Complete Butcher Array

Question

I've been given a partially complete Butcher Array, but I seem to be missing some other condition to find out the rest of the values.

This is the Butcher Array I've been given:

Using the conditions about row sums and consistency I get:

$ c_1 = a_{11} + a_{12} => a_{12} = -1/4$;

$ c_2 = a_{21} + a_{22} => c_2 - a_{21} = 5/12$;

$b_1 + b_2 = 1 => b_1 = 1/4$

So I have to be missing some condition for $c_2$ and $a_{21}$.

Answer 1

These are only the conditions for order 1. Next for order 2 you need $$ \sum b_ic_i=\frac12\implies \frac34 c_2=\frac12\implies c_2=\frac23 $$ which also determines the remaining $a_{2,1}=\frac14$.

Next check if by chance also the third order conditions $$ \sum b_ic_i^2=\frac13\text{ and }\sum b_ia_{i,j}c_j=\frac16 $$ are satisfied.