Speaker: Vera Thuemmler
Title: Numerical Computation of Relative Equilibria of PDEs
Affiliation: Bielefeld University, Germany
Date: Friday, September 22, 2006.
Place and Time: Room 101 - Love Building, 3:35-4:30 pm.
Refreshments: Room 204 - Love Building, 3:00 pm.

Abstract. Relative equilibria are special solutions of partial differential equations which are stationary in an appropriate comoving frame of reference. Such solutions are often found in biological and chemical systems, e.g. when describing pattern formation of reaction-diffusion systems. Examples are traveling waves in 1d, planar and spiral waves in 2d and scroll waves in 3d. Such patterns are receiving increased attention by experimentalists in neurobiology.
In this talk we present a numerical method for the computation of relative equilibria, which makes it possible to observe phenomena which are visible after a transient phase only and which can not be handled by direct numerical simulation. A traveling wave for example would leave the finite domain of computation after short time. The main idea of the method is to split the dynamics into two parts: one part which is related to the symmetry of the equation, which in case of traveling waves is translation, and the remaining part, which in case of traveling waves describes the formation of the shape. This shape finally becomes stationary thereby reducing computational effort considerably.
We demonstrate the method on two mathematical models which describe different aspects of electrical signal propagation in cardiac tissue: the FitzHugh-Nagumo equations and the Complex Ginzburg-Landau equations. Both models show traveling waves in 1d and spiral waves in 2d, which are are believed to be responsible for pathological cardiac arrhythmias. Besides demonstrating that the methods works, we will discuss the problems which can arise due to the introduction of large convective terms by the method.