Numerical Analysis of equivariant evolution equations
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.
