# Runge-Kutta Order Question

Consider the Runge-Kutta method defined by (sorry hard to make this matrix)

$$\begin{array}{c|cccc} 0 & 0 & 0& 0 & 0& \\ \frac{1}{2} & \frac{1}{2}& 0 & 0 & 0 & \\ 1 & 0& 1& 0& 0& \\ 1 & 0& 0 & 1 & 0 & \\ \hline & \frac{1}{6}& \frac{2}{3} & 0 & \frac{1}{6} \end{array}$$

a) Write the formula which corresponds to this Runge-Kutta matrix.

b) Using the order conditions (4.14) and (4.15), find the order of this method.(See pictures for 4.14/4.15)

So this is what I have so far,

a) the table corresponds to,

$y_{n+1}=y_n+h[\frac{K_1}{6}+\frac{2K_2}{3}+\frac{1K_4}{6}]$ with,

$K_1= f(t_n,y_n)$

$K_2= f(t_n+\frac{h}{2},y_n+\frac{h}{2}K_1)$

$K_3= f(t_n+h,y_n+hK_2)$

$K_4= f(t_n+h,y_n+hK_3)$

and Local truncation error of the method is $0(h^5)$.

b) This is where I'm stuck. I'm very certain that I did part a) correct but I'm not sure about part b.