The methods of Gellert and of Brusa and Nigro are Padé approximant methods

Abstract

SUMMARYWe discuss properties of methods proposed recently by Gellert' and by Brusa and Nigro' for solving systems of second-order ordinary differential equations. In both cases, we extend the derivations to obtain whole families of methods with improved stability properties (roughly equivalent to L-stability for first-order systems) and we show that obtaining improvement in stability properties results in a deterioration in accuracy and/or implementation properties. Also we bring out the relationship (and in some cases equivalence) of these methods to others proposed in the literature. In particular, we show that the methods proposed by the above authors are essentially equivalent to well-known methods of long standing, namely Pad6 approximant methods for equivalent first-order systems of ordinary differential equations.