论文标题
runge-kutta方法的弱阶段订单条件的代数结构
Algebraic Structure of the Weak Stage Order Conditions for Runge-Kutta Methods
论文作者
论文摘要
当应用于严格的问题时，runge-kutta（RK）方法可能会降低有序。对于与时间无关的运算符的线性问题，如果该方法满足某些弱阶段秩序（WSO）条件，则可以避免降低订单，而这些条件比传统的阶段秩序条件少限制。本文概述了WSO的第一个代数理论，并建立了将RK方案的WSO与其秩序和全部阶段数量和DIRK方案的阶段和阶段数相关联的一般秩序障碍。在几种情况下，构造的边界很清晰。该理论以正交不变子空间和相关的最小多项式为特征。随后，与WSO的RK方法结构的必要条件被证明是用于构建此类方案的实际使用。
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.
