I am reading about implicit Runge-Kutta methods as well as collocation methods, and I am struggling to understand the conditions of "relaxed constraints" of implicit methods shown on page 3 of this article. (The relaxed conditions are simply that values $ c_1,c_2 ... c_s$ are distinct and $(\sum b_i =1) \land b_i \neq 0 $)
From the article, $A(p)$ implies that Runge-Kutta methods will be accurate up to $O(h^p)$
How does one derive the equations for constraints $B(p),C(p),D(p)$ and $E(p,d)$? And what does each constraint mean?
In Theorem 1, it states that $A(p) \implies B(p)$. However, collocation methods have a theory on superconvergence that will have methods of order $p>s$ if $B(p)$ is satisfied. Is this theory only tied specifically to collocation methods, or can I use $B(p)$ to determine the order for any Runge-Kutta?