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?
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$\begingroup$ Look up Heun's 2nd order method, lots of tableaux should be available. Or write it down in steps, then read off the coefficients of the method. See math.auckland.ac.nz/~butcher/ODE-book-2008/Tutorials/… $\endgroup$– Lutz LehmannJan 24, 2021 at 19:11
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$\begingroup$ See math.stackexchange.com/questions/3308626 where a Butcher tableau was constructed from a rather convoluted method description. math.stackexchange.com/questions/1532838 is another such case. $\endgroup$– Lutz LehmannJan 25, 2021 at 13:10
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1 Answer
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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.
