A global negative result on algebraic stability and a special positive result on linear stability of generalized IRK methods

Abstract

Generalized implicit Runge-Kutta methods (generalized IRK methods) were introduced by Butcher in 1981. Each generalized IRK method is equivalent to a spline-collocation method of multivalue type. The concept of algebraic stability for general linear methods was introduced by Burrage and Butcher in 1980. First we give the above methods in their form as general linear methods. We show that a large, and important, subclass of the above methods has no algebraically stable members. We consider the linear stability of the one-stage and two-stage two-value spline-collocation methods of multivalue type. We determine for which collocation points these methods are convergent, stable at infinity and A-stable