Determine Butcher Table from a method

Question

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?

Answer 1

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.