# Determine Butcher Table from a method

If I have a method, for example $$y_{n+1} = y_n + \frac{h}{2}[f(t_n,y_n)+f(t_{n+1},y_n+hf(t_n,y_n)) ]$$, how would I go about determining its Butcher Table to write it as a Runge-Kutta method?

Usually we can observe that we have a linear combination of terms of $$f$$, plus a $$y_n$$ that is always there. What I personally observe first are the coefficients in the linear combination. The coefficients of these terms are the $$b$$'s in the final RK form (excluding $$h$$ from coefficients). Here you have 1/2 for both terms.
Secondly I find it simplest to identify the $$c$$'s. They reveal themselves from the time points which the $$f$$'s are evaluated at, with respect to $$t_n$$. Here you have $$t_n+0$$ and $$t_{n+1} = t_n + h\cdot 1$$, so the c's are $$0, 1$$. Each $$c_i$$ corresponds to a $$k_i$$.
Each $$k_i$$ is defined through a set of $$a$$'s. Then I find it simplest to start with the lowest $$c$$, usually zero (could be negative though), call this $$c_1$$. The $$k$$ could be implicitly defined, where this would be clear if any term of $$f$$ was to be evaluated in $$y_{n+i}, i>0$$ on the form you specified above. Your method is clearly explicit with $$k_1 = f(t_n, y_n + h \sum_j a_{1, j}k_j) = f(t_n, y_n)$$, so $$a_{1, j} = 0$$ for all $$j$$. This gves both $$a$$ and $$k$$ for this $$c$$.
Then move on to $$k_2$$ and $$c_2$$. Here we find that $$a_{2, 1} = 1$$, and the rest zero. Then we are done.