involv-
ing
u
i
and
u
`1
i
). We could equally well use the backward difference at grid point
.x
i
;t
`C1
/. That would result in the equivalent scheme
u
`C1
i
D
u
`C1
2
u
`C1
i
C
u
`C1
iC1
u
i
t
i1
C
f
`
i
;
(7.118)
x
2
where the unknown values are at time level `
C
1:
u
`C1
i1
,
u
`C1
i
,and
u
`C1
iC1
.Inthe
following we stick to (7.117).
The finite difference scheme implied by (7.117) is known as a backward Euler
scheme (where “backward” refers to the backward difference in time). Some also
call this scheme the fully implicit scheme or the implicit Euler scheme.
7.5.2
The Linear System of Equations
With scheme (7.90), we simply solved for the new unknown. Solving (7.117) with
respect to
u
i
gives
1
C
2
u
i
u
i1
C
t f
`
i
t
x
2
C
t
x
2
D
u
`1
i
C
u
iC1
:
The problem now is that
u
i1
and
u
iC1
are also unknown! It is impossible to derive
an explicit formula for
u
i
containing values at the previous time level `
1 only.
There are three unknown quantities:
u
i1
,
u
i
,and
u
iC1
. These three quantities are
coupled in scheme (7.117). A fruitful reordering of (7.117) involves collecting all
the unknown quantities on the left-hand side and all the known quantities on the
right-hand side:
t
x
2
t
x
2
u
i1
C
.1
C
2˛/
u
i
u
iC1
D
u
`1
i
C
t f
`
i
:
