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.
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The research on which this paper is based was supported by research fund of the university of Tabriz under No. 27-1216-3.
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Abdi, A., Hojjati, G. An extension of general linear methods. Numer Algor 57, 149–167 (2011). https://doi.org/10.1007/s11075-010-9420-y
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DOI: https://doi.org/10.1007/s11075-010-9420-y