Algebraic Structure of the Weak Stage Order Conditions for Runge-Kutta Methods
Abhijit Biswas, David Ketcheson, Benjamin Seibold, David Shirokoff
TL;DR
This work develops the first algebraic theory of Weak Stage Order (WSO) for Runge-Kutta methods, connecting WSO to $A$-invariant subspaces and associated minimal polynomials. It derives general order barriers that relate WSO $q$ to the classical order $p$, the number of stages $s$, and the number of distinct abscissas $n_c$, with refined bounds for DIRK schemes that can be sharp. The authors introduce spaces $Y$ and $K_q$ to recast WSO as an orthogonality condition, establish lower bounds on their dimensions via rational approximations and subspace arguments, and develop a stability-function formulation in an orthogonal polynomial basis tied to a linear functional. They also present necessary conditions on the polynomial $P(x)$ governing $K_q$ for DIRKs, illustrate sharpness through concrete DIRK examples, and outline practical consequences for constructing high-WSO schemes, including guidance for future SDIRK/ERK extensions and open theoretical questions.
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.