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Unformatted text preview: ains to examine those roots of (5.6.10) that have unit magnitude but are not
simple. Suppose that i(0) is such a root with multiplicity , then, as indicated in (5.6.8b),
the solution yn would contain the terms n in, n2 in, : : : , n ;1 in. With j i(0)j = 1, these
terms are unbounded as n ! 1. Hence, the LMM cannot be stable.
The preceding arguments suggest that the stability of a LMM is linked to the following
condition. De nition 5.6.7. A LMM satis es the root condition (or the condition of zero stability if the roots of ( ) = 0 are inside the unit circle or simple on the unit circle in the complex
-plane. Theorem 5.6.1. Satisfaction of the root condition is necessary and su cient for the
stability of a LMM. Proof. The key aspect of the proof have been presented. A more thorough analysis is
given in Gear 10], Section 10.1.
Example 5.6.2. The leap frog method of Example 5.1.1 has
2 = ;1 with k = 2 thus, using (5.6.7b) 0 =1 1 = 0, and ( ) = 2 ; 1:
The roots of this equation are 1(0) = 1 and 2(0) = ;1. Both roots have unit modulus
and are simple on the unit circle in the complex plane hence, the leap frog scheme
satis es the root condition and is stable.
While the leap frog scheme is stable, it is clear from Example 5.1.1 that solutions
grow rapidly when they should be decaying. Clearly a ner distinction is needed and this
prompts the concepts of strong and weak stability.
34 De nition 5.6.8. A LMM is weakly stable if it is stable, but has more than one root of
( ) = 0 on the unit circle in the complex -plane. De nition 5.6.9. A LMM is strongly stable if all roots of ( ) = 0 are inside the unit circle in the -plane except for the principal root 1 (0) = 1. Example 5.6.3. The results of Example 5.6.2 con rm that the leap frog scheme is
weakly stable. Parasitic solutions of a weakly stable scheme may either grow or decay for
nonzero values of h as n increases, depending on whether they move outside or inside
of the unit circle. In Example 5.6.1, we saw that the parasitic solution of the leap f...
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