Integration processes based on Radau quadrature formulas

Introduction.In a previous paper [1] certain properties of Runge-Kutta processes were derived.As an example of the use of these results a three stage implicit process was found in which the 17 conditions necessary for fifth-order accuracy were satisfied.Since this was achieved by a special choice of only 12 parameters the existence of various relationships between the equations that must be satisfied is suggested.Subsequently some of these relationships were found [2] and a set of processes was derived such that the process of v stages has order of accuracy p = 2v.The associated quadrature formula in this case is the y-point Gauss-Legendre formula.In this paper we shall refer to this type of integration process as a G-process.In a G-process the vectors g(1, gt2, • • •, g'"' which correspond to the different stages of the process are defined only implicitly and in practice they would need to be evaluated iteratively.The subject of the present paper, another class of implicit Runge-Kutta process, is based on the Radau quadrature formulas [3].It will be seen that these have certain advantages over the G-processes in that the accuracy achieved for the same number of implicit stages is higher.Results proved in [1] and [2] will be assumed here and the same notations will be