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# 54b and is ohp2 thus p p jdp j jyn yn1j jdp1j n n the

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Unformatted text preview: nal function evaluation is needed to estimate the error in the (lower) p th-order method. However, the derivation of such formula pairs is ; 46 not simple since the order conditions are nonlinear. Additionally, it may be impossible to obtain a p + 1-order method by adding a single stage to an s-stage method. Formulas, nevertheless, exist. Example 3.5.2. The forward Euler method is embedded in the trapezoidal rule predictor-corrector method. The tableaux for these methods are 00 0 11 0 1/2 1/2 00 1 The two methods are k1 = f (tn 1 yn 1) ; k2 = f (tn 1 + h yn 1 + hk1 ) ; ; ; 1 yn = yn 1 + hk1 2 yn = yn 1 + h (k1 + k2): 2 Example 3.5.3. There is a three-stage, second-order method embedded in the classical fourth-order Runge-Kutta method. Their tableaux are 0 1/2 1/2 1/2 0 1/2 1001 1/6 1/3 1/3 1/6 ; ; 0 1/2 1/2 1/2 0 1/2 0 01 These formulas are k1 = f (tn 1 yn 1) ; ; k2 = f (tn 1 + h=2 yn 1 + hk1=2) ; k3 = f (tn 1 + h=2 yn 1 + hk2=2) ; ; k4 = f (tn 1 + h yn 1 + hk3) ; ; 2 yn = yn 1 + hk3 ; 47 ; 4 yn = yn 1 + h (k1 + 2k2 + 2k3 + k4): 6 ; Example 3.5.4. Fehlberg 14] constructed pairs of explicit R...
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