Lecture 5

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Unformatted text preview: 440 j 4277 -7923 9982 -7298 2877 -475 19087 60480 Table 5.3.2: Coe cients of the Adamas-Bashforth method (5.3.8c) and the local error (5.3.9b) for orders one through six 1]. Problems 18 1. Show that (5.3.8b) can be written in the form yn = yn;1 + h k;1 X j =0 (;1)j fn;1;j k;1 X i=j i i j and that this leads to (5.3.8c). 5.4 Implicit Methods: Adams-Moulton Methods Implicit Adams formulas, called Adams-Moulton methods, are derived in the same manner as the explicit formulas of Section 5.3. In this case, we approximate f by a k ; 1 st degree interpolating polynomial Pk;1(t) that satis es Pk;1(tn;i) = f (tn;i y(tn;i)) i = 0 1 : : : k ; 1: (5.4.1) A formula for Pk;1(t) and its error follow from (5.3.3a) and (5.3.3b) upon replacement of n ; 1 by n. This yields i;1 k;1 X rifn Y i!hi j=0(t ; tn;j ) i=0 Pk;1(t) = Y f ( k)( y( )) k;1(t ; t ): Ek;1(t) = n;j k! j =0 (5.4.2a) (5.4.2b) Again letting = (t ; tn;1)=h and substituting (5.4.2) into (5.3.2), we nd y(tn) = y(tn;1) + h h Letting Z 1 0 Z 1X k;1 0 (;1)k hk i = Z 1 0 (;1)i ; i+ 1 rifnd + i=0 ; + 1 y(k+1)( )d : k (;1)i ; i+ 1 d : we nd the Adams-Moulton formula as yn = yn;1 + h 19 k;1 X i=0 i i r fn : (5.4.3a) (5.4.3b) y y(t) f n-k+1 t n-k t n-2 yn fn yn-1 f n-1 f n-2 t n-1 tn t Figure 5.4.1: Information needed for a k-value Adams-Moulton method. The local error is, as usual, a bit more complex with implicit formulas. We'll skirt this by using the local discretization error k;1 y(tn) ; y(tn;1) ; X rif n= n i h i=0 or = (;1)k hk Z 1 0 ; + 1 y(k+1)( )d = hk y(k+1)( ) k k 2 (tn;1 tn): (5.4.3c) As shown in Figure 5.4.1, the solution yn;1 and function values fn, fn;1, : : : , fn;k+1 are needed to compute yn. Unlike the Adams-Bashforth formula, the index k does not indicate the number of steps of the method. Methods with k = 1 and 2 are one-step methods and those with k 3 are k ; 1-step methods. We might call the k th AdamsMoulton method a k-value method since it involves k values of f . Like a k-step AdamsBashforth methods, a k-value Adams-Moulton method is of order k (cf. (5.4.3c)). For linear problems, (5....
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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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