Low Stage and High Order Explicit Runge--Kutta Methods via $Q$- and $D$-Conditions: Several Construction Details

This note provides additional details on the construction of the $Q$/$D$-space framework for sufficient order conditions of explicit Runge–Kutta (ERK) methods. Specifically, it presents a general version of the sufficiency theorem, several examples illustrating the verification of the sufficient conditions, a detailed construction of an ERK scheme of order $p=10$, the assembly of the associated linear systems, a complexity analysis of the construction algorithm, and tables of coefficients for the constructed ERK methods.