Embedded (4, 5) pairs of explicit 7-stage Runge-Kutta methods with FSAL property

TL;DR

This paper investigates the structure and classification of embedded (4, 5) pairs of explicit 7-stage Runge-Kutta methods with FSAL property, identifying five families and special simplifying cases.

Contribution

It provides a comprehensive analysis and classification of embedded (4, 5) Runge-Kutta pairs with FSAL, including known and new families, under specific simplifying assumptions.

Findings

Identifies five 4-dimensional families of methods

Includes known families satisfying simplifying assumptions

Analyzes exceptional cases and their properties

Abstract

The general case of embedded (4, 5) pairs of explicit 7-stage Runge--Kutta methods with FSAL property (a_7j = b_j, 1 <= j <= 7, c_7 = 1) is considered. Besides exceptional cases, the pairs form five 4-dimensional families. The pairs within two (already known) families satisfy the simplifying assumption sum_j a_ij c_j = c_i^2 / 2, i >= 3.