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scheme grew when h < 0 and decayed when h > 0. All parasitic solutions of a strongly
stable method decay for su ciently small values of h as n increases, since there modulus
is less than unity when h = 0.
Example 5.6.4. Let us examine the stability of the Adams-Bashforth and AdamsMoulton methods as h ! 0. Recall (Sections 5.3 and 5.4) that the Adams' methods have
the form (5.1.2) with 0 = 1, 1 = ;1, and i = 0, i = 2 3 : : : k, i.e., yn = yn;1 + h k
i=0 i fn;i : The coe cient 0 = 0 for the (explicit) Adams-Bashforth methods and is nonzero for the
(implicit) Adams-Moulton methods.
Using the test equation (5.6.4) yn = yn;1 + h
Assuming that yn is proportional to n k
i=0 i yn;i: leads to (5.6.7) which, in this case, is ( );h ( )=0
( ) = k;1( ; 1) ( )= k
i=0 i k ;i : Thus, the roots of the rst characteristic polynomial ( ( ) = 0) are
1 =1 2 = 3 35 = ::: = k = 0: The parasitic roots i(0), i = 2 3 : : : k, are all zero for both the Adams-Bashforth and
Adams-Moulton methods thus, these methods are strongly stable.
Computation is performed at nite values of h. The root condition and the notions
of strong and weak stability all involve reasoning in the limit of vanishingly small step
sizes. This shortcoming encourages us to further extend concepts of stability. De nition 5.6.10. A LMM is absolutely stable for those values of h where the roots
of (5.6.7a) satisfy j (h )j 1 when applied to the test problem (5.6.4).
De nition 5.6.11. A LMM is relatively stable for those values of h where the parasitic
roots of (5.6.7) are less in magnitude than the principal root. The boundary of the region of absolute stability of a LMM may be calculated by the
boundary locus method as described in Chapter 3 for one-step methods. Complex values
of having unit modulus are written in polar form
and (5.6.7a) is imposed. The boundary of the region of absolute stability follows by
2 0 ]:
h = (e i )
(e ) The regions of absolute stability of the Adams-Bashforth, Adams-Moulton, and backward di ere...
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