Complete Algebraic Characterization of A-Stable Runge–Kutta Methods

Abstract

Important stability concepts for Runge–Kutta methods are I-, A-, and B-stability. For these properties there exist very similar algebraic characterizations. The characterization of B-stability is known for S-irreducible methods. In this paper, an algebraic characterization of I-stability and A-stability related to the coefficients of the method is deduced without any assumption on the Runge–Kutta methods. The corresponding linear dynamic system and its transfer function is considered. The positive real lemma characterizes the passivity of the system or equivalently the positive realness of the transfer function by the Lyapunov equation. Dropping the assumption of controllability and observability a generalization is possible using the Kalman canonical decomposition. Interpreting the modified stability function of a Runge–Kutta method as the transfer function, the positive real lemma yields a complete algebraic characterization of A-stable Runge–Kutta methods.