# 112a with k1 f tn 1 yn 1 k2 f tn 1 h yn 1 hk1 3112b

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Unformatted text preview: + f (tn yn 1 + hf (tn 1 yn 1))]: 2 ; ; ; ; ; ; (3.1.13a) This too, can be written as a two-stage formula yn = yn 1 + hf (tn 1 yn 1) ^ (3.1.13b) ^ yn = yn 1 + h f (tn 1 yn 1) + f (tn yn)]: 2 (3.1.13c) ; ; ; ; ; ; The formula (3.1.13a) is reminiscent of trapezoidal rule integration. The combined formula (3.1.13b,c) can, once again, be interpreted as a predictor-corrector method. Thus, as shown in Figure 3.1.2, the explicit Euler method is used to predict a solution at tn and the trapezoidal rule is used to correct it there. We'll call (3.1.12, 3.1.13) the trapezoidal rule predictor-corrector however, it is also known as the improved tangent, improved polygon, modi ed Euler, or Euler-Cauchy method ( 15], Chapter 2). Using De nition 2.1.3, we see that the Taylor's series method (3.1.4) and the RungeKutta methods (3.1.11) and (3.1.13) are consistent to order two since their local errors are all O(h3) (hence, their local discretization errors are O(h2)). Problems 1. Solve the IVP y = f (t y...
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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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