The conditions for a Runge--Kutta method to be of order $p$ with $p\ge 5$ for a scalar non-autonomous problem are a proper subset of the order conditions for a vector problem. Nevertheless, Runge--Kutta methods that were derived historically only for scalar problems happened to be of the same order for vector problems. We relate the order conditions for scalar problems to factorisations of the Runge--Kutta trees into "atomic stumps" and enumerate those conditions up to $p=20$. Using a special search procedure over unsatisfied order conditions, new Runge--Kutta methods of "ambiguous orders" five and six are constructed. These are used to verify the validity of the results.

翻译：龙格- 库塔方法对于卡路里非自治问题来说,以5美元计价5美元的顺序是矢量问题顺序条件的一个适当子集,然而,历史上仅对卡路里问题得出的龙格- 库塔方法对于矢量问题来说,恰好是相同的顺序。我们把龙格- 库塔树的阶梯问题的顺序条件与对龙格- 库塔树的系数化联系起来,并列出这些条件,最高达20美元。对未满足的秩序条件采用特别搜索程序,并建立了新的龙格- 库塔方法,即“不协调命令”的5和6,用于核实结果的有效性。