Stable Runge-Kutta integrations for differential systems with semi-stable equilibria (Q1889955)

The behavior of Runge-Kutta schemes applied to ordinary differential equations near semi-stable equilibria is analyzed. Here semi-stable means that in any neighbourhood of the equilibrium there are stable and unstable orbits. The authors look for E-stable Runge-Kutta schemes, i.e., methods admitting a cone which is positively invariant for the discrete dynamics and the additional property that all discrete orbits in this cone tend towards the equilibrium. The main result of this paper is that stiffly accurate Runge-Kutta schemes are shown to be E-stable. Finally, the well known Robertson problem is used as test example.