Unlike the adams bashforth formula the index k does

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Unformatted text preview: 4.3a,b) only requires one function evaluation per step however, iteration will generally be needed for nonlinear problems. Expanding (5.4.3b) by evaluating i , i = 0 1 : : : k ; 1, according to (5.4.3a), we nd 1 1 19 3 yn = yn;1 + h(1 ; 1 r ; 12 r2 ; 24 r3 ; 720 r4 ; 160 r5 + 2 20 k k;1 r ; 1)fn : :::+ (5.4.4) Additional results, given in Hairer et al. 12], Section III.1, are reproduced in Table 5.4.1. j01 2 3 4 5 6 7 8 1 1 19 3 863 275 33953 1 ; 1 ; 12 ; 24 ; 720 ; 160 ; 60480 ; 24192 ; 3628800 j 2 Table 5.4.1: Coe cients j for implicit Adams methods ( 12], Section III.1). Expanding the backward di erences in (5.4.4) using (5.2.15a,b), we nd speci c formulas for given choices of k. Those formulas for k = 1 2 3 4, and their local discretization errors follow. k = 1 : Backward Euler method yn = yn;1 + hfn (5.4.5a) = ; h y00 ( ): 2 (5.4.5b) n k = 2 : Trapezoidal rule yn = yn;1 + h(fn ; 1 rfn) 2 or yn = yn;1 + h (fn + fn;1 ) 2 n 1 = ; 12 h2 y000 ( ): k = 3 : Three-value (two-step) Adams-Moulton formula h yn = yn;1 + 12 (5fn + 8fn;1 + fn;2) n = ; 1 h3yiv ( ): 24 21 (5.4.6a) (5.4.6b) (5.4.7a) (5.4.7b) k = 4 : Four-value (three-step) Adams-Moulton formula h yn = yn;1 + 24 (9fn + 19fn;1 ; 5fn;2 + fn;3) n 19 = ; 720 h4 yv ( ): (5.4.8a) (5.4.8b) Once again, we present the coe cients of the Adams-Moulton methods yn = yn;1 + h k;1 X j =0 j fn;j (5.4.9) and their error coe cients (5.4.3c) for k = 1 2 : : : 6, in Table 5.4.2. k j=0 1 23 45 k 1 1 1 ;2 j 1 2 2j 1 1 ; 12 1 3 12 j 5 8 -1 ; 24 19 4 24 j 9 19 -5 1 ; 720 3 5 720 j 251 646 -264 106 -19 ; 160 863 6 1440 j 475 1427 -798 482 -173 27 ; 60480 Table 5.4.2: Coe cients of the Adams-Moulton method (5.4.9) and their local error coe cients (5.4.3c) for orders one through six 1]. Example 5.4.1. Comparing Tables 5.3.2 and 5.4.2, we see that the error coe cient of the Adams-Moulton method of order k is smaller than that of the Adams-Bashforth method of the same order. Thus, with comparable derivatives, the implicit methods should produce more accuracy than the explicit methods. Let's ex...
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This document was uploaded on 03/16/2014 for the course CSCI 6820 at Rensselaer Polytechnic Institute.

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