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(d) We would like to use the second order Runge-Kutta method to compute an approximation y_{n=1..N} of the solution at time steps t_{n=1..N} y_1 = y(0) y_{n+1} = y_n + h f(t_n + frac{h}{2}, y_n + frac{h}{2} f(t_n, y_n)), where h = t_{n+1} - t_n. i. By looking at the test problem { y(t)' = lambda y(t), lambda < 0, 0 le t le 1 { y(0) = 1. show that the above Runge-Kutta Method is Absolutely stable under the condition |1 + hlambda + frac{h^2lambda^2}{2}| < 1. ii. For which lambda should the method be stable here ? iii. How would you define h to ensure stability ?

          (d) We would like to use the second order Runge-Kutta method to compute an approximation y_{n=1..N} of the solution at time steps t_{n=1..N}

y_1 = y(0)
y_{n+1} = y_n + h f(t_n + frac{h}{2}, y_n + frac{h}{2} f(t_n, y_n)),

where h = t_{n+1} - t_n.
i. By looking at the test problem

{ y(t)' = lambda y(t), lambda < 0, 0 le t le 1
{ y(0) = 1.

show that the above Runge-Kutta Method is Absolutely stable under the condition

|1 + hlambda + frac{h^2lambda^2}{2}| < 1.

ii. For which lambda should the method be stable here ?
iii. How would you define h to ensure stability ?

$(d) We would like to use the second order Runge-Kutta method to compute an approximation yn=1..N of the solution at time steps tn=1..N y1 = y(0) yn+1 = yn + h f(tn + frach2, yn + frach2 f(tn, yn)), where h = tn+1 - tn. i. By looking at the test problem y(t)' = lambda y(t), lambda < 0, 0 le t le 1 y(0) = 1. show that the above Runge-Kutta Method is Absolutely stable under the condition |1 + hlambda + frach^2lambda^22| < 1. ii. For which lambda should the method be stable here ? iii. How would you define h to ensure stability ?$

Added by Angela J.

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