# Proving a R-K method's order

1. Dec 8, 2008

### Jopi

1. The problem statement, all variables and given/known data
$$x'=f(t,x)$$
$$k_1=f(t_n,x_n)$$
$$k_2=f(t_n+2h/3,x_n+2h k_1/3)$$
$$k_3=f(t_n+2h/3,x_n+h(k_1+3k_2)/6$$
$$x_{n+1}=x_n + h(k_1+ k_2+2k_3)/4$$

Prove that the above Runge-Kutta method is of third order by examining the problem
$$x'=x, \; x(0)=1$$.

2. Relevant equations

3. The attempt at a solution
I have to prove that the difference between the exact solution (which is exp(x)) and the solution given by the RK is of magnitude h4, where h is the step size. I'm just being very thick-skulled here, I can program this in Matlab but I can't do it on paper.
Can someone just show me how I calculate k1, k2 and k3? I don't completely understand the notation used. What is f(tn,xn)?