2N-storage Runge-Kutta methods: Order conditions, general properties and some analytic solutions

TL;DR

This paper investigates low-storage 2N-storage Runge-Kutta methods, deriving explicit constraints, correcting existing formulas, and presenting new schemes with third-order accuracy and rational coefficients, supported by numerical analysis.

Contribution

It introduces explicit 2N-storage constraints, corrects a key formula, and provides new high-order schemes with rational coefficients for efficient numerical integration.

Findings

Derived explicit 2N-storage constraints for the first time.

Corrected Williamson's formula for coefficient conversion.

Presented and numerically tested new four- and five-stage schemes.

Abstract

Low-storage Runge-Kutta schemes of Williamson's type, so-called 2N-storage schemes, are examined. Explicit 2N-storage constraints are derived for the first time and used to establish new relations between the entries of the Butcher tableau. An error in the Williamson's formula for converting coefficients between the standard and 2N-storage formats in the special case is pointed out and corrected. The new relations are used to derive a closed-form solution for four- and five-stage 2N-storage methods with the third order of global accuracy. Several new four- and five-stage schemes with rational coefficients are presented and numerically examined for illustration.