52a as h2 h ny jdnj jcnjhp1 jy1 2 np j 353a

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Unformatted text preview: e local error estimate (3.5.3a) or (3.5.3b) may be added to yn or yn 2, respectively, to obtain a higher-order method. For example, using (3.5.3b), h ; y h=2 h= y(tn) = yn 2 + yn2p ; n + O(hp+2): 1 Thus, we could accept h ; y h=2 h= yn 2 = yn 2 + yn2p ; n ^h= 1 ; as an O(hp+2) approximation of y(tn). This technique, called local extrapolation, is also a bit risky since we do not have an error estimate of yn 2. We'll return to this topic in ^h= Chapter 4. Embedding, the second popular means of estimating local (or local discretization) errors, involves using two one-step methods having di erent orders. Thus, consider calculating two solutions using the p th- and p + 1 st-order methods p yn = yn 1 + h p(tn 1 yn 1 h) ; ; ; p dp = Cnhp+1 n (3.5.4a) p dp+1 = Cn+1hp+2: n (3.5.4b) and p yn+1 = yn 1 + h ; p+1 (tn 1 ; yn 1 h) ; (The superscripts on yn and dn are added to distinguish solutions of di erent order.) The local error of the p-order solution is p p p p jdp j = jyn ; y(tn)j = jyn ; yn+1 + yn+1 ; y(tn)j: n 45...
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