Numerical methods for equivariant evolution equations
by
Wolf-Jürgen Beyn
Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, D-33501 Bielefeld, Germany
Coauthors: Vera Thümmler

We discuss a general method of solution for such equations (called equisplit) that takes advantage of the equivariance. equisplit splits the dynamics into an ODE on the Lie group coupled to a PDE, that differs from the original equation by a forcing term. The splitting is determined by a set of algebraic constraints (phase conditions) the number of which is given by the dimension of the Lie group. The method is particularly useful when solving the evolution equation near relative equilibria where the solution of the forced PDE is expected to become constant.

Noncompact Lie groups, such as the Euclidean group, arise when the underlying domain is unbounded (for example a strip or the whole space). For numerical calculations it is then necessary to truncate to a finite domain and to introduce asymptotic boundary conditions. In this case the longtime behavior of the original and of the equisplit equation can differ substantially. We discuss the errors introduced by these approximations for several cases of localized and non-localized patterns such as traveling waves and spiral waves.

Date received: May 1, 2003

Copyright © 2003 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # calh-05.