Accuracy of Runge-Kutta method when no analytical solution

[jimmychoo]

Homework Statement

I have to find eigenvalues to
$$\frac{d^2y}{dx^2} + p^2 e^x y = 0,\, y(0)=0,y(1)=0$$
using the Runge-Kutta single step method to solve the ODE (splitting it up), with step length $h$and then another numerical method. This is not a problem. However, I need to be able to find the eigenvalues to a specified accuracy: so this is my question:

Is it possible to bound the error of the Runge-Kutta method (at a certain step length) for solving this ode without using the analytical solution?

The Attempt at a Solution

Runge-Kutta is a 4th order method, so I tried to model the error as $kh^5$ and using different step lengths solve for the constant, but this didn't work.
Maybe I'm just missing an analytical feature of this particular equation (if that's the case and anybody else can see it, don't give me the specifics of it!)

thanks for any help!

 

[mighty2000]

Thank you for your question. In order to bound the error of the Runge-Kutta method, we need to consider the local truncation error (LTE) and global error. The LTE is the error that occurs at each step of the numerical method, while the global error is the accumulation of all the LTEs over the entire interval.

For the specific ODE given, the LTE of the Runge-Kutta method can be approximated as kh^5, where k is a constant. However, this approximation may not hold for all step lengths and may vary depending on the specific equation being solved. Therefore, it is not possible to accurately bound the error without using the analytical solution.

One possible approach is to compare the results obtained from the Runge-Kutta method with a different numerical method, such as the Euler method, which has a known error bound. This can give an indication of the accuracy of the Runge-Kutta method for the given ODE.

Another approach is to use adaptive step size methods, such as the Runge-Kutta-Fehlberg method, which adjusts the step size based on the estimated error. This can help in achieving a desired level of accuracy without relying on the analytical solution.

I hope this helps. Best of luck with your calculations.