Biomedical Engineering Reference
In-Depth Information
Now, as before, if
u
n
is a perturbation of
˜
u
n
, this perturbation satisfies
ˆ
u
n
+
1
−
u
n
t
+
θλ
˜
u
n
+
1
+
(1
−
θ
)
λ
˜
u
n
=
0.
(15.17)
Clearly, the perturbation at
t
=
t
n
+
1
can be expressed as
−
−
θ
λ
t
1
(1
)
u
n
+
1
=
u
n
.
(15.18)
1
+
θλ
t
A
The factor
A
is called the amplification factor. To have a stable integration scheme
the magnitude of
u
n
, i.e. the pertur-
bation should not grow as time proceeds. Hence, stability requires
u
n
+
1
should be smaller than the magnitude of
˜
˜
|˜
u
n
+
1
|≤|˜
u
n
|
,
which holds if the amplification factor
1.
Fig.
15.2
shows the amplification factor
A
as a function of
|
A
|≤
λ
t
with
θ
as a
parameter. For 0
0.5 the integration scheme is conditionally stable,
meaning that the time step
t
has to be chosen sufficiently small related to
λ
.
In the multi-variable case the above corresponds to the requirement that, in case
0
≤
θ<
0.5,
λ
t
should be small compared to the eigenvalues of the matrix
A
.
For 0.5
≤
θ
≤
1 the scheme is unconditionally stable, hence for any choice of
t
a stable integration process results.
≤
θ<
1.5
1
0.5
θ
=
0.75
A
0
θ
=
0.5
−0.5
θ
=
0.25
−1
θ
=
0
−1.5
0
1
2
3
4
5
λ
Δ
t
Figure 15.2
Amplification factor
A
as a function of
λt
for various values of
θ
.
