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# This can be done using q x jq qf r n1 1 j fn1j 538a j

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Unformatted text preview: X j fn;j (5.3.8c) j;1 : (5.3.8d) j =1 i The formula (5.3.8d) now has the form of (5.1.2a). The parameters j of (5.3.6a) and j of (5.3.8d) are independent of the problem and can be evaluated in advance of the computation. Thus, using (5.3.6a) 0 = Z 1 0 ; 0 16 d =1 Z ;; 1 Z d =1 2 0 Z1 0 Z 1 ; (; ; 1) 5 ; d= d = 12 : 2= 2 2 0 0 Continuing in this manner and using the results in (5.3.7a) yields 5 95 yn = yn;1 + h(1 + 1 r + 12 r2 + 3 r3 + 251 r4 + 288 r5 + : : : + 2 8 720 1 = 1 d= 1 k;1r k ; 1 )f n;1 : (5.3.9a) Additional results, given in Hairer et al. 12], Section III.1, are reproduced in Table 5.3.1. j0123 4 5 6 7 8 1 5 3 251 95 19087 5257 1070017 j 1 2 12 8 720 288 60480 17280 3628800 Table 5.3.1: Coe cients j for Adams-Bashforth methods ( 12], Section III.1). You may recall the second mean value theorem for integrals which states that if p( ) 2 C 0 a b] and q( ) is integrable and does not change sign on a b] then Z b a p( )q( )d = p( ) Examining (5.3.4b), reveals that ; Z b a q( )d 2 (a b): : = ( + 1) : : k!( + k ; 1) k does not change sign for 2 0 1]. Thus, we can use the second mean value theorem with (5.3.7b) and write the local error in the more explicit form Z1 ; d = k hk+1y(k+1)( ) k hk+1 y (k+1) ( ) dn = (;1) 2 (tn;1 tn): (5.3.9b) k 0 Thus, the error coe cient k is also known from, e.g., Table 5.3.1. Contrast this with discretization error formulas for Runge-Kutta methods that were extremely complex. Formulas and their local discretization errors for k = 1 2 3 4, follow. k = 1 : Euler's method yn = yn;1 + hfn;1 n = h y00 ( ) 2 17 (5.3.10a) (5.3.10b) k = 2 : Two-step Adams-Bashforth formula yn = yn;1 + h(fn;1 + rfn;1=2) or yn = yn;1 + h (3fn;1 ; fn;2) 2 5 = 12 h2y000( ): n k = 3 : Three-step Adams-Bashforth formula h yn = yn;1 + 12 (23fn;1 ; 16fn;2 + 5fn;3) n = 3 h3yiv ( ): 8 k = 4 : Four-step Adams-Bashforth formula h yn = yn;1 + 24 (55fn;1 ; 59fn;2 + 37fn;3 ; 9fn;4) n = 251 h4 yv ( ): 720 (5.3.11a) (5.3.11b) (5.3.12a) (5.3.12b) (5.3.13a) (5.3.13b) The coe cients of the Adams-Bashforth methods (5.3.8c) and the local error coe cient according to (5.3.9b) are repeated in Table 5.3.2 for k = 1 2 : : : 6 1]. k j=1 2 3 4 5 6 k 1 1 1 j 2 5 2 2j 3 -1 12 3 3 12 j 23 -16 5 8 251 4 24 j 55 -59 37 -9 720 95 5 720 j 1901 -2774 2616 -1274 251 288 6 1...
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