SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 48-72, February 2024.
Abstract. 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.

中文翻译：

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龙格-库塔法弱级阶条件的代数结构

SIAM 数值分析杂志，第 62 卷，第 1 期，第 48-72 页，2024 年 2 月。
摘要。龙格-库塔 (RK) 方法在应用于刚性问题时可能会表现出阶数降低。对于具有与时间无关的算子的线性问题，如果该方法满足某些弱阶段顺序（WSO）条件（比传统阶段顺序条件限制更少），则可以避免降阶。本文概述了 WSO 的第一个代数理论，并建立了一般阶障碍，将 RK 方案的 WSO 与其阶数以及全隐式和 DIRK 方案的级数联系起来。在几个场景中都表明所构建的边界是尖锐的。该理论用正交不变子空间和相关的最小多项式来表征 WSO。由此产生的 RK 方法与 WSO 结构的必要条件被证明对于构建此类方案具有实际用途。