On the construction of second derivative diagonally implicit multistage integration methods for ODEs

Second derivative diagonally implicit multistage integration methods (SDIMSIMs) as a subclass of second derivative general linear methods (SGLMs) have been divided into four types, depending on the nature of the differential system to be solved and the computer architecture that is used to implement these methods. In this paper, we describe the construction of SDIMSIMs for all types with Runge–Kutta stability property. Examples of ( p , q , r , s ) SDIMSIMs are given with p = q = r = s ⩽ 4 , where p is the order, q is the stage order, r is the number of external stages, and s is the number of internal stages of the method. Efficiency of the constructed methods is shown by numerical experiments.