111a or yn 12 yn 1 h f tn 2 1 yn 1 yn

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Unformatted text preview: .11c) This is the midpoint rule integration formula that we discussed earlier. The ^ on yn 1=2 indicates that it is an intermediate rather than a nal solution. As shown in Figure 3.1.1, we can regard the two-stage process (3.1.11b,c) as the result of two explicit Euler steps. The intermediate solution yn 1=2 is computed at tn + h=2 in ^ the rst (predictor) step and this value is used to generate an approximate slope f (tn + h=2 yn 1=2) for use in the second (corrector) Euler step. According to Gear ^ 15], this method has been called the Euler-Cauchy, improved polygon, Heun, or modi ed Euler method. Since there seems to be some disagreement about its name and because of its similarity to midpoint rule integration, we'll call it the midpoint rule predictor-corrector. ; ; ; 2. Select b2 = 1=2, then a = c = 1 and b1 = 1=2. According to (3.1.5), this RungeKutta formula is yn = yn 1 + h (k1 + k2 ) 2 ; 5 (3.1.12a) with k1 = f (tn 1 yn 1) ; k2 = f (tn 1 + h yn 1 + hk1): ; ; ; (3.1.12b) Again, eliminating k1 and k2, yn = yn 1 + h f (tn 1 yn 1)...
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