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HWK 2, Due Tue Feb 22

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Make your write-up clean and concise. I do not need to see all you algebra. Imagine that you are writing an example for a book. The main steps have to come out clearly but all the details should not be shown.

1. Plot the stability diagram for Runge-Kutta 2nd order (RK2) as well as for RK3 and RK4. Plot all three diagrams on the same figure. Identify the intercepts with the real and imaginary axes [plots cannot be done analytically. Hint: |G|=1 means G=e ^{i p}, find z=ah as a function of the parameter p]
2. Find the stability diagrams for Backward Euler y_n+1 =y_n + h f(t_n+1,y_n+1) and for Crank-Nicholson y_n+1 =y_n + 0.5 h [ f(t_n,y_n) + f(t_n+1,y_n+1)] . Briefly comment on these methods, what is involved to compute y_n+1 ? [Stability diagrams can be found analytically]
3. What is the long time behavior of solutions to y'= cos(t y) ? Try to find an asymptotic expansion as we did in class for y'=t² - y².