Lecture 3

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Unformatted text preview: 2 z(tn 1 ) 0)j = O(h): ; (3.4.8d) Substituting (3.4.8b,c,d) into (3.4.8a) jenj jen 1j + h Ljen 1 j + (h) + (h)]: ; ; (3.4.9) Equation (3.4.9) is a rst order di erence inequality with constant (independent of n) coe cients having the general form jenj Ajen 1j + B (3.4.10a) A = 1 + hL (3.4.10b) B = h (h) + (h)]: (3.4.10c) ; where, in this case, The solution of (3.4.10a) is n; jenj Anje0j + A ; 11 B A n 0: Since e0 = 0, we have (1 + hL)n ; 1 h (h) + (h)] hL jenj or, using (2.1.10) eLT ; 1 (h) + (h)]: L Both (h) and (h) approach zero as h ! 0 therefore, jenj h 0 ! n lim Nh=T zn = z(tn ): !1 Thus, zn converges to z(tn ), where z(t) is the solution of (3.4.4). If the one-step method satis es the consistency condition (3.4.3), then z(t) = y(t). Thus, yn converges to y(tn), n 0. This establishes su ciency of the consistency condition for convergence. 41 In order to show that consistency is necessary for convergence, assume that the onestep method (3.4.1a) converges to the solution of the IVP...
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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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