Contractivity of Runge-Kutta methods

In this paper we present necessary and sufficient conditions for Runge-Kutta methods to be contractive. We consider not only unconditional contractivity for arbitrary dissipative initial value problems, but also conditional contractivity for initial value problems where the right hand side function satisfies a circle condition. Our results are relevant for arbitrary norms, in particular for the maximum norm.For contractive methods, we also focus on the question whether there exists a unique solution to the algebraic equations in each step. Further we show that contractive methods have a limited order of accuracy. Various optimal methods are presented, mainly of explicit type. We provide a numerical illustration to our theoretical results by applying the method of lines to a parabolic and a hyperbolic partial differential equation.

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