We learned earlier that Taylor's Series gives us a reasonably good approximation to a function, especially if we are near enough to some known starting point, and we take enough terms.
However, one of the drawbacks with Taylor's method is that you need to differentiate your function once for each new term you want to calculate. This can be troublesome for complicated functions, and doesn't work well in computerised modelling.
Carl Runge (pronounced "roonga") and Wilhelm Kutta (pronounced "koota") aimed to provide a method of approximating a function without having to differentiate the original equation.
Their approach was to simulate as many steps of the Taylor's Series method but using evaluation of the original function only.
We begin with two function evaluations of the form:
`F_2=hf(x+alpha h,y+beta F_1)`
The `alpha` and `beta` are unknown quantities. The idea was to take a linear combination of the `F_1` and `F_2` terms to obtain an approximation for the `y` value at `x = x_0+h`, and to find appropriate values of `alpha` and `beta`.
By comparing the values obtains using Taylor's Series method and the above terms (I will spare you the details here), they obtained the following, which is Runge-Kutta Method of Order 2:
As usual in this work, the more terms we take, the better the solution. In practice, the Order 2 solution is rarely used because it is not very accurate.
A better result is given by the Order 3 method:
This was obtained in a similar way to the earlier formula, by comparing Taylor's Series results.
The most commonly used Runge-Kutta formula in use is the Order 4 formula (RK4), as it gives the best trade-off between computational requirements and accuracy.