# 23 with s 1 as yn yn 1 hb1 f tn 1 c1h y1 331a y1

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Unformatted text preview: ries and substituting the result into the above expression yields ; ; y(tn) = y(tn 1) + hb1 f + c1 hft + ha11 ffy + O(h2)]: ; Comparing the terms of the above series with the Taylor's series h2 (f + ff ) + O(h3) y(tn) = y(tn 1) + hf + 2 t y of the exact solution yields 1 b1 = 1 a11 = c1 = 2 : Substituting these coe cients into (3.3.1), we nd the method to be an implicit midpoint rule ; yn = yn 1 + hf (tn 1 + h=2 Y1) ; ; Y1 = yn 1 + h f (tn 1 + h=2 Y1): 2 The tableau for this method is ; ; 19 (3.3.2a) (3.3.2b) 1 2 1 2 1 The formula has similarities to the midpoint rule predictor-corrector (3.1.11) however, there are important di erences. Here, the backward Euler method (rather than the forward Euler method) may be regarded as furnishing a predictor (3.3.2b) with the midpoint rule providing the corrector (3.3.2a). However, formulas (3.3.2a) and (3.3.2b) are coupled and must be solved simultaneously rather than sequentially. Example 3.3.2. The two-stage method having maximal order four presented in th...
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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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