Runge-Kutta (RK) methods may exhibit order reduction when applied to stiff problems. For linear problems with time-independent operators, order reduction can be avoided if the method satisfies certain weak stage order (WSO) conditions, which are less restrictive than traditional stage order conditions. This paper outlines the first algebraic theory of WSO, and establishes general order barriers that relate the WSO of a RK scheme to its order and number of stages for both fully-implicit and DIRK schemes. It is shown in several scenarios that the constructed bounds are sharp. The theory characterizes WSO in terms of orthogonal invariant subspaces and associated minimal polynomials. The resulting necessary conditions on the structure of RK methods with WSO are then shown to be of practical use for the construction of such schemes.

翻译：Runge-Kutta (RK)方法在应用于刚性问题时可能出现阶降低现象。对于时间无关算子的线性问题，若该方法满足某些弱阶段阶（WSO）条件（其限制性弱于传统阶段阶条件），则可避免阶降低。本文首次提出WSO的代数理论，建立了全隐式与对角隐式RK方法的WSO与其阶数及阶段数之间的一般阶障碍，并在多个场景中证明所构造的界是紧的。该理论通过正交不变子空间及其关联的最小多项式刻画WSO，进而推导出具有WSO的RK方法结构所需满足的必要条件，并表明这些条件对构造此类方法具有实际应用价值。