Preprint of the project: SFB 343: Discrete structures in mathematics

Topics in numerical linear algebra and discrete dynamical systems

99-047 Wolf-Jürgen Beyn, Johannes Schropp.
Runge-Kutta Discretizations of Singularly Perturbed Gradient Equations


We analyze Runge-Kutta discretizations applied to singularly perturbed gradient systems. It is shown in which sense the discrete dynamics preserve the geometric properties and the longtime behavior of the underlying ordinary differential equation. If the continuous system has an attractive invariant manifold then numerical trajectories started in some neighbourhood (the size of which is independent of the step-size and the stiffness parameter) approach an equilibrium in a nearby manifold. The proof combines invariant manifold techniques developed by Nipp and Stoffer for singularly perturbed systems with some recent results of the second author on the global behavior of discretized gradient systems. The results support the favorable behavior of ODE methods for stiff minimization problems.