On the stability of adaptations of Runge-Kutta methods to systems of delay differential equations

Abstract

This paper is concerned with the adaptation of Runge-Kutta methods to initial value problems for systems of delay differential equations. At present, three main types of interpolation procedures can be distinguished in the literature for adapting Runge-Kutta methods to delay differential equations, viz. Hermite interpolation with respect to the gridpoints, interpolation procedures that use continuous extensions of the Runge-Kutta method, and the interpolation procedures that have been introduced by in 't Hout (1992). In this paper, we are interested in the stability of the corresponding three types of adaptations of the class of Runge-Kutta methods. We give a survey of the stability results that can be obtained from the literature up to now, relevant to the testproblem U′( t) = LU( t) + MU( t − τ) ( t ≥ 0), where L, M denote constant square matrices, and τ > 0. In addition, we also derive two new results.