Numer Algor (2011) 57:149–167 DOI 10.1007/s11075-010-9420-y ORIGINAL PAPER An extension of general linear methods Ali Abdi · Gholamreza Hojjati Received: 12 January 2010 / Accepted: 11 August 2010 / Published online: 21 August 2010 © Springer Science+Business Media, LLC 2010 Abstract General Linear Methods (GLMs) were introduced as the natural generalizations of the classical Runge–Kutta and linear multistep methods. An extension of GLMs, so-called SGLMs (GLM with second derivative), was introduced to the case in which second derivatives, as well as first derivatives, can be calculated. In this paper, we introduce the definitions of consistency, stability and convergence for an SGLM. It will be shown that in SGLMs, stability and consistency together are equivalent to convergence. Also, by introducing a subclass of SGLMs, we construct methods of this subclass up to the maximal order which possess Runge–Kutta stability property and A-stability for implicit ones. Keywords General linear methods · Ordinary differential equation · Stability · Consistency · Convergence Mathematics Subject Classification (2010) 65L05 1 Introduction Traditional numerical methods for solving an initial value problem generally fall into two main classes: linear multistep (multivalue) and Runge–Kutta The research on which this paper is based was supported by research fund of the university of Tabriz under No. 27-1216-3. A. Abdi · G. Hojjati (B ) Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran e-mail: ghojjati@yahoo.com, ghojjati@tabrizu.ac.ir A. Abdi e-mail: a_abdi@tabrizu.ac.ir