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