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

Numerische Analyse äquivarianter Evolutionsgleichungen

13-020 Wolf-Jürgen Beyn, Denny Otten, Jens Rottmann-Matthes.
Stability and Computation of Dynamic Patterns in PDEs

Nonlinear waves are a common feature in many applications such as the spread of epidemics, electric signaling in nerve cells, and excitable chemical reactions. Mathematical models of such systems lead to time-dependent PDEs of parabolic, hyperbolic or mixed type. Common types of such waves are fronts and pulses in one, rotating and spiral waves in two, and scroll waves in three space dimensions. These patterns may be viewed as relative equilibria of an equivariant evolution equation where equivariance is caused by the action of a Lie group. Typical examples of such actions are rotations, translations or gauge transformations. The aim of the lectures is to give an overview of problems related to the theoretical and numerical analysis of such dynamic patterns. One major theoretical topic is to prove nonlinear stability and relate it to linearized stability determined by the spectral behavior of linearized operators. The numerical part focusses on the freezing method which uses equivariance to transform the given PDE into a partial differential algebraic equation (PDAE). Solving these PDAEs generates moving coordinate systems in which the above-mentioned patterns become stationary.