Explicit Nordsieck second derivative general linear methods for ODEs

Abstract

The paper deals with the construction of explicit Nordsieck second derivative general linear methods with \(s\) stages of order \(p\) with \(p=s\) and high stage order \(q=p\) with inherent Runge–Kutta or quadratic stability properties. Satisfying the order and stage order conditions together with inherent stability conditions leads to methods with some free parameters, which will be used to obtain methods with a large region of absolute stability. Examples of methods with \(r\) external stages and \(p=q=s=r-1\) up to order five are given, and numerical experiments in a fixed stepsize environment are presented.

doi: https://doi.org/10.1017/S1446181122000049