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# 313 h 0 yn yn1 24 55fn1 59fn2 37fn3 9fn4 dn

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Unformatted text preview: Ms that does not rely on the predictor and corrector formulas having the same order. The development will also illustrate the appropriate information to be saved between time steps. We'll present the results in general but illustrate the approach when the third-order Adams-Bashforth method is used as a predictor and when the fourth-order Adams-Moulton method is as a corrector. Thus, from (5.3.12a) and (5.4.8a), we have h (0) yn = yn;1 + 12 (23fn;1 ; 16fn;2 + 5fn;3) (5.7.5a) h( ( yn ) = yn;1 + 24 (9fn ;1) + 19fn;1 ; 5fn;2 + fn;3) Let us write these methods in a vector form by introducing 2 3 2 yn;1 yn 0 0 6 hyn;1 7 6 hyn yn;1 = 6 hy0 7 yn = 6 hy0 4 n;2 5 4 n;1 0 0 hyn;3 hyn;2 3 7 7 5 (5.7.5b) (5.7.6a) 0 where yn = fn, etc. The vector yn;1 contains the information needed to calculate yn and yn contains the information that is transferred to the next step. With this notation, the predicted solution (5.7.5a) is or 2 (0) yn 0 6 hyn(0) 60 4 hyn;1 0 hyn;2 3 2 23 16 5 1 ; 7 6 0 12 ;12 12 7=6 3 3 1 5 40 1 0 0 00 1 (0) yn = Byn;1: 49 32 yn;1 7 6 hyn;1 7 6 00 5 4 hyn;2 0 0 hyn;3 3 7 7 5 (5.7.6b) (5.7.6c) The rst row of (5.7.6b) is just the predictor formula (5.7.5a). The third and fourth rows are obvious identities. The second row can be developed from the Newton interpolation formula (5.3.5a), which we repeat here for convenience, Pk;1(t) = k;1 X (;1)i ; i i=0 0 riyn;1 (5.7.7a) where = t ; tn;1 (5.7.7b) h and k = 3 for the present example. Recall that Pk;1(t) provides an approximation of y0(t) = f (t y(t)) on the interval tn;k tn] thus, setting = 1 (t = tn ) in (5.7.7a) yields 0 yn(0) = Pk;1(tn) = But (cf. (5.3.4c,d)) thus, k ;1 X 0 (;1)i ;1 riyn;1: i i=0 (;1)i ;1 i 0 (0) yn = Setting k = 3 for the example at hand k;1 X i=0 =1 0 riyn;1: 0 0 0 0 0 0 0 yn(0) = yn;1 + (yn;1 ; yn;2) + (yn;1 ; 2yn;2 + yn;3) or 0 0 0 0 yn(0) = 3yn;1 ; 3yn;2 + yn;3: In a similar manner, we write the corrector as ( ( ( yn ) = yn ;1) + cg(yn ;1)) = 1 2 ::: : (5.7.8a) ( The vector c and the scalar function g(yn ;1)) must be determined so that (5.7.8) agrees with the origina...
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