Preprint des Projektes: SFB 701: Spektrale Strukturen und Topologische Methoden in der Mathematik - Projekt B3

Numerische Analyse äquivarianter Evolutionsgleichungen

06-030 Vera Thümmler.
Asymptotic stability of discretized and frozen relative equilibria

In this paper we prove nonlinear stability results for the numerical approximation of relative equilibria of equivariant parabolic partial differential equations. Relative equilibria are solutions which are equilibria in an appropriately comoving frame and occur frequently in systems with underlying symmetry. By transforming the PDE into a corresponding PDAE via a freezing ansatz [3] the relative equilibrium can be analyzed as a stationary solution of the PDAE. As a main result we obtain that nonlinear stability properties are inherited by the numerical approximation with finite differences using appropriate boundary conditions. This is a generalization of the results in [15] and is illustrated by numerical computations for the quintic complex Ginzburg Landau equation.