What is Ralston's method?

Ralston's second order Runge-Kutta method approximates the solution to various ordinary differential equations (ODEs).

This method involves dividing the ODE into smaller time steps and calculating the approximate solution iteratively at each step. Ralston's method utilizes two evaluation functions to estimate the solution at each point.

Working example

Consider the following second order Runge-Kutta ODE:

where

We are supposed to approximate the solution for $y$. Ralston's method uses the following equation to approximate the solution:

along with the two evaluation functions mentioned below:

As mentioned earlier, Ralston's method involves iterative estimation of the solution. Each iteration utilizes the results from the previous iteration to refine the approximation.

$0th$ iteration

First, we'll solve for $k_{1}$. Since $y_{0}=1.0$ at $x_{0}=0$ (given) the value for $k_{1}$ will be:

Now for $k_{2}$ it will be:

Since we have $x_{0} = 0, k_{1} = 2, h=0.5, \textnormal{and } y_{0} = 1.0$, then the value for above equation will become:

Now that we have $k_{1} = 2 \textnormal{ and } k_{2}=1.233$, we can approximate $y_{1}$using the Ralston's equation as follows:

$1st$ iteration

With the value of $y_{1}$obtained from the 0th iteration, we can calculate $y_{2}$ in the 1st iteration using the same equations provided by Ralston's method as follow:

Just plug in the values and evaluate. This activity is left as an exercise for you.

Note: You can keep iterating this process as long as you want. The more you iterate, the better the approximation you will get.

Conclusion

In most cases, you will be given a specified number of iterations for Ralston's method. Increasing the number of iterations brings you closer to the optimal solution and improves accuracy. However, it's important to note that with more iterations, the computational complexity also increases. Thus, finding a balance between accuracy and computational efficiency is crucial when determining Ralston's method's appropriate number of iterations.

Check out 4th order Runge-Kutta method.