# 113 are consistent to order two since their local

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Unformatted text preview: ) = y ; t2 + 1 0 y(0) = 1=2 on 0 < t 1 using the explicit Euler method and the midpoint rule. Use several step sizes and compare the error at t = 1 as a function of the number of evaluations of f (t y). The midpoint rule has twice the number of function evaluations of the Euler method but is higher order. Which method is preferred? 6 yn yn-1 ^ slope f(t n-1 + h/2,y n-1/2 ) ^ yn-1/2 h/2 h/2 t n-1 tn t Figure 3.1.1: Midpoint rule predictor-corrector (3.1.11b,c) for one time step. average slope yn yn-1 ^ slope f(t n,y n) ^ yn h t n-1 tn t Figure 3.1.2: Trapezoidal rule predictor-corrector (3.1.13b,c) for one time step. 3.2 Explicit Runge-Kutta Methods We would like to generalize the second order Runge-Kutta formulas considered in Section 3.1 to higher order. As usual, we will apply them to the scalar IVP (3.1.1). Runge-Kutta methods belong to a class called one-step methods that only require information about the solution at time tn 1 to calculate it at tn . This being the case, it's po...
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