Low Stage High Order Explicit Runge--Kutta Methods via Q- and D-Conditions: General Theory and Efficient Recursive Construction

TL;DR

This paper introduces a new theoretical framework and recursive construction method for explicit Runge--Kutta methods, enabling efficient creation of high-order methods with fewer stages.

Contribution

It generalizes classical order conditions using Q- and D-spaces and provides a recursive procedure to construct high-order ERK methods with optimized stage counts.

Findings

Constructs ERK methods of arbitrary even order using structured linear systems.

Produces methods with stage count comparable to classical families but with improved linear term.

Offers a systematic approach for designing methods with better stability and short-time accuracy.

Abstract

Constructing explicit Runge--Kutta (ERK) methods with as few stages as possible for a given order is a classical problem in numerical analysis. In this work, we introduce a $Q$/$D$-space framework of sufficient order conditions for ERK methods. This framework generalizes Butcher's classical simplifying assumptions by reformulating them in terms of simplified $Q$- and $D$-spaces defined through their residual vectors. It yields sufficient conditions which, together with $B(p)$, ensure order $p$. It also leads to a recursive construction procedure for ERK methods of arbitrary even order, in which the Butcher coefficients are obtained from two structured linear systems. For every even order $p\ge 4$, the construction produces ERK methods with stage number $s(p)=(p^2-2p+8)/4$. This stage count has the same leading term as that of the classical Gragg families, while improving the linear term. The free parameters retained by the construction further provide a systematic framework for designing methods with enhanced stability and short-time accuracy.