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# E an evaluation of f 0 example 571 a method that uses

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Unformatted text preview: q. Thus, the predictor satis es (0) yn = y(tn) + O(hp+1) 47 and the corrector satis es yn = y(tn) + O(hq+1): The th iteration of the corrector satis es ( yn ) = y(tn) + O(hp+1+ ) + O(hq+1): Thus, each corrector iteration decreases the e ect of the predicted solution by one until the order of the corrector formula is reached. Since corrector iterations should be minimized, the order of the predictor is typically chosen as either q or q ; 1. When the predictor and corrector formulas both have the same order q, the di erence between them can be used to estimate the local error. Neglecting all errors prior to tn and assuming the solution of the ODE to be su ciently di erentiable, the local error of the predicted and corrected solutions satisfy (0) y(tn) ; yn = Cq hq+1y(q+1)(tn ) + O(hq+2) and ( y(tn) ; yn ) = Cq hq+1y(q+1) (tn) + O(hq+2): The constants Cq and Cq are known error coe cients of the predictor and corrector, respectively. Subtracting the above two equations to eliminate the exact solution y(tn) yields ( (0) (Cq ; Cq )hq+1 y(q+1)(tn) = yn ) ; yn + O(hq+2): Thus, for example, the local error of the corrected solution may be estimated as ( Cq hq+1ynq+1) Cq (y( ) ; y(0)): n Cq ; Cq n (5.7.4) Example 5.7.2. Consider the use of the fourth-order Adams-Bashforth method (5.3.13), h (0) yn = yn;1 + 24 (55fn;1 ; 59fn;2 + 37fn;3 ; 9fn;4) dn = 251 h5yv (tn) + O(h6): 720 as a predictor and the fourth-order Adams-Moulton method (5.4.8), h( ( yn ) = yn;1 + 24 (9fn ;1) + 19fn;1 ; 5fn;2 + fn;3) 48 19 dn = ; 720 h5 yv (tn) + O(h6): With p = q = 4, C4 = 251=720 and C4 = ;19=720. Using (5.7.4), the local error of the Adams-Moulton corrector may be estimated as 19 ( (0) ( y(tn) ; yn ) ; 270 (yn ) ; yn ): Error estimates obtained by (5.7.4) are used to control step sizes so that speci ed local error tolerances are satis ed. Chosen step sizes must also satisfy appropriate (relative or absolute) stability conditions and ensure the convergence of the iterative scheme. There is a second method of estimating the local errors of LM...
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