34 5. IMPLICIT INTEGRATION
5.3 STABILITY ANALYSIS
We can investigate the stability properties of this implicit Euler integration by looking at the
same test equation, as discussed in Chapter 3. Analog to before, we discretize the continuous
time equation. is time, let’s look at the behavior of the backward Euler scheme. e discretiza-
tion of the test equation is given by
y
kC1
D y
k
C hf .t
kC1
; y
kC1
/
D y
k
C hy
kC1
(5.6)
or after grouping terms, we find
.
1 h
/
y
kC1
D y
k
:
(5.7)
e next time step is then computed as
y
kC1
D
1
.
1 h
/
y
k
:
(5.8)
Just like before, induction brings us to the following expression:
y
k
D
1
.
1 h
/
k
y
0
:
(5.9)
We assumed that the exact solution of the equations we are solving for will be bounded
when time goes to infinity. is was expressed using the condition that Re.h/ is non-positive.
e time step h is always positive so this is equivalent to Re./ being non-positive. e require-
ment for the discretized solution to be bounded is
ˇ
ˇ
ˇ
ˇ
1
1 h
ˇ
ˇ
ˇ
ˇ
< 1; Re./ < 0: (5.10)
e remarkable thing is that this condition is satisfied for any positive time step h and
real . e implicit Euler method will be unconditionally stable. e stability of the results
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