Unformatted text preview: n h for any xed
^
such that j i(h )j < 1, i = 1 2 : : : k, for h < h. These arguments suggest that we can
study the stability characteristics of the LMM by examining the roots of
( )= k
X
i=1 32 i i =0 (5.6.10) which is the solution of (5.6.7a) when h = 0.
If the LMM is consistent then (5.6.7d) implies that unity is a root of (5.6.10). From
(5.6.9) we see that this corresponds to the exact solution of the test equation, i.e.,
1 (0) = 1: (5.6.11) We have noted that roots i(0), i = 2 3 : : : k, that are less than unity will remain so for
su ciently small values of h . Thus, the limiting stability (as h ! 0) is determined by
those roots that are greater than or equal to unity in modulus. Any root having larger
than unit modulus will produce unbounded solutions and must by unstable according
to De nition 5.6.6. It, therefore, remains to investigate the behavior of those roots that
have unit modulus when h = 0. Let i be such a root and let us initially assume that it
is simple. Supposing that jh j 1, let us expand i(h ) as the series in powers of h
i (h ) = i(0) + ih + O((h )2): (5.6.12a) Substituting (5.6.12a) into (5.6.7a)
( i(0) + ih + O((h )2)) ; h ( i(0) + ih + O((h )2)) = 0:
Expanding and
( i(0)) + h i 0( i(0)) ; ( i(0))] + O((h )2) = 0: Now, ( i(0)) = 0 according to (5.6.10). Thus, to satisfy (5.6.7a) for su ciently small
h , the coe cient of h in the above expression must vanish. This yields
( i(0)) :
(5.6.12b)
i= 0
( i(0))
Remark 1. The principal root 1 (0) = 1. This, with the consistency conditions
(5.6.7d) further implies
(1) = 1:
(5.6.12c)
1= 0
(1)
The series expansion suggests the existence of a constant c such that j i(h )j j i(0)j(1 + ch )
33 ^
h < h: Taking the nth power and recalling that j i(0)j = 1 yields j in(h )j j(1 + ch )nj jech nj jec T j C 0tT ^
h < h: Therefore, the parasitic solutions only grow by a bounded amount for nite times when
roots of unit magnitude are simple. Hence, the LMM is stable (but not absolutely stable)
for the test equation in this case.
It rem...
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 Spring '14
 JosephE.Flaherty
 Numerical Analysis, The Land, yn, Tn, Numerical ordinary differential equations

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