Providing a single point spectrum for Runge-Kutta schemes of high stage order based on perturbed collocation

Abstract

The coefficient matrix of a Runge-Kutta method of order p has to contain a block of size m ≥ p so that the respective method also has a stage order of p. If m = p and additionally a single point spectrum of the block is demanded all block coefficients and the respective relative step sizes are completely defined through that aside from a common factor, see e. g. [1]. This results in fixed stability properties for the stability functions of the block stages. To avoid such an imposition and to provide more freedom in the choice of coefficients we consider in this work a block of size m = p + 1 and derive conditions on a single point spectrum by means of a perturbed collocation ansatz. Our results also cover methods with an explicit first stage and imply classical results related to non-perturbed collocation. Additionally, we present some results that may prove beneficial in the actual construction of methods with a stage order of p and a block size of m = p + 1.