General linear methods are extended to the case in which second derivatives, as well as first derivatives, can be calculated. Methods are constructed of third and fourth order which are A-stable, possess the Runge-Kutta stability property and have a diagonally implicit structure for efficient implementation.

General linear methods in the Nordsieck form have been introduced for the numerical solution of Volterra integral equations. In this paper, we introduce general linear methods of order p and stage order q = p for the numerical solution of Volterra integral equations in general form, rather than Nordsieck form. A-and V 0 (α)-stable methods are constructed and applied on stiff problems to show their efficiency.

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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