Application of Fourier series in deriving stability polynomial 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) with Runge-Kutta stability property (RKS). The conditions arise from RKS which are a system of polynomial equations can not be produced by symbolic manipulation packages for the methods of order p ≥ 5. In this paper, we describe an approach to construct SDIMSIMs with RKS property by using some variant of the Fourier series method. We construct explicit and implicit SDIMSIMs of order 5 and 6 which respectively are appropriate for both non-stiff and stiff differential systems and show their efficiency by applying to some well-known problems