3. (24 points, 6 each) Consider the following method for solving the ODE y = f(y,t) y = yn + f(yu,ta) (2) Yu+1 = y +hlaf

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3 24 Points 6 Each Consider The Following Method For Solving The Ode Y F Y T Y Yn F Yu Ta 2 Yu 1 Y Hlaf 1 (25.61 KiB) Viewed 34 times

3. (24 points, 6 each) Consider the following method for solving the ODE y = f(y,t) y = yn + f(yu,ta) (2) Yu+1 = y +hlaf (y..tu+1) + (1 - a) f(...)) where 0 Sasi (a) Apply this method to y = iwy, where w is a real number, and find the optimal value of a for stability. What is the largest time step you can take with this optimal value of a? (b) With the value of a obtained in part (a), we solve the system y' = iwy with y(0) = 1 and step size h=1/w. What are the amplitude and phase error after 100 stepx? (c) Now find the value of a that gives you maximum possible accuracy (d) For the value of a obtained in part (e), what are the stability characteristics of the method when applied to the ODE / www real)