Abstract
Recently, Kondrat and Jacques [1] gave an extended two-step fourth order A-stable method; the reason noted for giving a two-step method is that a one-step method of the type is not possible.
Firstly, we show that it is possible to obtain a two-parameter family of such methods which are one-step, fourth order and A-stable; and, there exists a one-parameter sub-family of these methods which are, in addition, L-stable. Secondly, from among the extended two-step methods, it is natural to ask if the well-known Simpson's rule can be stabilized; we present an A-stabilized version of Simpson's rule.
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