Numerische Approximation und Spektrale Analysis unendlich-dimensionaler Dynamischer Systeme
03-022 Wolf-Jürgen Beyn, Vera Thümmler.
Freezing solutions of equivariant evolution
equations
In this paper we develop numerical
methods for integrating general evolution equations
ut= F(u), u(0)=u0, where F
is defined on a dense subspace of some Banach space (generally
infinite dimensional) and is equivariant with respect to the
action of a finite dimensional (not necessarily compact) Lie
group. Such equations typically arise from autonomous PDE's on
unbounded domains that are invariant under the action of the
Euclidean group or one of its subgroups. In our approach we
write the solution u(t) as a composition of the action
of a time dependent group
element with a 'frozen solution' in the given Banach space. We
keep the 'frozen solution' as constant as possible by
introducing a set of algebraic constraints (phase conditions)
the number of which is given by the dimension of the Lie group.
The resulting PDAE (Partial Differential Algebraic Equation) is
then solved by combining classical numerical methods, such as
restriction to a bounded domain with asymptotic boundary
conditions, half-explicit Euler methods in time and finite
differences in space. We provide applications to reaction
diffusion systems that have traveling wave or spiral solutions
in one and two space dimensions.
