$4^{\text{th}}$ order Runge-Kutta method

Question

I would like to know the motivation behind the choice of numbers or coefficients in front of $k_1$, $k_2$, $k_3$ and $k_4$ in $4^{\text{th}}$ order Runge-Kutta method. There are many choices of the coefficients one can make. However, $\frac16$, $\frac13$, $\frac13$, $\frac16$ are the most popular set. Can anyone explain this point?

Answer 1

Consider a differential equation $y' = f(x,y)$ with initial condition $y(0)=0$, and write out the series solution to order $x^5$ (in terms of the coefficients of the bivariate Taylor series of $f$ at $(0,0))$. Then compare with what you get from the Runge-Kutta scheme with coefficients $k_1, \ldots, k_4$. In order for the Runge-Kutta to agree with the series solution to order $x^5$, there will be a set of equations to solve. It will turn out that $k_1 = 1/6$, $k_2 = 1/3$, $k_3 = 1/3$, $k_4 = 1/6$ are the solution.

Answer 2

Runge Kutte forms an infinite family of ODE solvers. The coefficients come from something called a Butcher Tableau. The most basic form of Runge Kutte is Eulers method. Later there were better methods made with the mid-point method then Heun's method. Then Kutta gave an explanation of $4th$ order methods. The evaluation of the stages give the tableau. There is a lecture here on derivation for RK4 which is typically done in numerical analysis.