Isomeric trees and the order of Runge--Kutta methods

TL;DR

This paper explores the order conditions of Runge--Kutta methods for scalar and vector problems, introduces a novel classification using isomeric trees, and constructs new methods with ambiguous orders five and six.

Contribution

It relates scalar and vector order conditions via isomeric trees, and develops new Runge--Kutta methods with ambiguous orders using a specialized search.

Findings

Enumeration of order conditions up to p=20

Construction of new methods with ambiguous orders five and six

Verification of theoretical results through method development

Abstract

The conditions for a Runge--Kutta method to be of order $p$ with $p\ge 5$ for a scalar non-autonomous problem are a proper subset of the order conditions for a vector problem. Nevertheless, Runge--Kutta methods that were derived historically only for scalar problems happened to be of the same order for vector problems. We relate the order conditions for scalar problems to factorisations of the Runge--Kutta trees into "atomic stumps" and enumerate those conditions up to $p=20$. Using a special search procedure over unsatisfied order conditions, new Runge--Kutta methods of "ambiguous orders" five and six are constructed. These are used to verify the validity of the results.