Stability and Dynamics of Numerical Methods for Nonlinear Ordinary Differential Equations

Stability of numerical methods for nonlinear autonomous ordinarydifferential equations is approached from the point of viewof dynamical systems. It is proved that multistep methods (withnonlinear algebraic equations exactly solved) with bounded trajectoriesalways produce correct asymptotic behaviour, but this is notthe case with Runge-Kutta. Examples are given of Runge-Kuttaschemes converging to wrong solutions in a deceptively ‘smooth’manner and a characterization of such two-stage methods is presented.PE(CE)^m schemes are examined as well, and it is demonstratedthat they, like Runge-Kutta, may lead to false asymptotics.