Application of Fourier series in deriving stabilitypolynomial of multivalue methods for ODEs

We construct second derivative diagonally implicit multistage integration methods(SDIMSIMs) as a subclass of second derivative general linear methods (SGLMs) withRunge–Kutta stability property (RKS). The conditions arise from RKS which area system of polynomial equations can not be produced by symbolic manipulationpackages for the methods of order p ≥ ۵. In this paper, we describe an approachto construct SDIMSIMs with RKS property by using some variant of the Fourierseries method. We construct explicit and implicit SDIMSIMs of order ۵ and ۶ whichrespectively are appropriate for both non–stiff and stiff differential systems and showtheir efficiency by applying to some well–known problems.