Lab for the Runge Kutta Method

Modue for the Runge Kutta Method

Runge-Kutta Method of Order 4 for O.D.E.'s. To approximate the solution of the initial value problem with
by using the formula ,
where

Report to be handed in.

Computer Problems.

Exercise 1. Solve the I.V.P. with y(0) = 1 over [0.0, 0.4].
Use the Runge-Kutta method with 2 subintervals of [0.0, 0.4] to get a numerical approximation to the solution.
Show the calculations for K1, K2, K3, K4 for each step.
Compare the Runge-Kutta solution with the known analytic solution .

Exercise 2. Solve the I.V.P. with y(0) = 0 over [0.0, 10.0].
Use the Runge-Kutta subroutine and 10 subintervals of [0,10] to get your answer.
Plot the solution using the 11 points you just computed in the window {{0,10},{0,1}}.
Report the last point which is the numerical approximation to (10, y(10)).

Exercise 3. Solve the I.V.P. with y(0) = 0 over .
Use the Runge-Kutta subroutine and 10 subintervals of [0,10] to get your answer.
Plot the solution using the 21 points you just computed in the window {{0,10},{0,1}}.
Report the last point which is the numerical approximation to (10, y(10)).

Exercise 4. Solve the I.V.P. with y(0) = 0 over .
Use the Runge-Kutta subroutine and 10 subintervals of [0,10] to get your answer.
Plot the solution using the 51 points you just computed in the window {{0,10},{0,1}}.
Report the last point which is the numerical approximation to (10, y(10)).

Exercise 5. Solve the I.V.P. with y(0) = 0 over .
Use the Runge-Kutta subroutine and 10 subintervals of [0,10] to get your answer.
Plot the solution using the 101 points you just computed in the window {{0,10},{0,1}}.
Report the last point which is the numerical approximation to (10, y(10)).

Remark. The value of y(10) is known to be:

Exercise 6. Read the textbook and see if you can find out something about how small the step size should be in the Runge-Kutta method to guarantee an accurate solution.

(c) John H. Mathews, 1998