Algebraic Structure of the Weak Stage Order Conditions for Runge-Kutta Methods

TL;DR

This paper develops the first algebraic theory of weak stage order (WSO) for Runge-Kutta methods, establishing bounds and conditions that help avoid order reduction in stiff problems.

Contribution

It introduces an algebraic framework for WSO, relates it to scheme order and stages, and provides practical conditions for constructing RK methods with WSO.

Findings

Established sharp bounds relating WSO to order and stages.

Characterized WSO using orthogonal invariant subspaces.

Provided necessary conditions for designing RK schemes with WSO.

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.