Collection of PhD theses of (former) members of the research group

PhD thesis

jr210 Jens Rottmann-Matthes.
Computation and stability of patterns in hyperbolic-parabolic systems

This work is devoted to the stability analysis and numerical long time-simulation of relative equilibria in partial differential equations. The simplest examples of nontrivial relative equilibria − or patterns − are traveling waves. These arise in a wide range of applications like nerve axon equations or reaction-diffusion models. Due to their relevance, the stability of traveling waves has been considered by many authors. Important contributions are made by [Evans, 1972--1975], [Sattinger, 1976], [Henry, 1981], and [Bates and Jones, 1989], to name just a few.
A major difficulty with the numerical long-time simulation of relative equilibria arises from the fact that the relevant part of the solution typically leaves the computational domain. Here the freezing method is a possibility to counter this problem. It is presented in a very general setting and justified for a large class of equations. Originally the method was introduced independently by [Beyn and Thümmler, 2004] and [Rowley et al., 2003]. Its basic idea is to split the evolution into a symmetry and a shape part. In practice, the method leads to a partial differential algebraic equation (PDAE).
We consider traveling waves in hyperbolic Cauchy problems of the form

and also in coupled hyperbolic-parabolic problems

Such systems are also considered in [Kreiss et al., 1994]. We present stability theorems for these equations which show the exponential stability of traveling waves (fronts and pulses). Moreover, the rate of convergence is obtained from the spectrum of the linearization. Our method of proof is a new combination of a nonlinear change of coordinates as in [Henry, 1981] and the Laplace transform technique as in [Kreiss et al., 1994]. The coordinate change leads to a hyperbolic(-parabolic) PDAE system that is closely related to the freezing method. For this system existence results and resolvent estimates are shown so that the Laplace transform can rigorously be applied and yields stability. In particular, our proof clarifies the treatment of the asymptotic phase.
Furthermore, we prove that the freezing method can be applied to (1) and (2) and show that it converges exponentially and provides asymptotically with time the correct speed and profile of the traveling wave. This is important from a numerical point of view and extends results by [Thümmler, 2005] to another class of equations.
In the final chapter, the theory is applied to several important examples (like the FitzHugh-Nagumo system) and the freezing method is tested for these. The freezing method is also tested for systems which are not covered by the theory. A particular interesting example is the viscous Burgers' equation for which the freezing method enables us to observe metastability numerically.

  • Publisher:Shaker Verlag, Aachen (2010)