77b 2 01 0 h dpdtt dpd2 ryn 2 2 1r2yn 00 setting

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Unformatted text preview: ess pressure on storage, we may store an extra derivative. Considering (5.7.5b) once again, let us approximate the solution derivative more accurately using a third-degree polynomial. Thus, using (5.7.11) with l = k = 4 01 0 0 0 P3 = yn + ( ; 1)ryn + 1 ( ; 1) r2 yn + 6 ( ; 1) ( + 1)r3yn: 2 53 Di erentiating while using (5.7.7b) 3 0 0 0 h dPdt(t) = dPd3( ) = ryn + 1 (2 ; 1)r2yn + 1 (3 2 ; 1)r3yn: 2 6 Setting = 1 00 0 01 0 hyn = ryn + 1 r2 yn + 3 r3yn: 2 Di erentiating again 2 2 P P 0 0 h2 d dt32(t) = d d 3 ( ) = r2 yn + r3 yn: 2 Setting = 1 000 0 0 h2yn = r2 yn + r3 yn: Di erentiating once again iv 0 h3yn = r3 yn: Thus, enlarging the Nordsieck vector by one, we have 2 32 yn 1 0 6 hyn 7 6 0 6 2 00 7 6 6 h yn=2 7 = 6 0 6 3 000 7 6 4 h yn =3! 5 4 0 iv h4 yn =4! 0 32 000 0 100 0 76 76 0 1=2 1=4 1=6 7 6 76 0 0 1=6 1=6 5 4 0 0 0 1=24 yn 0 hyn 0 rhyn 0 r2 hyn 3 0 r hyn 3 7 7 7: 7 5 We've already seen that the local error of a k th-order multistep method has the form dn = Ck hk+1y(k+1)(tn) + O(hk+2): (5.7.12a) The Nordsieck vector (5.7.10) contains approximations of yn and its rst k ; 1 derivatives. However, by enlarging it to store the k th derivative (Example 5.7.5), we can construct an error estimate. Thus, let 2 k 0 00 ( an = yn hyn h yn : : : h ! ynk)]T 2! k and take a backward di erence at two consecutive time steps to obtain ran = an ; an;1 : 54 (5.7.12b) The last component of this vector ran k+1 is an approximation of hk+1y(k+1)(tn)=k!. The local error can, thus, be approximated as dn Ck k!ran k+1: (5.7.12c) Example 5.7.6. The local discretization error of the fourth-order Adams-Moulton method (5.7.5b) is given by (5.4.8b) as 19 dn = ; 720 h5 yv (tn) + O(h5): Di erencing the last component of the Nordsieck vector of Example 5.7.5 at tn and tn;1 gives h4 iv iv h5 iv iv h5 ran 5 = 24 (yn ; yn;1) = 24 yn ; yn;1 24 yv (tn): h Substituting into the discretization error formula dn ; 19 ran 5: 30 Suppose we seek to keep jdnj , then, either upon completion of a successful step or upon failure of a step, we c...
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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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