A new way of deriving implicit Runge-Kutta methods based on repeated integrals

TL;DR

This paper introduces a novel approach to deriving coefficients for implicit Runge-Kutta methods using repeated integrals, resulting in new methods that are compared with existing ones through numerical experiments.

Contribution

It presents a new methodology based on repeated integrals for deriving implicit Runge-Kutta methods, producing both new and known Butcher tableaux.

Findings

New implicit Runge-Kutta methods derived

Methods show competitive performance in numerical tests

Comparison with standard methods highlights advantages

Abstract

Runge-Kutta methods have an irreplaceable position among numerical methods designed to solve ordinary differential equations. Especially, implicit ones are suitable for approximating solutions of stiff initial value problems. We propose a new way of deriving coefficients of implicit Runge-Kutta methods. This approach based on repeated integrals yields both new and well-known Butcher's tableaux. We discuss the properties of newly derived methods and compare them with standard collocation implicit Runge-Kutta methods in a series of numerical experiments, including the Prothero-Robinson problem.