Stability and Dynamics of Numerical Methods for Nonlinear Ordinary Differential Equations

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Abstract
Stability of numerical methods for nonlinear autonomous ordinary differential equations is approached from the point of view of dynamical systems. It is proved that multistep methods (with nonlinear algebraic equations exactly solved) with bounded trajectories always produce correct asymptotic behaviour, but this is not the case with Runge-Kutta. Examples are given of Runge-Kutta schemes 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 demonstrated that they, like Runge-Kutta, may lead to false asymptotics.

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