2N-storage Runge-Kutta methods: c-reflection symmetry and factorization of the Butcher tableau

TL;DR

This paper investigates the structure and symmetry properties of 2N-storage Runge-Kutta methods, revealing c-reflection symmetry and factorization of the Butcher tableau, with numerical validation and new scheme constructions for higher orders.

Contribution

It introduces the concept of c-reflection symmetry in 2N-storage Runge-Kutta methods and explores the factorization of the Butcher tableau, providing new schemes and insights for method development.

Findings

Factorization of the Butcher tableau into special matrices.

Existence of c-reflection symmetry in methods of order less than five.

Numerical construction of new (5,4), (6,4), and (8,4) schemes.

Abstract

Low-storage Runge-Kutta schemes of Williamson's type, so-called 2N-storage schemes, are further examined as a follow-up to the recent work. It is found that the augmented Butcher tableau factorizes into a product of matrices with special properties. Those properties reveal that the 2N-storage methods of the order of global accuracy less than five possess a symmetry, called c-reflection symmetry, i.e. most methods exist in pairs. A transformation that relates the Butcher tableaux of the pairs is found and the fact that the c-reflected method satisfies the same order conditions as the original one is proven. Numerical evidence that validates the analytic results is presented. Branches of solutions for (5,4) methods, first explored by Carpenter and Kennedy, are constructed numerically. Four new (5,4) schemes with coefficients expressed in radicals and one with rational coefficients are examined for illustration. Eight new (6,4) schemes, some of which can be expressed in rationals or radicals, and one (8,4) scheme, are studied to understand the practical implications of the c-reflection symmetry for methods with higher number of stages. In the absence of closed-form analytic solutions for 2N-storage Runge-Kutta methods of order four and above, the general symmetry properties, as well as some specific analytic solutions presented here, may help in development and optimization of 2N-storage schemes.