Unformatted text preview: cult to obtain for LMMs. Theorem 5.6.4. (i) No explicit LMM is Astable, (ii) the order of an Astable LMM cannot exceed two, and (iii) the secondorder Astable LMM with the smallest error coe cient is the trapezoidal rule.
Proof. cf. Dahlquist 8]. A di erent proof of (ii) and (iii) appears in Wanner et al.
17]. This a major negative result of numerical ODEs. No such restriction exists for implicit
RungeKutta methods and arbitrarily highorder Astable methods may be constructed.
In order to obtain high orders of accuracy with LMMs, we will have to relax our
stability demands and give up a portion of the lefthalf of the h plane. De nition 5.6.13. A numerical method is A( )stable,
absolute stability contains the in nite wedge 2 0 =2) if its region of W = fh j ; < ; arg(h ) < g:
42 Im(h λ
) Im(h λ
)
hλ α Re(h λ
) Re(h λ
) Figure 5.6.6: Regions of Astability (left) and A( )stability (right).
The notion of A( )stability was introduced by Widlund 18] and is discussed in Gear
10], Chapter 11 and Hairer and Wanner 13], Section V.2. An example of the wedge W
is shown in Figure 5.6.6. A(0+)stable methods have regions of absolute stability that
contain the negative real axis. A( =2)stable methods are Astable. For a given with
Re( ) < 0, the point h either lies inside W for all positive h or outside it for all positive
h. If we know that all of the eigenvalues of a sti system lie inside of a certain wedge
W , then an A( )stable method can be used without any stability restriction on h.
Widlund 18] showed that A( )stable LMMs of orders one through four exist with
arbitrarily close to =2. Grigorie and Schroll 11] showed that kstep LMMs of order k
also exist for all k (cf. 13], Section V.2).
Gear 10], Chapter 11, describes another means of relaxing Astability by introducing
the notion of sti stability, which combines both stability and accuracy. De nition 5.6.14. A numerical method is sti y stable if it is absolutely stable in the
region R1 = fh j Re(h ) DL g and accurate in the region R2 = fh j DL &...
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 Spring '14
 JosephE.Flaherty
 Numerical Analysis, The Land, yn, Tn, Numerical ordinary differential equations

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