Environmental Engineering Reference
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Eq.(4.6) shows that the system is stable if where de-
notes the real part of a complex variable. In what follows we examine
the stability of the LSS-PC algorithm.
Let us consider the forward Euler formula first. Applying the forward
Euler formula to (4.5) gives
where
To have a bounded response for a given initial condition
when
the following condition must be met
Let
where
and
denotes the real domain, we have
In order to have a stable numerical solution, the step size must be such
that is confined within the unit circle centered at (-1,0), as shown in
Fig.4.1. It is seen that for large the step must be small enough in
order to ensure numerical stability of the forward Euler formula. The
following important conclusions are drawn with respect to the stability
of forward Euler formula:
If
the system itself is unstable, from
we have
forward Euler formula yields unstable response.
If (4.9) is violated, even though the system itself is stable, i.e.
the computed response from the forward Euler formula will be di-
verging.
In a very like manner, one can shown that the stable region of back-
ward Euler formula is given by
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