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65 relationship between consistency convergence the

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Unformatted text preview: cult to obtain for LMMs. Theorem 5.6.4. (i) No explicit LMM is A-stable, (ii) the order of an A-stable LMM cannot exceed two, and (iii) the second-order A-stable 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 Runge-Kutta methods and arbitrarily high-order A-stable 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 left-half 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 A-stability (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 A-stable. 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 k-step LMMs of order k also exist for all k (cf. 13], Section V.2). Gear 10], Chapter 11, describes another means of relaxing A-stability 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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