The Effect of Freezing and Discretization to the Asymptotic Stability of Relative Equilibria

In this paper we prove nonlinear stability results for the numerical approximation of relative equilibria of equivariant parabolic partial differential equations in one space dimension. 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 [2] the relative equilibrium can be analyzed as a stationary solution of the PDAE. The main result is the fact that nonlinear stability properties are inherited by the numerical approximation with finite differences on a finite equidistant grid with appropriate boundary conditions. This is a generalization of the results in [14] and is illustrated by numerical computations for the quintic complex Ginzburg Landau equation.